Electromagnetism Physics 15b Lecture #1 Coulomb’s Law Electric Charge, Force, and Energy Purcell 1.1–1.6

Today’s Goals Electric charge Quantization, charge symmetry   Conservation  

Coulomb’s Law Forces between two charged objects  Inverse-square law   CGS units vs. SI units  

Superposition Principle  

2-body  N-body of charges

Work and energy  

Conservative force

Charles-Augustin de Coulomb (1736–1806)

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Four Forces There are 4 fundamental forces in Nature Gravity   Long distance. Keeps planets and satellites in orbits   Classical model (Newtonian) simple and accurate   Modern model (General Relativity) complex and more accurate   Electromagnetic force   Long distance. Responsible for most daily things   Classical model (Maxwell) simple, accurate, and identical to the modern model   Strong nuclear force   Short distance. Keeps protons and neutrons in atomic nuclei   Weak nuclear force   Short distance. Responsible for nuclear b-decays  

Electrical Charge Objects don’t always respond to electricity  

Something must be added  or, they must be “charged”

Experiments have told us: Charges can be positive or negative (Franklin)   They come in small same-sized units (Millikan)   Each unit is so small, and the number of units in human-size object so large (~1023), it looks like continuous  

Intrinsic property of the elementary particles Proton is positively charged   Electron is negatively charged by the same amount  

e = “Elementary Charge”

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Charge Symmetry What is “positive” (or “negative”) is just a convention  

Franklin could have named them oppositely

Elementary particles have oppositely-charged anti-particles Particle Proton

Antiparticle +e

Electron –e Neutron

0

Antiproton

–e

Positron

+e

Antineutron

0

Physical laws are almost unchanged if the positive and negative charges are replaced

Small catch: Universe made exclusively of particles Just so, i.e. it started that way and stayed that way?   Only in our neighborhood, i.e. this galaxy?   Small violation of the charge symmetry?  

Conservation of Charge Charge cannot be created or destroyed  

You can only move them from one object to another

Total charge of an isolated system is conserved True even in extreme conditions where particle are destroyed and created   Example from high-energy collision of an electron and a positron   

e + e − → π +π −π +π −π +π −

Demo: charge conservation

An event from the CLEO Experiment Courtesy of A. Foland

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Coulomb’s Law Electric force (F) between two charges (q1 and q2) can work over a distance (r) Smaller r  stronger F   Larger r  weaker F  

F1

Coulomb found

F2 = k  

q1q2 r212

q1

q2

r21

F2

Proportional to both charges

rˆ21

Along the line connecting the charges Inversely proportional to the (distance)2

NB: definition of the unit vector

rˆ ≡

r r

Parallel to the original r, with the length

rˆ = 1

Units Two systems of units: CGS vs. SI (Systèm Internationale) CGS

SI

Fundamental Units cm, g, s m, kg, s, ampere (A)  

In CGS, Coulomb’s Law (with k = 1) defines the unit of charge

F[dyne] =  

q1[esu]q2 [esu] r21[cm]2

1esu = 1cm 1dyne electrostatic unit

In SI, the unit of charge is coulomb (C) = ampere x second

F[N] = k  

q1[C]q2 [C]

More often:

r21[m]2

k≡

1 4πε 0

k = 8.9875 × 109 N ⋅ m 2 C2 ≈ 9.0 × 109 N ⋅ m 2 C2

ε 0 = 8.8542 ×10−12 C2 N ⋅ m 2 permittivity of vacuum

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Conversion Table SI Units

CGS Units

Energy

1 joule (J)

= 107 erg

Force

1 newton (N)

= 105 dyne

Electric Charge

1 coulomb (C)

= “3” x 109 esu

Electric Current

1 ampere (A)

= “3” x 109 esu/sec

Electric Potential

“3” x 102 volt (V) = 1 statvolt

Electric Field

“3” x 104 V/m

= 1 statvolt/cm

Elementary Charge e

1.6 x 10–19 C

= 4.8 x 10–10 esu

“3” = 2.99892458 (exact)   See Table E.1 (p.475) of Purcell for more  

We will stick with CGS most of the time (as Purcell does)  

Real-world examples may have to use SI

Inverse-Square Laws Electrostatic force and gravity share the r-dependence

Fe = k  

q1q2 r

2

Fg = G

m1m2 r

Inverse-square laws

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Much of what you know about gravity will apply to electricity

⎧k = 1

Difference 1: constants ⎨

−8 2 2 ⎪⎩G = 6.672 × 10 dyne ⋅ cm g

1 esu

1 esu

1g

1g

1 dyne 0.00000007 dyne

1 cm

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Electrostatic Force vs. Gravity Difference 2: signs Charge or Mass Force Electrostatic positive or negative attractive or repulsive Gravity only positive only attractive Since electrostatic force is so strong, large objects tend to contain roughly equal numbers of protons and electrons  

Net charge is zero, or quite small

Electrostatic force is important inside or between atoms  

That’s physics and chemistry

Gravity is important for astronomical objects

Superposition Principle Suppose we have many charges distributed in space  

What is total force on charge Q from all the other charges?

Principle of Superposition Just add them up as vectors   For 3 charges (picture right)

q1q3 r312

rˆ31 +

q2 q3 r322

N

Fj = ∑ Fjk = q j ∑ k =1 k≠ j

Force on the j-th charge

k =1 k≠ j

q3 F31

rˆ32

F32

Generally for N charges N

r32

r31

 

F3 = F31 + F32 =

q2

q1

qk rjk2

rˆ jk

F3 Sum over k, skipping j

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Work against Gravity “Work” is defined (in physics) by W = ∫ force × distance  

Slope and gravity from 15a:

© J.E. Hoffman

W = mg sin θ1 ⋅ L21 + h 2 = mg

h L +h 2 1

2

W = mg sin θ 2 ⋅ L22 + h 2

L21 + h 2

= mgh  

= mg

h L +h 2 2

2

L22 + h 2

= mgh

Work against gravity does not depend on the angle of the slope  Gravity is a conservative force

Conservative Force With a conservative force (such as gravity) Work is path-independent : depends on the initial and final positions, but not on the path taken between   Work is reversible : WAB = –WBA   You can “store” your work and get it back later   Energy U can be defined as a function of the positions, so that WAB = U(B) – U(A)   For gravity, U(h) = mgh  

Electrostatic force is conservative It must be  Same inverse-square law as gravity! In fact any force in the form F = f (r) rˆ is conservative   Will have some more to say in Chapter 2    

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Two Charges Bring two charges from very far away to a distance r12

q1

 

r12

q2

ds

F r=∞

r

Work is

W = ∫ F ⋅ ds =

r12

q1q2 qq rˆ ⋅ (− drrˆ ) = 1 2 2 r r12 r=∞

∫

Using r = ∞ as the reference point (U = 0) we can define the electrostatic energy of the two-charge system as

U=    

q1q2 r12

Note that it doesn’t matter which charge is moved It doesn’t matter if the path is not straight

One More Charge Bring a third charge to the system

q1

r13

r12

q3 r23

q2

   

ds

F32 F31

r=∞

W = ∫ (F31 + F32 ) ⋅ ds = ∫ F31 ⋅ ds + ∫ F32 ⋅ ds

Superposition Principle allows us to do the integration in pieces Each piece is the energy of a two-charge system, which is pathindependent

Total energy for the three-charge system is

U=

q1q2 q1q3 q2 q3 + + r12 r13 r23

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N-Charge System Electrostatic energy of an N-charge system is

q j qk 1 N U = ∑∑ 2 j =1 k ≠ j rjk    

The double sum runs j = 1 … N and k = 1 … N, but skipping j = k ½ takes care of the double-counting of, e.g. (j, k) = (1, 2) and (2, 1)

Remember: U = 0 is defined as “when all charges are very far away from each other”

Summary Electric charge = Source and recipient of electric forces  

Positive and negative; conserved; and quantized

Coulomb’s Law F2 =  

q1q2 r212

rˆ21

F1

q1

q2

r21

Inverse-square law same as gravity

Multiple charges  Superposition Principle Fj = Electrostatic force is conservative

∑F k≠ j

F2

jk

Electrostatic energy depends only on the positions of the charges   For 2-body: For N-body: N qq  

U=

q1q2 r12

U=

1 j k ∑ ∑ 2 j =1 k ≠ j rjk

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