CHAPTER 21. ELECTRIC CHARGE AND ELECTRIC FIELD • •

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Review vector addition, geometry and calculus (differentiation and integration) Four fundamental forces: o Gravitational force  involves “mass” o Electromagnetic force  involves “electric charge” o Strong force (or Nuclear force) o Weak force

A. Electric charge • Electric charge is a scalar; it has no direction. • SI unit of electric charge is coulomb (C) • Three properties of electric charge: 1. Dichotomy property  The electric charge is either “positive” (+) or “negative” (–).  Like charges repel; opposite charges attract. 2. Conservation property  The algebraic sum of all the electric charges in any closed system is constant.  In charging, charge is not created nor destroyed; it is only transferred from one body to another.  This is a “universal” conservation law. 3. Quantization property  The magnitude of charge of the electron or proton is a natural unit of charge.  Basic unit of charge  e = 1.602×10-19 C a. Charge of 1 proton = +e = 1.602×10-19 C b. Charge of 1 electron = –e = –1.602×10-19 C  Every observable amount of electric charge is always an integer multiple of this basic unit. • Other keywords: o Electrostatics – involves electric charges that are at rest (i.e. speed is zero) in the observer’s reference frame o Atom – composed of electron, proton, and neutron o Neutral atom – atom with zero net charge (# of electrons = # of protons) o Positive ion (cation) – atom with positive net charge (lost one or more electrons) o Negative ion (anion) – atom with negative net charge (gained one or more electrons) o Ionization – gaining or losing of electrons B. Types of materials in terms of electric conduction • Conductors o Objects that permit the easy movement of electrons through them o Ex: most metals, copper wire, earth o In metals, the mobile charges are always negative electrons  “sea of free electrons” o The earth can act as an infinite source or sink of electrons  “grounding” • Insulators o Objects that does NOT permit the easy movement of electrons through them o Ex: most nonmetals, ceramic, wood, plastic, rubber, air o The charges within the molecules of an insulator can shift slightly  “polarization”

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© Physics 72 Arciaga Semiconductors o Objects with properties between conductors and insulators o Ex: silicon, diodes, transistors Superconductors o Objects with zero resistance against the movement of electrons o Ex: some compounds at very low temperatures

C. Ways of charging a material • Charging by rubbing – charge of “charger” changes; electrons transfer • Charging by contact – charge of “charger” changes; electrons transfer • Charging by induction (without grounding) – charge of “charger” does NOT change • Charging by induction (with grounding) – charge of “charger” does NOT change – negative “charger” induces a positive charge (positive “charger” induces a negative charge) • Charging by polarization – charge of “charger” does NOT change – charged object can still attract a neutral object by polarization D. Coulomb’s law • “The amplitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.” qq 1 q1q 2 • Mathematically: Fe = k 1 2 2 = 4πε o r 2 r ; where Fe = magnitude of the electric force between two point charges q1 and q2 = electric charges of the two point charges r = distance between the 2 point charges k = proportionality constant = 1/4πεo εo = permittivity of free space (permittivity of vacuum)  NOTES: 1. The direction of Fe is along the line joining the two point charges. 2. The electric force on q1 by q2 is equal in magnitude but opposite in direction to the electric force on q2 by q1. [Recall: Newton’s third law of motion] 3. It is an “inverse square law.” [Compare: Newton’s law of gravity] 4. k = 1/4πεo = 8.988×109 N⋅m2/C2 5. εo = 8.854×10-12 C2/N⋅m2 6. If there are more than two point charges, use the “principle of superposition of forces”. Use vector addition (not scalar addition). 7. For atomic particles, the electric force is much greater than the gravitational force. E. Electric field and electric forces • Electric field:   Fe  E= q test  ; where E = electric field at a particular position qtest = charge of a “test” charge placed at the particular position  Fe = net electric force experienced by the test charge at the particular position NOTES: 1. Electric field is a vector. 2. Electric field is an “intermediary” for the electric force; an “aura” of electric charges.

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© Physics 72 Arciaga 3. A charged body experiences an electric force when it “feels” an electric field created by other charged bodies. 4. Compare it with the gravitational field. 5. SI unit of electric field is newton per coulomb (N/C). Electric force experienced by a point charge due to a given electric field:    Fe = qE  ; where Fe = net electric force experienced by a point charge at a particular position q = charge of a point charge placed at the particular position  E = electric field at the particular position   NOTES: 1. Fe and E are in the same direction if q is positive.   2. Fe and E are in the opposite direction if q is negative. Electric field created by a point charge:  1 q  E= rˆ 4πε o r 2  ; where E = electric field created by a point charge q = charge of the point charge r = distance from the point charge ˆ  r = unit vector pointing away from the point charge (i.e. “radially outward”) NOTES: 1. E points away from a positive charge.  2. E points toward a negative charge. 3. The electric field by a point charge is an “inverse-square relation”. 4. If there are more than one point charge, use the “principle of superposition of electric fields”. Use vector addition (not scalar addition). 5. Other keywords: o Source point – location of the point charge that creates the electric field o Field points – locations at which the electric field are being determined o Vector field – infinite set of vectors drawn in a region of space o Uniform field – constant vector field (i.e. magnitude and direction are constant) Electric field created by a continuous distribution of charge: 1. Use principle of superposition of electric fields; perform an integration! 2. Imagine the continuous distribution of charge as composed of many point charges. 3. Sometimes symmetry analysis makes the solution easier. 4. Other keywords: o linear charge density [λ] – charge per unit length (C/m) o surface charge density [σ] – charge per unit area (C/m2) o volume charge density [ρ] – charge per unit volume (C/m3)

F. Electric field lines (also called “lines of force”) • Electric field lines o imaginary line or curve drawn so that its tangent at any point is in the same direction of the electric field vector at that point o tangent at an electric field line  determines direction of the electric field o spacing of electric field lines  determines magnitude of the electric field  electric field lines are closer together  indicates strong electric field  electric field lines are farther apart  indicates weak electric field o electric field lines never intersect o electric field magnitude can vary along one electric field line

© Physics 72 Arciaga G. Electric dipoles • Electric dipole o a pair of point charges with equal magnitude and opposite sign separated by a particular distance o ex: water molecule, polar molecules, TV antenna • Electric dipole moment    p = qd rˆp ; where p = electric dipole moment of a dipole q = magnitude of the electric charge (of a charge) in the dipole d = separation distance between the two charges rˆp = unit vector pointing from the negative to the positive charge NOTE: Electric dipole moment is a vector: a. magnitude = |qd| b. direction = from the negative to the positive charge • Torque of an electric dipole in a uniform electric field      τ = p×E ; where τ = torque experienced by an electric dipole in an electric field  p = electric dipole moment of a dipole  E = electric field NOTE: Torque is a vector [recall Physics 71]: a. magnitude = pE sinθ   ; where θ = small (tail-to-tail) angle between p and E b. direction = use right-hand rule [recall Physics 71] • Potential energy of an electric dipole in a uniform electric field    U = −p • E ; where U = potential energy experienced by an electric dipole in an electric field  p = electric dipole moment of a dipole  E = electric field NOTE: Potential energy is a scalar [recall Physics 71]: a. magnitude = –pE cosθ   ; where θ = small (tail-to-tail) angle between p and E • Equilibrium concepts [recall Physics 71]  If both the net force and the net torque on an object are ZERO, then that object is in EQUILIBRIUM; otherwise, that object is NOT in equilibrium.  If the potential energy of an object is a MINIMUM, then that object is in STABLE equilibrium. But if the potential energy of an object is a MAXIMUM, then that object is in UNSTABLE equilibrium.

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CHAPTER 22. GAUSS’S LAW

A. Electric flux • Electric flux – like a “flow” of the electric field through an imaginary surface • For a uniform electric field through a flat surface:        Φ E = E • A = E • nˆ A = E A cos θ ; where Φ E = electric flux  E = electric field   A = nˆ A = vector area

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nˆ = unit vector perpendicular to the area (“unit normal vector”)    θ = tail-to-tail angle between E and A (or equivalently, E and nˆ ) NOTES: 1. Electric flux is a scalar.  2. Electric flux is zero if E is parallel to the surface. 3. The “vector area” has: a. magnitude equal to the area of the surface; and b. direction perpendicular to the surface. General definition: For any electric field through any surface     Φ E =  E • dA =  E • nˆ dA =  E cos θ dA   NOTES: 1. This is called a “surface integral” of E • dA . 2. For a closed surface: a. unit vector nˆ points outward (by convention) b. electric flux is positive if “flowing” outward the closed surface c. electric flux is negative if “flowing” inward the closed surface

B. Gauss’s law • Qualitative statements of Gauss’s law: 1. The net electric flux through a closed surface is outward (/inward) if the net enclosed charge is positive (/negative). 2. The net electric flux through a closed surface is zero if the net enclosed charge is zero. 3. The net electric flux through a closed surface is unaffected by charges outside the closed surface. 4. The net electric flux through a closed surface is directly proportional to the net amount of enclosed charge. 5. The net electric flux through a closed surface is independent of the size and shape of the closed surface (if the net amount of enclosed charge is constant). • Mathematically:    Q  Φ E =  E • dA =  E • nˆ dA = enc εo NOTES: 1. The symbol



means “surface integral for a closed surface”.

2. The closed surface to be used is imaginary !!!  called a “Gaussian surface” 3. Two possible uses: a. Given a charge distribution, enclose it with a proper Gaussian (imaginary) surface that utilizes the symmetry of the situation, then determine the electric field. b. Given an electric field, construct a Gaussian (imaginary) surface, then determine the charge distribution inside it.

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C. Conductors in electrostatics  • The electric field is zero ( E = 0 ) in the bulk material of a conductor. • Any excess charge resides entirely on the surface of the conductor; no charge can be found in the bulk material. • The electric field at the surface of the conductor is always perpendicular to the surface; there is no tangential or parallel component. • The electric field at the surface of the conductor has a magnitude equal to σ/εo. • The magnitudes of electric field and surface charge density on the surface of the conductor are higher at the “sharper” locations. • The electric field is discontinuous (in magnitude and/or direction) wherever there is a sheet of charge.

CHAPTER 23. ELECTRIC POTENTIAL

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A. Electric potential energy • Review of some important remarks [recall Physics 71]:  1. Work done by a force F on a particle that moves from position a to position b. b   Wa →b =  F • ds a

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2. Electric force is a conservative force. A conservative force has the following properties: a. The work it does on a particle is independent of the path taken by the particle and depends only on the initial and final positions. b. The total work it does on a particle is zero when the particle moves around any closed path, in which the initial and final positions are the same. c. The work it does on a particle is reversible, i.e. energy can always be recovered without loss. d. The work it does on a particle can be expressed as the difference between the initial and final values of a potential-energy function.  Wab = –∆U = –(Ub – Ua) = Ua – Ub ; where Wab = work done by a conservative force when a particle moves from position a to position b ∆U = change in the potential energy Ua and Ub = potential energies at positions a and b, respectively 3. Conservation of mechanical energy can be applied when only internal force and conservative force do work on the system. 4. A system tends to attain the lowest possible potential energy (i.e. it tends to attain a state of stable equilibrium). A charged particle in a uniform electric field:  U = Uo + qEh ; where U = electric potential energy of a charged particle in a uniform electric field q = electric charge of the charged particle E = magnitude of the uniform electric field  h = position of the charged particle against E Uo = reference potential energy (i.e. value of U at h = 0)  Wab = –∆U = qE(ha – hb) ; where ha and hb = positions at a and b, respectively NOTE: Compare with the gravitational potential energy (i.e. UGPE = Uo + mgh). Two point charges: 1 q1q 2  U = Uo + 4πε o r ; where U = electric potential energy of two point charges q1 and q2 = electric charges of the two point charges r = separation distance of the two point charges Uo = reference potential energy NOTES: 1. Commonly, Uo = 0. Meaning, U = 0 at r = . 2. U is negative if the two charges have opposite signs. 3. U is positive if the two charges have the same sign. 4. The above formula can also be used if one or both of the point charges is/are replaced by any spherically symmetric charge distribution (in that case, r is the distance between the centers).

© Physics 72 Arciaga 1 1 1 q1q 2  −  4πε o  ra rb  ; where ra and rb = separation distance of the two point charges at positions a and b, respectively A point charge with other point charges: q  q 1 1 q  U = Uo + q1  2 + 3 + ...  = U o + q1  i 4πε o  r12 r13 4πε o i r1i  ; where U = electric potential energy of a point charge with other point charges q1 = electric charge of the point charge q2, q3, … = electric charges of the other point charges r12, r13, … = separation distance of q1 from q2, q1 from q3, … Uo = reference potential energy NOTES: 1. Commonly, Uo = 0. Meaning, U = 0 if q1 is very far away from the other charges. 2. The above formula can also be used if any of the point charges is replaced by any spherically symmetric charge distribution (in that case, r is the distance between the centers). 3. The above formula is just scalar addition of electric potential energies.   1 1 1   Wa →b = −∆U = q1   qi  −    r1i,a r1i,b   4πε o i    Interpretations of the electric potential energy 1. The work done by the electric force when a charged particle moves from position a to position b is equal to (Ua – Ub).  Wby electric force = –∆U = Uinitial – Ufinal 2. The work that must be done by other external force to move the charged particle slowly from position a to position b is equal to (Ub – Ua).  Wby other external force = ∆U = Ufinal – Uinitial 

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Wa →b = −∆U =

B. Electric potential • Electric potential – electric potential energy per unit charge – often called simply as “potential” U  V= ; where V = potential q test U = electric potential energy qtest = electric charge of a “test” charge   NOTES: 1. Compare with electric field ( E = Fe q test ). 2. Potential is a scalar. 3. SI unit of potential is volt (V) : 1 V = 1 J/C • By a uniform electric field:  V = Vo + Eh ; where V = potential in a uniform electric field E = magnitude of the uniform electric field  h = position against E Vo = reference potential (i.e. value of V at h = 0)

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© Physics 72 Arciaga By a point charge: 

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V = Vo +

1 q 4πε o r

; where V = potential by a point charge

q = electric charges of the point charge r = distance from the point charge Vo = reference potential NOTES: 1. Commonly, Vo = 0. Meaning, V = 0 at r = . 2. The above formula can also be used if the point charge is replaced by any spherically symmetric charge distribution (in that case, r is the distance from the center). By a collection of point charges:  1  q 2 q3 1 qi  V = Vo +  + + ...  = Vo +  4πεo  r2 r3 4πε o i ri  ; where U = potential by several point charges q2, q3, … = electric charges of the point charges r2, r3, … = separation distance from q2, from q3, … Vo = reference potential NOTES: 1. Commonly, Vo = 0. Meaning, V = 0 somewhere very far away from the point charges. 2. The above formula can also be used if any of the point charges is replaced by any spherically symmetric charge distribution (in that case, r is the distance from the centers). 3. The above formula is just scalar addition of potentials. By a continuous distribution of charge: 1 dq  V=  4πε o r NOTES: 1. The integration is done over the entire distribution of charge (length, area, or volume). 2. For finite distribution of charge, you can set V = 0 at r = ∞. 3. For infinite distribution of charge, you cannot set V = 0 at r = ∞. What you can do is to set V = 0 somewhere else.

C. Potential difference (or Voltage) • Some important relations:  Wab = –∆U = –(Ub – Ua) = Ua – Ub  Wab = –q∆V = –q(Vb – Va) = q(Va – Vb) = qVab b  b    Wa →b =  Fe • ds =  qE • ds a a b    Vab = Va − Vb =  E • ds a

; where Wab = work done by the electric force in moving a charged particle from position a to position b q = electric charge of the charged particle ∆U = change in the electric potential energy Ua and Ub = electric potential energies at positions a and b, respectively Va and Vb = potential at positions a and b, respectively Vab = Va – Vb = potential at a with respect to b (or “voltage between a and b”)  Fe = electric force  E = electric field

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© Physics 72 Arciaga  NOTES: 1. The E points toward decreasing V.  2. If E = 0 in a certain region, V is constant in that region (e.g. body of a conductor).  3. If E = 0 at a certain location, it does not necessarily mean that V = 0 at that location.  4. If V = 0 at a certain location, it does not necessarily mean that E = 0 at that location. Some common units:  Units of electric field  newton per coulomb (N/C)  volt per meter (V/m) : 1 V/m = 1 N/C  Units of energy  joule (J)  electron volt (eV) : 1 eV = 1.602×10-19 J

D. Equipotential surfaces • Equipotential surface – 3D surface on which the potential is the same at every point (V = constant) • Some notes: 1. Contour lines on a topographic map  curves of constant grav. potential energy per test mass Equipotential surfaces  curved surfaces of constant elec. potential energy per test charge  2. Electric field lines  curved lines (arrows) to represent E ;  E is not necessarily constant in an electric field line 3. Equipotential surfaces  curved surfaces to represent V; V is constant in an equipotential surface 4. Electric field line is perpendicular to equipotential surfaces. 5. Electric field points toward decreasing potential. 6. Magnitude of electric field is large in regions where equipotential surfaces are close to each other. E. Potential gradient • Gradient operator  ∂ ˆ ∂ ˆ ∂  ∇ = ˆi + j +k ∂x ∂y ∂z  NOTES: 1. ∇ = “gradient” operator (also called as “grad” or “del” operator) 2. A mathematical operation that can convert a scalar to a vector. 3. Utilizes “partial differentiation”. • Potential gradient    ∂V ˆ ∂V ˆ ∂V   E = −∇V = −  ˆi +j +k  ∂y ∂z   ∂x  E = electric field V = potential  ∇ = gradient operator  NOTES: 1. ∇V = “gradient of V” (also called as the “potential gradient”)  2. From the scalar V, a vector E can be obtained.  3. ∇V is directed toward the rapid decrease of V. 4. If V depends only on the radial distance [i.e. V = V(r)], then    ∂V   E = −∇V = −  rˆ  ; where rˆ = unit radial vector  ∂r 

CHAPTER 24. CAPACITANCE AND DIELECTRICS

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A. Capacitors • Capacitor – composed of two conductors separated by an insulator or vacuum – can store electric potential energy and electric charge • Capacitance – characteristic property of a capacitor – measure of the ability of a capacitor to store energy Q  C= Vab ; where C = capacitance of a capacitor Q = charge of the capacitor (i.e. charge on one conductor is +Q; and charge on the other is –Q) Vab = potential difference between the two conductors NOTES: 1. SI unit of capacitance is farad (F) : 1 F = 1 C/V 2. The capacitance depends on the insulator between the two conductors. [see Section D] 3. In vacuum, the capacitance depends only on the shape, configuration, and size of the capacitor. 4. In vacuum, the capacitance does NOT depend on the charge and potential difference of the capacitor. • Parallel-plate capacitor (in vacuum): A  C = εo d ; where C = capacitance of a parallel-plate capacitor in vacuum A = area of the parallel plates d = distance separation between the two parallel plates B. Connections of capacitors • Key idea: A connection of several capacitors can be replaced by a single capacitor with a certain “equivalent capacitance” (also called “effective capacitance” in other textbooks). • Capacitors in series connection −1

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 1  1 1  Ceq =  + + + ...   C1 C 2 C3  ; where Ceq = equivalent capacitance of a series connection C1, C2, C3, … = capacitances of the capacitors in the series connection NOTES: 1. Ceq is less than any of C1, C2, C3, … 2. Qseries = Q1 = Q2 = Q3 = … [i.e. equal charges] 3. Vseries = V1 + V2 + V3 + … [i.e. sum of potential differences] Capacitors in parallel connection  Ceq = C1 + C 2 + C3 + ... ; where Ceq = equivalent capacitance of a parallel connection C1, C2, C3, … = capacitances of the capacitors in the parallel connection NOTES: 1. Ceq is greater than any of C1, C2, C3, … 2. Qparallel = Q1 + Q2 + Q3 + … [i.e. sum of charges] 3. Vparallel = V1 = V2 = V3 = … [i.e. equal potential differences]

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C. Energy stored in capacitors • Capacitors can store electric potential energy and electric charge. • Two equivalent interpretations of energy storage in capacitors: 1. Energy stored is a property of the charge in the capacitor 1 Q2 1 1  U= = CV 2 = QV 2 C 2 2 ; where U = electric potential energy stored in a capacitor Q = charge of the capacitor V = potential difference across the capacitor C = capacitance of the capacitor NOTES: 1. These assign U = 0 if the capacitor is uncharged (Q = 0). 1 Q2 2. Work needed to charge the capacitor: Wch arg e = U = 2 C 2. Energy stored is a property of the electric field produced by the capacitor 1  u = εo E 2 2 ; where u = electric energy density stored in a capacitor (in a vacuum) E = electric field in the capacitor NOTES: 1. Electric energy density is electric potential energy per unit volume: u = 2. Total electric potential energy: U =



U volume

u dv

volume

D. Dielectrics • Dielectric – a nonconducting material (i.e. insulator) – usually inserted between the plates of a capacitor • Characteristic properties associated with a dielectric: 1. Dielectric constant: o symbol: K o pure number; dimensionless; no units o in general, K ≥ 1 o for vacuum, K = 1 o for air (at 1 atm), K = 1.00059 ≈ 1 o for Mylar, K = 3.1 2. Permittivity: o symbol: ε o ε = Kεo o SI unit is C2/N⋅m2 or F/m o εo = permittivity of free space (permittivity of vacuum) o in general, ε ≥ εo o for air (at 1 atm), ε ≈ εo 3. Dielectric strength: o dielectric strength – maximum electric field (magnitude) that a dielectric can withstand without the occurrence of “dielectric breakdown” o dielectric breakdown – phenomenon at which the dielectric becomes partially ionized and becomes a conductor

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© Physics 72 Arciaga Effects of inserting a dielectric in the capacitor: 1. Separates the two plates even at very small distances 2. Increases the maximum possible potential difference between the plates (because some dielectrics have higher dielectric strength than air) 3. Increases the capacitance of the capacitor  Cw = KCwo 4. Decreases the potential difference between the plates when Q is kept constant  Vw = Vwo / K 5. Decreases the electric field when Q is kept constant (because of “polarization” and “induced charges” in the dielectric)  Ew = Ewo / K 6. Decreases the electric potential energy stored when Q is kept constant (because the electric field fringes do work on the dielectric)  Uw = Uwo / K  uw = uwo / K = ½ εEw2 ; where K = dielectric strength of the inserted dielectric ε = permittivity of the inserted dielectric Cw, Cwo = capacitances with and without the inserted dielectric Vw, Vwo = potential differences with and without the inserted dielectric Ew, Ewo = electric fields with and without the inserted dielectric Uw, Uwo = electric potential energies with and without the inserted dielectric uw, uwo = electric energy densities with and without the inserted dielectric Remark: In solving problems about capacitors, you must determine whether the “voltage” or the “charge” is constant. Here are some common situations: 1. capacitor is directly connected to a battery (or emf source)  implies constant voltage 2. charged capacitor is isolated (i.e. not connected to anything)  implies constant charge

© Physics 72 Arciaga CHAPTER 25. CURRENT, RESISTANCE, AND ELECTROMOTIVE FORCE A. Current • Remarks about conductors (particularly metals): 1. In electrostatics, a. electric field is zero within the material of the conductor. b. the free electrons move randomly in all directions within the material of the conductor; comparable with the motion of gas molecules. c. there is no net current in the material of the conductor. 2. In electrodynamics, a. electric field is nonzero within the material of the conductor. b. the free electrons move with a drift velocity in the opposite direction of the electric field (aside from the random motion described in 1b). c. there is a net current in the material of the conductor. • Current o any motion of charge from one region to another o rate of flow of charge (i.e. charge flowing per unit time) o moving charges: a. metals – electrons b. ionized gas (plasma) – electrons, positive ions, negative ions c. ionic solution – electrons, positive ions, negative ions d. semiconductors – electrons, holes (sites of missing electrons) o direction of current flow = same direction as the electric field in the conductor = same direction as the flow of positive charge = opposite direction to the flow of negative charge o mathematically: dQ  I= = n q v d A ; where I = current flowing through an area dt dQ = net charge flowing through the area dt = unit time n = concentration of the charged particles (i.e. number of particles per unit volume) q = charge of the individual particles vd = drift speed of the particles A = cross-sectional area NOTES: 1. Current is a scalar; not a vector. 2. SI unit of current is ampere (A) : 1 A = 1 C/s 3. If there are different kinds of moving charges, the total current is the sum of the currents due to each kind of moving charge. • Current density o current per unit area o mathematically:     J = nqv d ; where J = current density n = concentration of charged particles q = charge of the individual particles  vd = drift velocity of the particles

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NOTES: 1. Current density is a vector. 2. Magnitude: J = I / A = n q v d 3. Direction: same direction as the electric field in the conductor (see “direction of current flow” described above) 4. SI unit of current density is ampere per meter squared (A/m2) 5. If there are different kinds of moving charges, the total current density is the sum of the current densities due to each kind of moving charge. Two classifications of current: 1. Direct current – direction of current is always the same (i.e. does not change) 2. Alternating current – direction of current continuously changes

B. Resistivity E • ρ= J

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; where ρ = resistivity of a material

E = magnitude of electric field in the material J = magnitude of current density in the material NOTES: 1. Resistivity is a scalar; not a vector. 2. Summary: a. perfect conductors: ρ = 0 b. (nonperfect) conductors: low ρ c. insulators: high ρ d. semiconductors: ρ between conductor and insulator e. superconductos: ρ = 0 (at temperatures below a critical temperature Tc) 3. Conductivity – reciprocal of resistivity (i.e. σ = 1/ρ) 4. A material with high resistivity has low conductivity. ρ = ρo 1 + α ( T − To )  ; where ρ = resistivity of a conductor at a temperature T To = reference temperature (usually To = 20 oC or 0 oC) ρo = resistivity of the conductor at the reference temperature To α = temperature coefficient of resistivity NOTES: 1. The above equation is an equation of a line. 2. The above equation is only an approximation valid for small temperature range (usually up to ≈100 oC). 3. Summary: a. most conductors (especially metals): α > 0 [i.e. ρ increases if T increases] b. manganin: α = 0 [i.e. ρ does not change with T] c. graphite: α < 0 [i.e. ρ decreases if T increases] d. semiconductors: α < 0 [i.e. ρ decreases if T increases]

C. Resistance V L • R = =ρ I A

; where R = resistance of a conductor V = potential difference between the ends of the conductor I = current flowing through the conductor ρ = resistivity of the conductor L = length of the conductor A = (cross-sectional) area of the conductor

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NOTES: 1. R = V/I is a definition of resistance for any conductor. 2. SI unit of resistance is ohm (Ω): 1 Ω = 1 V/A 3. SI unit of resistivity is ohm⋅meter (Ω⋅m): 1 Ω⋅m = 1 V⋅m/A R = R o 1 + α ( T − To ) 

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; where R = resistance of a conductor at a temperature T To = reference temperature (usually To = 20 oC or 0 oC) ρo = resistance of the conductor at the reference temperature To α = temperature coefficient of resistance NOTES: 1. The above equation is an equation of a line. 2. The above equation is only an approximation valid for small temperature range (usually up to ≈100 oC). 3. In most conductors, the temp. coeff. of resistivity is equal to the temp. coeff. of resistance (especially if the length and area do not change much with temp.). Resistor – a circuit element or device that is fabricated with a specific value of resistance between its ends

D. Ohm’s law • Ohm’s law: o “At a given temperature, the current density flowing through a material is nearly directly proportional to the electric field in that material.”   o Mathematically: J ∝ E (or equivalently, I ∝ V ) NOTE: This is not actually a “law” because it is obeyed only by some materials (i.e. not all). • Two classifications of materials: 1. Ohmic material (or linear material) o material that obeys Ohm’s law o ex: resistors, metals, conductors  o at constant temperature, its ρ and R are constant (i.e. do not depend on E or V) o its I-V curve (i.e. current vs. voltage plot) is a straight line passing through the origin 2. Nonohmic material (or nonlinear material) o material that does not obey Ohm’s law o ex: semiconductors, diodes, transistors  o at constant temperature, its ρ and R vary (i.e. depends on E or V) o its I-V curve (i.e. current vs. voltage plot) is not a straight line, or a straight line but does not pass though the origin E. Circuits • Circuit – a path for current • Two classifications: 1. Incomplete circuit o also called open loop or open circuit o no steady current will flow through it (i.e. current eventually stops or dies) 2. Complete circuit o also called closed loop or closed circuit o a steady current will flow through it (i.e. current does not stop or die) o needs a source of emf

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F. Electromotive force • Electromotive force o something that can make the current flow from lower to higher potential energy o abbreviation: emf o symbol:  o it’s not a force; it’s a “potential” (i.e. potential energy per unit charge) o SI unit of emf is volt (V) • Source of emf o any device that can provide emf (i.e. potential or voltage) o ex: battery, electric generator, solar cell, fuel cell, etc. o can transform a particular for of energy (ex: chemical, mechanical, thermal, etc.) into electric potential energy o two classifications: 1. ideal source of emf  no internal resistance  provides a constant voltage across its terminals (called “terminal voltage”)  Vab = ; where Vab = terminal voltage provided by the source of emf  = emf in the source of emf 2. real (or nonideal) source of emf  has an internal resistance  provides a terminal voltage that depends on the current and resistance  Vab =  – Ir  ; where Vab = terminal voltage provided by the source of emf  = emf in the source of emf I = current through the source of emf r = internal resistance in the source of emf NOTES: 1. For an ideal source of emf, the terminal voltage is always equal to . 2. For a real source of emf, the terminal voltage becomes equal to  only when there is no current flowing (i.e. open circuit). • Some keywords: 1. Ammeter – a device that measures the current passing through it – must be connected in series to a circuit element or device – ideal ammeter = has zero resistance inside (so that there is no potential difference across its terminals) 2. Voltmeter – a device that measures the potential difference (or voltage) across its terminals – must be connected in parallel to a circuit element or device – ideal voltmeter = has infinitely large resistance inside (so that there is no current passing through it) 3. Short circuit – a closed circuit in which the terminals of a source of emf are connected directly to each other – creates very large current that can damage the devices in the circuit !!!

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G. Energy and power in electric circuits • Recall: Power = energy per time = rate of energy change or flow • P = IV ; where P = power delivered to or extracted from a circuit element or device I = current passing through the device V = voltage across the terminals (or ends) or the device NOTES: 1. Power is delivered to a resistor. A resistor dissipates energy (transforms electric potential energy into thermal energy or heat).  P = IV = I2R = V2/R 2. Power can be extracted from a source of emf. A source of emf provides energy (transforms chemical energy, mechanical energy, fuel energy, etc. into electric potential energy).  P = IV = I( – Ir) = I – I2r 3. Power can be delivered to a source of emf (ex: charging of batteries).  P = IV = I( + Ir) = I + I2r

CHAPTER 26. DIRECT-CURRENT CIRCUITS

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A. Connections of resistors • Key idea: A connection of several resistors can be replaced by a single resistor with a certain “equivalent resistance” (also called “effective resistance” in other textbooks). •

Resistors in series connection  R eq = R1 + R 2 + R 3 + ... ; where Req = equivalent resistance of a series connection R1, R2, R3, … = resistances of the resistors in the series connection NOTES: 1. Req is greater than any of R1, R2, R3, … 2. Iseries = I1 = I2 = I3 = … [i.e. equal currents] 3. Vseries = V1 + V2 + V3 + … [i.e. sum of potential differences]

•

Resistors in parallel connection −1

 1  1 1  R eq =  + + + ...   R1 R 2 R 3  ; where Req = equivalent resistance of a parallel connection R1, R2, R3, … = resistances of the resistors in the parallel connection NOTES: 1. Req is less than any of R1, R2, R3, … 2. Iparallel = I1 + I2 + I3 + … [i.e. sum of currents] 3. Vparallel = V1 = V2 = V3 = … [i.e. equal potential differences] B. Kirchhoff’s rules • Keywords: o Junction (or node) – any point in a circuit where three or more conductors meet o Loop – any closed conducting path in a circuit •

Kirchhoff’s junction rule (or Kirchhoff’s current law): o “The algebraic sum of the currents into any junction is zero.” o Mathematically:  I = 0 (at any junction) NOTES: 1. This is a consequence of conservation of electric charge. 2. At any junction, Iin = Iout.

•

Kirchhoff’s loop rule (or Kirchhoff’s voltage law): o “The algebraic sum of the potential differences in any closed loop is zero.” o Mathematically:  V = 0 (for any closed loop) NOTES: 1. This is a consequence of conservation of energy. 2. Consider voltage rise and voltage fall carefully.

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Problem-solving tips: 1. Usually, you first have to assume the direction of the current in each branch of the circuit. If the calculated current in the end is positive, then the assumed direction is correct (but if the calculated current is negative, then the assumed direction is opposite to the correct one). 2. Recall that current flows from high to low potential across a resistor. 3. Using Kirchhoff’s rules, setup a number of independent equations equal to the number of unknowns. Usually, you first apply the junction rule to all the junctions; then, complete the number of equations by applying the loop rule.

C. R-C circuits • Charging a capacitor:  Q = Cε 1 − e − t / RC = Q F 1 − e− t / τ

(

(

)

dQ ε − t / RC = e = Io e− t / τ dt R ; where Q and I = charge on and current through the capacitor, respectively t = time R and C = resistance and capacitance, respectively ε = terminal voltage (of the ideal emf source) QF = final charge on the capacitor = C ε Io = initial current = ε /R τ = time constant (or relaxation time) = RC NOTES: 1. Charge in the capacitor exponentially increases with time: a. t = 0: Q = 0 b. t = τ: Q = QF(1 – 1/e) = 0.63 QF c. t = : Q = QF 2. Current (magnitude) through the capacitor exponentially decreases with time: a. t = 0: I = Io b. t = τ: I = Io/e = 0.37 Io c. t = : I = 0 3. Recall the voltages across the resistor and capacitor: VR = IR and VC = Q/C 4. Rule of thumb a. Transient voltage across a charging capacitor is zero if it has no initial charge  like a short circuit element b. At steady-state of a fully-charged capacitor, current is zero  like an open circuit element Discharging a capacitor:  Q = Qo e − t / τ Q dQ  I= = − o e− t / RC = Io e − t / τ dt RC ; where Qo = initial charge on the capacitor Io = initial current = –Qo/RC τ = time constant (or relaxation time) = RC NOTES: 1. Charge in the capacitor exponentially decreases with time: a. t = 0: Q = Qo b. t = τ: Q = Qo/e = 0.37 Qo c. t = : Q = 0 2. Current (magnitude) through the capacitor exponentially decreases with time: a. t = 0: I = Io b. t = τ: I = Io/e = 0.37 Io c. t = : I = 0 3. Recall the voltages across the resistor and capacitor: VR = IR and VC = Q/C 4. Rule of thumb a. At steady-state of a fully-discharged capacitor, current is zero  like an open circuit element



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)

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I=

CHAPTER 27. MAGNETIC FIELD AND MAGNETIC FORCES • •

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In understanding the concepts of magnetism, I strongly suggest that you compare, contrast, or find analogies with the concepts of electricity (Chaps. 21 and 22). Please review the “cross product” (vector product) that you learned from Physics 71 !!!

A. Magnetic pole • Key ideas: 1. A permanent magnet has a north pole and a south pole. 2. North pole repels north pole, but attracts south pole. South pole repels south pole, but attracts north pole. 3. No experimental evidence of a magnetic monopole. Poles always appear in pairs. 4. A bar magnet sets up a magnetic field. • The earth is a magnet:  North geographic pole  it is actually (near) a south magnetic pole  South geographic pole  it is actually (near) a north magnetic pole B. Magnetic field • Analogy:  1. Electric field ( E )  produced by electric charges that may be at rest or moving    exerts an electric force ( F = qE ) on another electric charge that may be at rest or moving  2. Magnetic field ( B )  produced by moving electric charges (i.e. current)     exerts a magnetic force ( F = qv × B ) on another electric charge that must be moving • Direction of magnetic field: o same direction where the north pole of the compass needle points to o for a permanent magnet, the magnetic field points out of its north pole and into its south pole (but inside the magnet, the field points from the south to the north) NOTES: 1. SI unit of magnetic field is tesla (T): 1 T = 1 N/A⋅m 2. Another common unit of magnetic field is gauss (G): 1 G = 10-4 T C. Magnetic force on a moving charged particle • Mathematically:      F = qv × B ; where F = magnetic force on a moving charged particle q = electric charge of the moving charged particle  v = velocity of the charged particle  B = (external) magnetic field acting on the charged particle NOTES: 1. Magnetic force is a vector.  Magnitude: F = |q|v⊥B = |q|vB⊥  perpendicular components !!!    Direction: use right-hand rule  perpendicular to both v and B 2. The direction of the magnetic force depends on the sign of q and the directions   of both v and B .   3. Compare with the electric force ( F = qE ).

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© Physics 72 Arciaga Implications: o The magnetic force can never do work on a charged particle. o The magnetic force can only change the direction but not the magnitude (i.e. speed) of the velocity of a charged particle. Remark: When a situation involves both the electric force and the magnetic force, be careful on how you use the “principle of superposition”. o You can add together all electric fields (vector addition) o You can add together all magnetic fields (vector addition) o You can add together electric forces and magnetic forces (vector addition) o Never add electric fields with magnetic fields !!!

D. Magnetic field lines • Key ideas:  Magnetic field lines represent magnetic field in space.  Direction: The magnetic field is tangent to the magnetic field line at a particular point.  Magnitude: The closer (i.e. denser) the magnetic field lines are, the stronger the magnetic field is at that region.  Different magnetic field lines do not intersect. • Compare with the concept of electric field lines. • Recall: The north pole of a compass needle points toward the same direction as the magnetic field at that position. E. Magnetic flux • Similar idea as the electric flux (ΦE). It is like a “flow of magnetic field” though a surface. • Mathematically:     Φ B =  B • dA =  B • nˆ dA =  B⊥ dA =  B dA ⊥

NOTES: 1. Magnetic flux ( Φ B ) is a scalar.  2. Magnetic flux is zero if B is parallel to the surface.  3. Recall the “unit normal vector” ( nˆ ) and the “vector area” ( A )  (Chap. 22) 4. SI unit of magnetic flux is weber (Wb): 1 Wb = 1 Tm2 5. Sometimes the magnetic field is also called “magnetic flux density” (i.e. flux per unit area).

F. Gauss’s law for magnetism • Similar idea as the Gauss’s law for electrostatics  (Chap. 22) • Mathematically:     Φ B =  B • dA =  B • nˆ dA = 0

NOTES: 1. Magnetic flux through any closed surface is zero !!! 2. This is because of the absence (at least experimentally) of magnetic monopole. 3. This implies that magnetic flux lines always form closed loops (but not necessarily circular loops). A magnetic field line has no end points.

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G. Motion of charged particles • Recall the following (from Physics 71):   1. Newton’s 2nd law:  F = ma

2. Circular motion: a C = •

v2 ; v = rω ; r

ω=

2π = 2πf T

Examples: 1. Circular motion  uniform magnetic field; velocity has perpendicular component only mv  cyclotron radius (or Larmor radius or gyroradius) : r = qB v qB = r m



angular speed : ω =



cyclotron frequency (or Larmor frequency or gyrofrequency): f =

ω 1 qB = 2π 2 π m

2. Helical motion  uniform magnetic field; velocity has perpendicular and parallel components mv ⊥  cyclotron radius (or Larmor radius or gyroradius) : r =  [v⊥ matters!] qB

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•

v⊥ q B =  [v⊥ matters!] r m



angular speed : ω =



cyclotron frequency (or Larmor frequency or gyrofrequency): f =



pitch : x P = v T = v 2π

m  [v|| matters!] qB

ω 1 qB = 2π 2 π m

3. Mirror motion  non-uniform magnetic field; magnetic mirror (or magnetic bottle) configuration Some applications: 1. Velocity selector (or velocity filter)  purpose: to select ions moving with the prescribed velocity  how: balance the electric force and the magnetic force E  example: vselect = B 2. Thomson’s e/m experiment  purpose: to determine the value of “e/m”  how: velocity selector with speed determined from conservation of mechanical energy e E2  example: = m 2VB2 3. Mass spectrometer  purpose: to determine the mass (or the species) of ions assuming |q| is known  how: cyclotron radius due to a uniform magnetic field rqB  example: m = v Remark: Do NOT memorize the above formulas !!! Just start thinking from the fundamentals and learn to derive the above formulas.

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H. Magnetic force on a current-carrying conductor • Straight wire:    F = IL × B ; where F = magnetic force on a current-carrying straight wire I = current flowing through the straight wire  L = “vector length” (see NOTE 2 below)  B = magnetic field acting on the straight wire    NOTES: 1. This comes from adding the magnetic force ( F = qv × B ) acting on all the charged particles in the conductor.  2. L – let us call it the “vector length”: a. direction: along the wire, same direction as the flow of current b. magnitude: equal to the length of the straight wire • Any shape:     F =  I dL × B

NOTES: 1. This is a “line integral”. 2. The integration is done throughout the length of the wire (not necessarily straight).

I. Current loop • Current loop – a conductor that forms a loop and has a current flowing through it • Magnetic dipole – any object that experiences a magnetic torque – most common example is a current loop – analogy: electric dipole • Magnetic dipole moment – property of a magnetic dipole – also called “magnetic moment” – analogy: electric dipole moment     µ = IA ; where µ = magnetic dipole moment I= current flowing through the current loop (i.e. magnetic dipole) A = vector area NOTES: 1. Magnetic dipole moment is a vector; same direction as the vector area (see Chap. 22) 2. Direction: use the right-hand rule  curl fingers to the direction of the current 3. Its “arrow head” is the north pole; while its “arrow tail” is the south pole. • Torque on a current loop (in a uniform magnetic field)      τ = µ×B ; where τ = torque acting on a current loop in a uniform magnetic field  µ = magnetic dipole moment of the current loop  B = magnetic field acting on the current loop    NOTES: 1. Compare with torque on an electric dipole ( τ = p × E ). 2. Torque on a magnetic dipole is NOT always zero, but the magnetic force on the current loop in a uniform magnetic field is always zero. 3. Recall: If both the net force and the net torque on an object are ZERO, then that object is in EQUILIBRIUM; otherwise, that object is NOT in equilibrium.

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© Physics 72 Arciaga Potential energy of a current loop (in a uniform magnetic field)    U = −µ • B ; where U = potential energy of a current loop in a uniform magnetic field  µ = magnetic dipole moment of the current loop  B = magnetic field acting on the current loop   NOTES: 1. Compare with potential energy of an electric dipole ( U = −p • E ). 2. Recall: If the potential energy of an object is a MINIMUM, then that object is in STABLE equilibrium. But if the potential energy of an object is a MAXIMUM, then that object is in UNSTABLE equilibrium. Remark: For multiple loops or conducting coils consisting of several plane loops that are close together (e.g. solenoid), all the magnetic force, magnetic dipole moment, torque, and potential energy increase by a factor of N (i.e. number of loops).

CHAPTER 28. SOURCES OF MAGNETIC FIELD • •

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Recall:  Electric field – produced by electric charges that may be at rest or moving  Magnetic field – produced by moving electric charges (including current) Please practice your right-hand rules.

A. Magnetic field of a moving point charge with constant velocity  µo qv × rˆ  • B= ; where B = magnetic field of a moving point charge with constant velocity 4π r 2 q = electric charge of the moving point charge  v = velocity of the point charge r = distance from the point charge rˆ = unit vector (indicates direction) µo = permeability of free space (permeability of vacuum) NOTES: 1. This expression is valid only for constant velocity (or approximately constant). 2. This is an “inverse square law”. 3. µo = 4π×10-7 T⋅m/A 4. If there are more than two moving point charges, use the “principle of superposition of magnetic fields”. Use vector addition (not scalar addition). B. Magnetic field of an infinitesimal current element   µ o I dL × rˆ  • dB = ; where B = magnetic field of an infinitesimal current element 2 4π r I = current through the current element  dL = infinitesimal vector length NOTES: 1. This is called the “Biot-Savart law”. 2. This is used for infinitesimal current element only. 3. To find the total magnetic field of a current element of any shape:    µ o I dL × rˆ  B =  dB = 4π  r 2  The integration is done over the entire length of the current element. C. Ampere’s law   •  B • dL = µ o Ienc

•

; where Ienc = net current enclosed by the integration path

NOTES: 1. This is a “line integral for a closed path”. 2. The sign of the current is determined by the right-hand rule. 3. Only enclosed current matters.  Q  4. Compare with Gauss’s law:  E • dA = enc εo How to use Ampere’s law: 1. Very useful only for highly symmetrical situations. 2. Create a closed path for integration; this path is usually imaginary. 3. Assign a direction for the integration along the path. 4. Determine the net enclosed current; be careful with the proper signs. 5. Use Ampere’s law to determine the magnetic field.

CHAPTER 29. ELECTROMAGNETIC INDUCTION

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A. Electromagnetic induction • Key idea: When the magnetic flux through a circuit or loop changes, then an emf and current are induced in the circuit or loop. • Keywords: electromagnetic induction, induced emf, induced current B. Faraday’s law • “The induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop.” dΦ B • Mathematically:  = − dt ; where  = induced emf in the circuit or loop ΦB = magnetic flux through the circuit or loop dΦB/dt = rate of change of the magnetic flux • Remarks: 1.  depends on the change of ΦB only.  [independent of the material of the circuit] 2. Induced current depends on  (hence ΦB) and resistance (since I = /R).  [depends on the material of the circuit]   3. Recall: Φ B = B • A = BA cos θAB  It can be possibly changed by the following: a) Changing magnitude of B b) Changing magnitude of A   c) Changing angle between B and A (i.e. orientation) 4.  is larger if the rate of change of ΦB is faster. 5. The “–” sign is related to the polarity of   related to right-hand rule and Lenz’s law 6. To know the polarity of , it is important to know whether ΦB is increasing or decreasing. 7. For a coil with N identical loops or turns under the same change of ΦB, the  = –N(dΦB/dt). C. Lenz’s law • “The direction of any magnetic induction effect is such as to oppose the cause of the effect.” • Remarks: 1. If ΦB increases  dΦB/dt is positive   is negative If ΦB increases   must create an induced magnetic field to decrease ΦB !!! 2. If ΦB decreases  dΦB/dt is negative   is positive If ΦB decreases   must create an induced magnetic field to increase ΦB !!! 3. Right-hand rule must be utilized. D. Motional electromotive force • Key idea: When a conductor (either loop or not) moves through a region of magnetic field, then an emf can be induced on the conductor depending on the orientation of the magnetic field, conductor, and its motion. • Keyword: motional emf    • d = ( v × B ) • dL ; where d = motional emf produced on the conductor  v = velocity of the conductor  B = (external) magnetic field  dL = infinitesimal vector length of the conductor

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© Physics 72 Arciaga Remarks:    1. d may be zero or nonzero depending on the orientation of v, B, and dL . 2. For a closed conducting loop (i.e. conductor is part of closed circuit):      =  ( v × B ) • dL  [closed line integral over the entire loop]

3. This is actually an alternate form of Faraday’s law for the case of moving conductors. 4. Motional emf is just a special case of induced emf for the case of moving conductors. E. Induced electric fields • Key idea: When the magnetic flux through a stationary loop changes, then an electric field is induced in that loop. • Keyword: induced electric field   dΦ B • = −  E • dL = – dt ; where dΦB/dt = rate of change of the magnetic flux through a stationary loop  E = induced electric field produced in the loop  dL = infinitesimal vector length along the loop  = induced emf on the loop • Remarks: 1. The above expression is actually an alternate form of Faraday’s law for the case of changing magnetic flux through stationary conductors. 2. The induced emf has an associated induced electric field. 3. Changing magnetic field creates an electric field !!! 4. Two classifications of electric field: a) Electrostatic field (also called conservative electric field) - electric field produced by stationary charge distributions - conservative  - causes an electric force qE b) Nonelectrostatic field (also called nonconservative electric field) - induced electric field produced by changing magnetic flux - nonconservative  - causes an electric force qE F. Generalized Ampere’s law • Displacement current: dΦ E  I D = εo dt

; where ID = displacement current through a region

ΦE = electric flux through the region dΦE/dt = rate of change of ΦE through the region NOTES: 1. “Fictitious” current invented by Maxwell to correct or complete Ampere’s law and to satisfy Kirchhoff’s junction rule. 2. No actual flow of charged particles in a “displacement current”. 3. For distinction, current with flow of charged particles is called “conduction current”. 4. Changing electric flux has an associated displacement current. 5. Example: in the region between the plates of a charging capacitor 6. Displacement current can produce a magnetic field just like the conduction current. 7. Changing electric field creates a magnetic field !!!

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© Physics 72 Arciaga Generalized Ampere’s law:     B • dL = µ o ( IC,enc + ID,enc )

; where IC,enc = enclosed conduction current ID,enc = enclosed displacement current

G. Maxwell’s equations of electromagnetism • These are not new equations. They were just summarized neatly by Maxwell to emphasize their significance, particularly in building the idea of “electromagnetic wave”. • Maxwell’s four equations for electromagnetism: 1. Gauss’s law for electric fields:  Q    E • dA = enc εo  Implications: o static charges create an electric field (i.e. conservative electric field) o electric field lines start from positive charges and end at negative charges o Coulomb’s law can be derived from the above expression 2. Gauss’s law for magnetic fields:     B • dA = 0 Implications: o magnetic monopoles do not exist o magnetic field lines have no start and end (i.e. they are closed loops) 3. Ampere’s law with Maxwell’s correction:     dΦ      B • dL = µo ( IC,enc + ID,enc ) = µo IC,enc +  εo dtE enc   Implications: o moving charges (i.e. conducton currents) create a magnetic field o varying electric fields create a magnetic field o Biot-Savart law can be derived from the above expression 4. Faraday’s law:   dΦ B   E • dL = − dt  Implication: o varying magnetic fields create an electric field (i.e. nonconservative electric field) Amazing remark: Equations 1 and 2 look similar!!! Equations 3 and 4 look similar!!! 

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CHAPTER 30. INDUCTANCE •

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Recall the concept of electromagnetic induction (Chap. 29).

A. Mutual inductance • Key idea: A time-varying current in a coil (or circuit) causes an induced emf and induced current in another coil (or circuit), depending on their mutual inductance. • Keywords: mutual inductance, mutually-induced emf • Mutual inductance: Φ Φ  M = N 2 B2 = N1 B1 I1 I2 ; where M = mutual inductance (between coils 1 and 2) N1, N2 = number of turns of coils 1 and 2, respectively ΦB1, ΦB2 = magnetic flux through each turn of coils 1 and 2, respectively I1, I2 = current in coils 1 and 2, respectively NOTES: 1. Mutual inductance is scalar. 2. It is a shared property of two separated and independent coils (i.e. M = M12 = M21). 3. Depends on the geometry of the 2 coils (i.e. size, shape, number of turns, orientation, and separation) and the “core” material enclosed by the coils (vacuum, air, iron, etc.) 4. Independent of the current. 5. SI unit of mutual inductance is henry (H): 1 H = 1 Wb/A = 1 V⋅s/A = 1 Ω⋅s = 1 J/A2 6. High mutual inductance means that the 2 coils highly affect each other. • Mutually-induced emf: dI  2 = − M 1 dt dI  1 = − M 2 dt ; where 1, 2 = mutually-induced emf in coils 1 and 2, respectively M = mutual inductance (between coils 1 and 2) I1, I2 = current in coils 1 and 2, respectively B. Self-inductance and inductors • Key idea: A time-varying current in a coil (or circuit) causes an induced emf and induced current in itself, depending on its self-inductance. • Keywords: self-inductance, self-induced emf • Self-inductance: Φ  L=N B I ; where L = self-inductance (of a coil) N = number of turns of the coil ΦB = magnetic flux through each turn of the coil I = current in the coil NOTES: 1. Self-inductance is scalar. 2. It is a self property of a single coil. 3. Depends on the geometry of the coil (i.e. size, shape, number of turns) and the core. 4. Independent of the current. 5. SI unit of self-inductance is the same as that of mutual inductance.

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© Physics 72 Arciaga Self-induced emf: dI   = −L dt

; where = self-induced emf in a coil L = self-inductance (of the coil) I = current in the coil

Inductor: o a circuit device that is designed to have a particular inductance (i.e. self-inductance) o also called “choke” o opposes any variation of current through the circuit o voltage across an inductor depends on the time-rate of change of the current dI  V=L dt ; where V = voltage across an inductor (i.e. Ventry – Vexit) L = inductance of the inductor dI/dt = time-rate of change of the current through the inductor NOTES: 1. V is zero if dI/dt is zero (i.e. constant current). 2. V is either a rise or a drop depending on the sign of dI/dt (i.e. whether the current is increasing or decreasing).

C. Magnetic-field energy • Inductors can store magnetic-field energy (or simply magnetic energy). • Magnetic-field energy: 1  U = LI2 ; where U = magnetic energy stored in an inductor 2 L = inductance of the inductor I = current through the inductor NOTES: 1. Energy in the inductor is constant if the current is constant. 2. The inductor is storing energy while the current is increasing. 3. The inductor is releasing energy while the current is decreasing. 4. Compare with a capacitor that can store or release electric-field energy (U = ½ Q2/C). 5. Compare with a resistor that always dissipates energy. • Magnetic energy density: U 1 B2  u= = ; where u = magnetic energy density in an inductor volume 2 µ o B = magnetic field produced by the inductor NOTES: 1. Energy stored in an inductor is proportional to the square of the magnetic field. 2. Compare with the electric energy density of a capacitor (u = ½ εoE2).

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D. The R-L circuit • Time constant for an R-L circuit L  τ= ; where τ = time constant of an R-L circuit R L = inductance of the inductor R = resistance of the resistor  High τ ⇔ slow growth or decay of current  Low τ ⇔ fast growth or decay of current • Current growth in an RL circuit (if connected with an emf source):  Exponential increase o t = 0  I = 0 and dI/dt = /L o t = τ  I = 0.63 Imax o t = ∞  I = Imax and dI/dt = 0  Rule of thumb o Transient current through a “charging” inductor is zero if it has no initial current  like an open circuit element o At steady-state of a “fully-charged” inductor, current is constant and voltage is zero  like a short circuit element • Current decay in an RL circuit (if disconnected from an emf source):  Exponential decrease o t = 0  I = Imax and dI/dt = –ImaxR/L o t = τ  I = 0.37 Imax o t = ∞  I = 0 and dI/dt = 0  Rule of thumb o At steady-state of a “fully-discharged” inductor, current and voltage is zero  like a short circuit element but no current • Remark: Compare the above situations with that of the R-C circuit (Chap. 26). E. The L-C circuit • Key ideas: a. Electrical oscillation happens in an L-C circuit. i. Charge and current oscillate back and forth  the oscillation is sinusoidal in time ii. Total energy is conserved; but energy transforms from electric-field energy to magnetic-field energy back and forth  the oscillation is sinusoidal-squared iii. When charge is full, current is zero. When current is full, charge is zero. b. Analogous to a mechanical oscillation (review your Physics 71). i. Position and velocity oscillate sinusoidally  simple harmonic motion ii. Total energy is conserved; but energy transforms from potential energy to kinetic energy back and forth iii. For more interesting analogies, see Table 30.1 of Young (but this is OPTIONAL). • Angular frequency of the electrical oscillation. 1  ω= ; where ω = angular frequency of an L-C electrical oscillation LC L = inductance of the inductor C = capacitance of the capacitor NOTES: 1. Recall from Physics 71: T = 2π/ω and f = ω/2π. 2. High L  slow oscillation [and Low L  fast oscillation] 3. High C  slow oscillation [and Low C  fast oscillation]

© Physics 72 Arciaga F. The L-R-C circuit • Key ideas: a. Resistance is analogous to friction; they both dissipate energy. b. Damped electrical oscillation happens in an L-R-C circuit. c. Analogous to the damped mechanical oscillation (review Physics 71). • Three cases: 1. Underdamped  happens for low R (i.e. R < 2 L C )  charge and current still “oscillate”  but they “die” with an exponential decay envelope  increasing R causes slower “oscillation” but quicker “death” 2. Critically damped  happens at moderate R (i.e. R = 2 L C )  charge and current do not “oscillate”  they “die” the quickest possible way 3. Overdamped  happens for high R (i.e. R > 2 L C )  charge and current do not “oscillate”  they “die” slower compared to the critically damped case  increasing R causes slower “death”

CHAPTER 31. ALTERNATING CURRENT •

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Start thinking or visualizing sinusoidal graphs. This skill will help you a lot!

A. Alternating current • Alternating current (ac) – direction of the current continuously changes • AC source – any device that supplies a sinusoidally varying voltage or current  V = Vmax cosωt  I = Imax cosωt ; where V and I = instantaneous voltage and current, respectively Vmax and Imax = voltage and current amplitudes, respectively ω = angular frequency t = time Example: In the Philippines, f = 60 Hz (i.e. ω = 377 rad/s). • Phasors o rotating “vectors” that can be used to represent sinusoidally carrying voltages and currents o they are only geometric tools for easier analysis of ac circuits o characteristics:  a phasor rotates counterclockwise with a constant angular speed (ω)  length of a phasor is equal to the amplitude value (Vmax or Imax)  projection of a phasor onto the horizontal axis is the instantaneous value (V or I)  angular displacement of the phasor is ωt after an elapsed time t o key idea: When using phasor diagram for ac circuit analysis, it is just like performing vector addition but taking the x-component of the final answer. • Average values: 1. Root-mean-square value (rms value) V  Vrms = max 2 I max  I rms = 2 ; where Vrms and Irms = rms values of voltage and current, respectively Vmax and Imax = voltage and current amplitudes, respectively Example: In the Philippines, Vrms = 110 V or 220 V (i.e. Vmax = 156 V or 311 V). 2. Rectified average value (rav) 2  Vrav = Vmax π 2  I rav = I max π ; where Vrav and Irav = rav of voltage and current, respectively Vmax and Imax = voltage and current amplitudes, respectively NOTES: 1. rms value is more commonly used instead of rav. 2. Some quick guides: a. ave. value of (A sinωt) or (A cosωt) over a period is zero. b. ave. value of (A sinωt) or (A cosωt) over a quarter-period is 2A/π. c. ave. value of (A2 sin2ωt) or (A2 cos2ωt) over a period is ½ A2. d. ave. value of (A cosωt sinωt) over a period is zero.

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B. Resistance and reactance • General forms:  I = Imax cos(ωt)  V = Vmax cos(ωt + φ)  Vmax = ImaxX  Vrms = IrmsX ; where φ = phase angle X = resistance or reactance (SI unit is ohm) NOTE: The phase angle is the angle of the voltage phasor with respect to the current phasor. • Resistor:  I = Imax cos(ωt)  V = IR = ImaxR cos(ωt)  Vmax = ImaxR NOTES: 1. φ = 0o  voltage is in phase with the current 2. XR = R  resistance 3. ω does not affect R • Inductor:  I = Imax cos(ωt) dI  V = L = I max ωL cos(ωt + 90o ) dt  Vmax = ImaxωL NOTES: 1. φ = 90o  voltage is out of phase with the current  voltage leads the current by 90o 2. XL = ωL  inductive reactance 3. ↑ω  ↑XL  ↓Imax  inductor “hates” high ω and “loves” low ω • Capacitor:  I = Imax cos(ωt) Q I  V = = max cos(ωt − 90o ) C ωC I  Vmax = max ωC NOTES: 1. φ = –90o  voltage is out of phase with the current  voltage lags the current by 90o 1 2. X C =  capacitive reactance ωC 3. ↑ω  ↓XC  ↑Imax  capacitor “loves” high ω and “hates” low ω

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C. The L-R-C series circuit • Recall the properties of a series connection: a. IL = IR = IC = Isource  instantaneous currents are equal (not amplitude) b. VL + VR + VC = Vsource  instantaneous voltages are added (not amplitude) • Apply the phasor diagram analysis. • Some important relations:  Vmax = ImaxZ 

Z = R 2 + (X L − X C ) 2

XL = ωL 1  XC = ωC  X − XC   φ = tan −1  L  R   ; where Z = impedance of the ac circuit (SI unit is ohm) φ = phase angle NOTES: 1. R, XL, XC, and Z are analogous. They are all measures of “resistance to current flow”. 2. If XL > XC, then φ > 0 (i.e. the voltage leads the current) 3. If XL < XC, then φ < 0 (i.e. the voltage lags the current) 4. If XL = XC, then φ = 0 (i.e. the voltage is in phase with the current) and resonance occurs ( see Section E). 

D. Power in ac circuits • Instantaneous power  P = IV ; where P, I, and V = instantaneous power, current, and voltage, respectively • Average power  Pave = ½ ImaxVmaxcosφ = IrmsVrmscosφ Pave = average power Vmax and Imax = voltage and current amplitudes, respectively Vrms and Irms = rms values of voltage and current, respectively φ = phase angle NOTES: 1. cosφ is called the power factor of the ac circuit. 2. For a pure resistor R connected to an ac source: • cosφ = 1 V2 1 • Pave = Imax Vmax = I rms Vrms = I 2rms R = rms 2 R 3. For a pure inductor L or a pure capacitor C connected to an ac source: • cosφ = 0 • Pave = 0 4. For a series L-R-C circuit connected to an ac source: • cosφ = R/Z 1 R R R 2 • Pave = Imax Vmax = I rms Vrms = I2rms R = Vrms 2 Z Z Z2

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E. Resonance in a series L-R-C circuit • Key ideas: 1. If a series L-R-C circuit is connected to an ac source, then there will be an electrical driven oscillation. (Analogous to the mechanical driven oscillation in Physics 71.) 2. If a series L-R-C circuit is connected to an ac source, electrical resonance can occur if the frequency of the source is the same as the natural frequency of the series L-R-C circuit. (Analogous to the mechanical resonance in Physics 71.) • Conditions for resonance in a series L-R-C circuit 1. XL = XC 1 2. ωsource = = ωnatural LC • What happens at resonance 1. Z is minimum (i.e. Z = R). 2. Imax is largest (i.e. Imax = Vmax/R). F. Transformers • Transformer – device that employs the idea of electromagnetic induction to step-up or step-down the voltage amplitudes from a primary coil to a secondary coil. • Important parts of a transformer: 1. Primary coil or winding – connects to an ac source 2. Secondary coil or winding – connects to a circuit or device 3. Core (usually iron) – ensures that almost all magnetic field lines from primary coil pass through the secondary coil • Important relations: Vmax,2 N 2  = Vmax,1 N1  Vmax,2Imax,2 = Vmax,1Imax,1 ; where Vmax,1 and Vmax,2 = voltage amplitudes in the primary and secondary coils, respectively Imax,1 and Imax,2 = current amplitudes in the primary and secondary coils, respectively N1 and N2 = number of turns in the primary and secondary coils, respectively NOTES: 1. The first relation comes from Faraday’s law.  For constant flux change, ↑turns  ↑induced emf 2. The second relation comes from conservation of energy.  For constant power, ↑voltage  ↓current 3. Step-up transformer:  Vmax,2 > Vmax,1 ⇔ N2 > N1 and Imax,2 < Imax,1 4. Step-down transformer:  Vmax,2 < Vmax,1 ⇔ N2 < N1 and Imax,2 > Imax,1

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CHAPTER 32. ELECTROMAGNETIC WAVES • •

Review the key concepts about mechanical waves (Physics 71) Important: Review the Maxwell’s equations and its implications (Chap. 29)

A. Electromagnetic waves • Wave o transports disturbance, energy, and momentum from one region to another o speed of the wave: v = λf ; where v = speed of a wave λ = wavelength of the wave f = frequency of the wave • Electromagnetic wave (EM wave) o also called electromagnetic radiation o a wave that can propagate even when there is no matter (i.e. vacuum) or no medium o predicted by the four Maxwell’s equations o consists of time-varying electric and magnetic fields (i.e. “waving” electric and magnetic fields) o produced by accelerating charges (e.g. transmitter antenna) • General characteristics of electromagnetic wave (as predicted by Maxwell’s equations) o speed in vacuum 1  c= = 3.00×108 m/s ; where c = speed of EM wave in vacuum εoµ o εo = permittivity of free space or vacuum µo = permeability of free space or vacuum

o speed in matter (i.e. not vacuum) or medium 1 c  v= = ; where v = speed of EM wave in a medium εµ n c = speed of EM wave in vacuum ε = permittivity of the medium µ = permeability of the medium n = index of refraction of the medium  Remarks: • v  c  nothing is faster than “c” • n  1  index of refraction is a property of matter • EM waves slow down when moving in a medium • commonly, for EM waves in a medium, replace εo, µo, and c by ε, µ, and v, respectively o transverse wave  the electric field, magnetic field, and direction of propagation of the EM wave are all perpendicular to each other    E × B points the direction of propagation of the EM wave o definite ratio of amplitude  E = cB ; where E = magnitude of the electric field B = magnitude of the magnetic field c = speed of EM wave in vacuum  Remark: • E >> B  magnitude of the magnetic field is usually small

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B. Energy and momentum in electromagnetic waves • Energy o Poynting vector in vacuum  1     S= E×B ; where S = Poynting vector of the EM wave in vacuum µo   E and B = electric and magnetic fields, respectively  Remarks: • Poynting vector points toward the direction of propagation of the EM wave   • S = EB/µo since the E and B are perpendicular in an EM wave • significance of Poynting vector: o energy flowing per unit time per unit area o power transfer per unit area 1 dU o S= A dt o Intensity of sinusoidal EM wave in vacuum 1 1  I = Save = Smax = E max Bmax 2 2µo ; where I = intensity of the sinusoidal EM wave in vacuum Save = average Poynting vector Smax = maximum Poynting vector Emax = electric field amplitude Bmax = magnetic field amplitude  Remark: • significance of intensity: o average energy flowing per unit time per unit area o average power transfer per unit area 1  dU  o I=   A  dt ave • Momentum o Radiation pressure of EM wave if totally absorbed S I  p rad = ave = ; where prad = radiation pressure by an absorbed EM wave c c o Radiation pressure of EM wave if totally reflected 2S 2I  p rad = ave = ; where prad = radiation pressure by a reflected EM wave c c  Remarks: • significance of radiation pressure: o average rate of momentum transfer per unit area o average force per unit area 1  dp  1 o p rad =   = Fave A  dt ave A • larger force is imparted by the EM wave to a surface it hits when it is reflected than when it is absorbed by the surface

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C. Electromagnetic spectrum Categories

Frequency (Hz)

Wavelength (m)

Radiowave Microwave Infrared Visible light Red Orange Yellow Green Blue Violet Ultraviolet X ray Gamma ray

3×108 ~3×108-1012 ~3×1011-1015

1 ~10-4-1 ~10-7-10-3

~405-480×1012 ~480-510×1012 ~510-530×1012 ~530-600×1012 ~600-700×1012 ~700-790×1012 ~3×1015-1017 ~3×1016-1021 3×1018

~625-740×10-9 ~590-625×10-9 ~565-590×10-9 ~500-565×10-9 ~430-500×10-9 ~380-430×10-9 ~10-9-10-7 ~10-13-10-8 10-10

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Applications radio (AM, FM), TV (UHF, VHF) cellphone, oven, radar, wi-fi camera focusing, stove, heat sensor

high-precision apps, eye surgery x-ray imaging, crystal structure analysis cancer treatment, sterilization

Remarks: o for any EM wave, v = λf always o in general, EM waves with higher frequency have shorter wavelengths o v = c = 3×108 m/s if EM wave moves in vacuum o some categories overlap in the spectrum o in general, EM waves with higher frequency have higher energy  Physics 73 o the range of values in the table above are just approximate values

CHAPTER 33. THE NATURE AND PROPAGATION OF LIGHT • • •

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Optics – branch of physics that deals with the behavior of light and other EM waves Geometric optics – focuses on ray analysis of light Physical optics – focuses on wave behavior of light

A. The nature of light • Light o usually refers to the visible portion of the electromagnetic spectrum o has a wave-particle duality (i.e. possesses both wave-like and particle-like properties) o particle-like – appropriate when discussing the emission and absorption of light  Plato, Socrates, Euclid, Newton, Einstein o wave-like – appropriate when discussing the propagation of light  Huygens, Maxwell, Hertz, Young, Fresnel, Fraunhofer • Some keywords to describe wave propagation a. wave front – locus of all adjacent points at which the phase of the wave is the same – distance between two adjacent wave fronts is equal to the wavelength b. spherical wave – produced by a point source – represented by spherical wave fronts centered at the point source c. plane wave – produced by a very far point source – represented by plane wave fronts d. ray – an imaginary line that indicates the direction of propagation of the wave – perpendicular to the wave fronts • Huygens’ principle o Geometrical method to determine the succeeding wave front from the preceding wave front o “Every point of a wave front may be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the wave.” B. Reflection and refraction • Key idea: light can be partially reflected and partially transmitted (refracted) at an interface between two media (i.e. materials) with different indexes of refraction • Some keywords: a. incident ray – ray that describes the wave coming to the interface b. reflected ray – ray that describes the wave reflected from the interface c. refracted ray – ray that describes the (bent) wave transmitted through the interface • Types of reflection 1. specular reflection – well-directed reflection from a smooth surface 2. diffused reflection – scattered reflection from a rough surface • Index of refraction (or refractive index) o recall: n = c/v  see Chap. 32 o dimensionless number that describes the medium o for vacuum, n =1 o for air, n = 1.0003 (i.e. n ≈ 1) o in general, n  1 o affects the speed, direction, and wavelength of the wave (but not the frequency)  high n, low speed [v = c/n]  high n, short wavelength [λ = λvacuum/n]

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© Physics 72 Arciaga Law of reflection o θi = θr ; where θi and θr = angle of incidence and angle of reflection, respectively o Remarks  angles are measured between the ray and the normal (i.e. perpendicular to surface)  reflected ray is at the same angle as the incident ray  also, the incident ray, reflected ray, and normal all lie in the same plane Law of refraction o ni sinθi = nt sinθt ; where θi and θt = angle of incidence and angle of refraction, respectively ni = index of refraction of primary medium (incident side) nt = index of refraction of secondary medium (refracted side) o Remarks  angles are measured between the ray and the normal (i.e. perpendicular to surface)  refracted ray is bent with respect to the incident ray a. bends toward the normal if light moves from low n to high n b. bends away from the normal if light moves from high n to low n  also, the incident ray, refracted ray, and normal all lie in the same plane

C. Total internal reflection (TIR) • Key idea: light can be totally reflected at an interface between two materials even if the second medium is transparent • Critical angle n o sin θcrit = t ; where θcrit = critical angle for TIR ni o Remarks  Two necessary conditions for TIR to happen a. n of primary medium is greater than n of second medium [ni > nt] b. angle of incidence is greater than or equal to the critical angle [θi  θcrit]  If TIR happens, then there is no refracted (transmitted) ray D. Dispersion • Dispersion – dependence of the index of refraction on the wavelength (in vacuum) of the wave • Dispersion curve – plot showing the dependence of n on λvacuum [n vs. λvacuum curve] • Key idea: ordinary white light, which is composed of EM waves with different wavelengths, can be separated into its different colors by dispersion (e.g. prism dispersion) • Remarks o λvacuum determines n, and n determines v o If n decreases as λvacuum increases, then  long λ are faster than short λ [recall: λ = λvacuum/n]  long λ have smaller deviation θ than short λ o If n increases as λvacuum increases, then  long λ are slower than short λ [recall: λ = λvacuum/n]  long λ have larger deviation θ than short λ

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© Physics 72 Arciaga Rainbow formation o dispersion + refraction + reflection o primary rainbow  single reflection inside the water droplet  bright, but thin  red has larger radius than violet o secondary rainbow  double reflection inside the water droplet  thick, but faint  violet has larger radius than red (i.e. reverse order of primary rainbow)

E. Polarization • Tip: When thinking about polarization, it helps a lot if you will imagine about components of an oscillating or rotating vector (in this case, the vector is the electric field) • Polarization – characteristic of all transverse waves – for EM waves, it describes the direction of oscillation of the electric field – this is different from the polarization you learned in Chap. 21 • Unpolarized light (or natural light) – light with no polarization (i.e. random direction) • Types of polarization 1. linearly polarized  electric field oscillates along a line  can be composed of two perpendicular wave components with phase difference equal to 0 or ±π 2. circularly polarized  tip of electric field traces a circle [looks like a rotating helix] a. right circularly polarized – clockwise rotation (as viewed opposite to direction of propagation) b. left circularly polarized – counterclockwise rotation (as viewed opposite to direction of propagation)  can be composed of two perpendicular wave components with same amplitude and phase difference equal to ±π/2 [i.e. quarter-cycle or quarter-wave difference] 3. elliptically polarized  tip of electric field traces an ellipse [looks like a rotating “distorted” helix]  can be composed of two perpendicular wave components with same amplitude and phase difference NOT equal to 0, ±π, or ±π/2  can be composed of two perpendicular wave components with different amplitudes and phase difference NOT equal to 0 or ±π • Methods of polarization 1. radiowave: a straight antenna creates a linearly polarized radiowave 2. radiowave: two perpendicular straight antennas with a phase-shifting network can create circularly or elliptically polarized radiowave 3. microwave: a grill-like array of conducting wires can transform any microwave into linearly polarized microwave 4. light: a quarter-wave plate birefringent material can transform a linearly polarized light to a circularly polarized light, and vice-versa  birefringent – a material with different indexes of refraction for different directions of polarization (e.g. calcite)  birefringence – behavior of birefringent materials

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© Physics 72 Arciaga 5. light: a polarizing filter (or polarizer) composed of a dichroic material can transform any light into linearly polarized light  dichroic – a material which absorbs a particular direction of polarization (e.g. Polaroids in sunglasses and cameras)  dichroism – behavior of dichroic materials  polarizing axis – the orientation of the transmitted linearly polarized light 6. light: reflection can cause partial or total polarization of light  key idea: the component of the electric field parallel to the interface is reflected MORE than the non-parallel component  Brewster’s law for the polarizing angle n • tan θpol = t ni ; where θpol = polarizing angle ni = index of refraction of primary medium (incident side) nt = index of refraction of secondary medium (refracted side) • If incident angle is equal to the polarizing angle, then a. reflected ray is completely linearly polarized parallel to the interface b. refracted ray is partially linearly polarized non-parallel to the interface c. reflected and refracted rays are perpendicular to each other Intensity after polarization o When an unpolarized light passes through a single polarizer, the intensity of the transmitted linearly polarized light is halved  Iline = ½ Iunpol o When a circularly polarized light passes through a single polarizer, the intensity of the transmitted linearly polarized light is halved  Iline = ½ Icirc o When a linearly polarized light passes through another polarizer, the intensity of the transmitted linearly polarized light depends on the orientation of the polarizing axis  Iout = Iin cos2φ ; where φ = angle between the directions of the incident linearly polarized light and the polarizing axis Iin = intensity of the incident light Iout = intensity of the transmitted light  Remarks • The above equation is called “Malus’s law”. • It can be used successively for a series of 2 or more polarizers. • A series of 2 polarizers is usually called a “polarizer-analyzer” setup. • If the 2 polarizing axes are aligned, then Iout = Iin. • If the 2 polarizing axes are perpendicular, then Iout = 0.

F. Scattering of light • Scattering – when light is absorbed and then re-radiated to different directions • Key idea: long wavelength is less scattered 1 o Iscattered ∝ 4 λ o The above relation is called Rayleigh’s scattering law • Explains why the sky is blue, sunsets are red, and clouds are white

CHAPTER 34. GEOMETRIC OPTICS • •

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Geometric optics – focuses on the ray analysis of light Review ray model of light, law of reflection, and law of refraction (Chap.33)

A. Keywords • Object o Point object o Extended object – can be thought of as composed of numerous point objects

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• • •

• •

• •

o o Image o o

Real object – when the light actually passes through the object Virtual object – … does not actually pass … Point image Extended image – can be thought of as formed by numerous point images

o Real image – when the light actually passes through the image  can be seen on a screen o Virtual image – … does not actually pass …  cannot be seen on a screen Location o Object distance – distance of the object from the mirror, surface, or lens o Image distance – … image … Magnification (lateral) o Magnified / Enlarged – when the image is larger than the lateral size of the object o Reduced / Diminished / Minified – … smaller … Orientation o Erect / Upright – when the image is in the same lateral direction as the object o Inverted – … opposite lateral direction … o Reversed – … opposite axial direction … Plane surface Spherical surface o Center of curvature o Radius of curvature o Vertex o Optic axis – axis connecting the vertex and the center of curvature o Focal point – location where light seems to converge/diverge due to the mirror or lens o Focal length – distance between the focal point and the mirror or lens Spherical mirror o Converging mirror / Concave mirror o Diverging mirror / Convex mirror Thin lens o Converging lens / Positive lens  Double-convex  Plano-convex  Meniscus o Diverging lens / Negative lens  Double-concave  Plano-concave  Meniscus

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B. Principal-ray diagram • Principal rays 1. “center” ray – no deviation through center 2. “vertex” ray – at vertex, equal angle with optic axis 3. “parallel” ray – parallel then to/away from focal point 4. “focal” ray – focal point then parallel • NOTES: o Key idea: Find the intersection of the principal rays to locate the image of the object. o Warning: Find the intersection of the outgoing parts of the rays !!! [not with the incoming] o Generally, two rays are enough. o Valid approximation only for cases involving paraxial rays (i.e. close and nearly parallel to the optic axis)  usually called “paraxial approximation” C. Sign rules • Object distance o Positive – if object is on the same side as the incoming light (i.e. “real object”) o Negative – … opposite side … (i.e. “virtual object”) • Image distance o Positive – if image is on the same side as the outgoing light (i.e. “real image”) o Negative – … opposite side … (i.e. “virtual image”) • Radius of curvature o Positive – if center of curvature is on the same side as the outgoing light o Negative – … opposite side … • Focal length o Positive – if the mirror or lens is converging o Negative – … diverging • Magnification o Positive – if the image is erect o Negative – … inverted D. Reflection at a plane or spherical surface (including mirror) • Basic equations o Object-image relation 1 1 2 1  + = = s s' R f o Magnification y' s'  m= =− y s ; where s, s’ = object and image distance, respectively R = radius of curvature f = focal length m = magnification y, y’ = (lateral) height of the object and image, respectively o NOTES:  Follow the “sign rules” properly !!! [see Section C]  The above equations are valid only for “paraxial approximation”  For spherical surfaces, f = ½ R  If the surface is plane, just make R =  (and f = 0)  Can be verified using the “principal-ray diagram” [see Section B]

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© Physics 72 Arciaga Plane mirror o s = –s’ [image and object are equidistant from the plane mirror] o m = 1 [image and object have the same size] o virtual, same-sized, erect (but reversed) image Spherical mirror o Convex mirror / Diverging mirror  virtual, reduced, erect image o Concave mirror / Converging mirror  if s > R : real, reduced, inverted image  if s = R : real, same-sized, inverted image  if R > s > f : real, magnified, inverted image  if s = f : image is at infinity  if s < f : virtual, enlarged, erect image

E. Refraction at a plane or spherical surface • Basic equations o Object-image relation na nb nb − na  + = s s' R o Magnification n s' y'  m= =− a y n bs ; where s, s’ = object and image distance, respectively R = radius of curvature m = magnification y, y’ = (lateral) height of the object and image, respectively na = index of refraction of the medium at incident-side nb = index of refraction of the medium at refracted-side o NOTES:  Follow the “sign rules” properly !!! [see Section C]  The above equations are valid only for “paraxial approximation”  If the surface is plane, just make R =  (and f = 0)  Cannot be verified using the “principal-ray diagram” in Section B  Use ray-tracing and Snell’s law of refraction for verification [see Chap. 33] • Plane surface s' s o =− nb na o m = 1 [image and object have the same size] o virtual, same-sized, erect image o If na > nb, the apparent depth is shallower than the true depth (e.g. water-to-air) • Spherical surface o Convex surface o Concave surface

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F. Thin lenses • Thin lens – can be thought of as composed of two refracting surfaces in close proximity • Basic equations o Object-image relation 1 1 1  + = s s' f o Magnification y' s'  m= =− y s o Lensmaker’s equation 1  n lens  1 1   = − 1 −  f  n sur   R1 R 2  ; where s, s’ = object and image distance, respectively f = focal length of the thin lens m = magnification y, y’ = (lateral) height of the object and image, respectively nlens = index of refraction of the lens nsur = index of refraction of the medium surrounding the lens R1 = radius of curvature of the “entry” side of the lens R2 = radius of curvature of the “exit” side of the lens o NOTES:  Follow the “sign rules” properly !!! [see Section C]  The above equations are valid only for the following conditions: • “paraxial approximation” • “thin lens approximation”  If one of the two surfaces is plane, just make R =  for that surface.  Can be verified using the “principal-ray diagram” [see Section B] but: • you cannot use the “vertex” ray • interpret the “center” ray as a ray passing through the center of the lens (not center of curvature) and goes on undeviated  Usually, the surrounding medium is air (nsur = 1).  The lensmaker’s equation relates the focal length of a lens with the index of refraction and radii of curvature of the two surfaces of the lens. o Important tip !!!  For a given series or combination of lenses (or mirrors), treat the image formed by a particular lens (or mirror) as the object of the next lens (or mirror) in the series. • Converging lens / Positive lens o Positive focal length o Thicker at the center than the edges (e.g. double convex, plano-convex, meniscus) • Diverging lens / Negative lens o Negative focal length o Thinner at the center than the edges (e.g. double concave, plano-concave, meniscus) • Optional reading assignment: o Camera – converging lens, film, detector, telephoto lens, wide-angle lens, zoom lens o Eye – cornea, crystalline lens, retina, myopia, hyperopia, astigmatism o Magnifier – converging lens o Microscope – objective lens, eyepiece (ocular) o Telescope – objective lens (or concave mirror), eyepiece

CHAPTER 35. INTERFERENCE •

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Physical optics – focuses on the wave behavior of light

A. Interference • Interference: o happens when two or more waves pass through the same region at the same time o overlapping of two or more waves o apply principle of superposition of waves • The principle of superposition of waves: o When two or more waves interfere, the actual displacement of any point in the medium at any time is equal to the sum of the displacements of the separate waves. • Monochromatic light o light of single color, single frequency, or single wavelength (e.g. laser) • Coherent sources o two monochromatic sources that have the same frequency and a constant phase difference (although not necessarily in phase with each other) o able to produce coherent waves or coherent light (e.g. laser) o only coherent sources interfere !!!  “coherence is needed for interference” • Special types of interference: 1. constructive interference  happens when two (or more) waves interfere “in-phase”  happens when the path difference between the two waves is an integral multiple of the wavelength o r2 – r1 = mλ (m = 0, ±1, ±2, …)  happens when the phase difference between the two waves is an integral multiple of 2π o φ2 – φ1 = m2π (m = 0, ±1, ±2, …)  results to reinforcement of wave amplitudes 2. destructive interference  happens when two (or more) waves interfere “180o out-of-phase”  happens when the path difference between the two waves is a half integral multiple of the wavelength o r2 – r1 = (m + ½)λ (m = 0, ±1, ±2, …)  happens when the phase difference between the two waves is a half integral multiple of 2π o φ2 – φ1 = (m + ½)2π (m = 0, ±1, ±2, …)  results to cancellation of wave amplitudes

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B. Double-slit interference of light (Young’s experiment) • Some simplifying approximations: o rays are almost parallel to each other  when the screen is very far from the slits o rays are near the central axis  when the considered angles are very small • Bright fringes (constructive interference) o r2 – r1 = d sinθ = mλ φ2 − φ1 r2 − r1 o = 2π λ mRλ o y= d ; where r2 – r1 = path difference φ2 – φ1 = phase difference (in radians) y = position of the bright fringe measured from the central axis m = any integer (m = 0, ±1, ±2, …) d = separation distance of the adjacent slits θ = angle of the ray with respect to the central axis λ = wavelength of the light R = distance of the screen from the slits • Dark fringes (destructive interference) o r2 – r1 = d sinθ = (m + ½)λ φ2 − φ1 r2 − r1 o = 2π λ (m + ½)Rλ o y= d ; where r2 – r1 = path difference φ2 – φ1 = phase difference (in radians) y = position of the dark fringe measured from the central axis • Intensity φ −φ   πd  o I = Imax cos 2  2 1  = I max cos 2  y  Rλ   2  ; where I = intensity Imax = maximum intensity y = position along the screen measured from the central axis • Remarks: Rλ o Separation of adjacent bright (or dark) fringes: ∆y = d  pattern is more spread if λ is high  … if d is low  … if R is high o Maximum intensity: Imax = 4⋅Islit y o For small angles, you can use: θ ≈ sin θ ≈ tan θ = R

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C. Interference in thin films • Caused by interference of reflected waves • Phase reversal (or 180o phase-shifting) o phenomenon in which the reflected wave shifts by half a cycle (i.e. 180o phase shift) o happens when the wave comes from a “fast speed” medium to a “slow speed” medium (e.g. when light travels from low n to high n, such as reflection at air-to-water interface) • Case I: If no phase reversal or if phase reversal occurs for both waves: o Constructive interference (at normal incidence)  2t = mλ’ (m = 0, ±1, ±2, …) o Destructive interference (at normal incidence)  2t = (m + ½)λ’ (m = 0, ±1, ±2, …) ; where t = thickness of the thin film λ’ = λvacuum / n = wavelength of the wave in the thin film medium n = index of refraction of the thin film medium • Case II: If phase reversal occurs for only one of the two waves: o Constructive interference (at normal incidence)  2t = (m + ½)λ’ (m = 0, ±1, ±2, …) o Destructive interference (at normal incidence)  2t = mλ’ (m = 0, ±1, ±2, …) • Important notes: o The conditions for constructive and destructive interferences in the two cases are opposite. o For λ’, use wavelength in the thin film medium; not just the wavelength in vacuum. o In anti-reflection coatings or thin films, consider destructive interference.

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CHAPTER 36. DIFFRACTION

A. Diffraction • Diffraction: o interference of many waves from many sources or from a continuous source o happens when a wave encounters apertures, barriers, or edges o apply Huygens’ principle and principle of superposition of waves  see Chap. 33 and 35 • Remark: There is really no fundamental distinction between interference and diffraction B. Multiple slits (ideal case) • Ideal case: Neglect width of the slits • Remarks: o Principal maxima 

separation does not depend on N (number of slits) : ∆y =

Rλ d

intensity is proportional to N2 : Imax = N2⋅Islit become narrower (sharper) as N increases ; where ∆y = separation of the adjacent principal maxima N = number of slits (N  2) d = separation distance of the adjacent slits λ = wavelength of the light R = distance of the screen from the slits Imax = maximum intensity in the pattern Islit = intensity passing through each slit o Secondary maxima  there will be (N – 2) secondary maxima in between adjacent principal maxima  there will ne (N – 1) minima in between adjacent principal maxima  no secondary maxima for a 2-slit setup  

C. Single-slit diffraction • Do not neglect width of the slits • Remarks: o Central bright maximum (i.e. bright band at the center) 2Rλ  width of CBM : w = a Rλ  half-width of CBM : w half = a ; where w = width of the CBM whalf = half-width of the CBM a = width of each slit o Dark fringes mRλ  positions : y = ; (m = ±1, ±2, …)  Note: m = 0 is not included !!! a

o Intensity





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2

  πa    sin  Rλ y    I = Imax   π a  y   Rλ  ; where I = intensity Imax = maximum intensity a = width of each slit The function “sin(x)/x” is called “sinc(x)”

C. Multiple slits (real case) • Real case: Do not neglect width of the slits • Key idea: The resulting intensity pattern is just a modulated version of the ideal case. Modulation is done by enveloping with the single-slit intensity pattern corresponding to the width of the slit [i.e. the sinc2(x) function shown above]. • Remarks: o separation of the bright fringes does not change by the modulation, except for some possible “vanishing” maxima o number of bright fringes within the CBM is related to the ratio of the slit separation to the slit width