Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field © Kari Eloranta 2016 Jyväskylän Lyseon lukio International Baccalaureate

March 29, 2016

© Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Charge

Electric Charge Q Electric charge Q Electric charge is the physical property of matter that causes electromagnetic interaction. The types of electric charge are negative and positive. In any process, electric charge is not created or destroyed. Law of Conservation of Electric charge The total electric charge of an isolated system remains constant. © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Conductors and Insulators

Electric Conductors and Insulators Metallic materials, such as copper, aluminium, and steel, are examples of electrical conductors. Electrical conductivity in metals can be understood in terms of free electrons in a metallic substance. Non-metallic materials, such as most plastics, rubber and wood, are electric insulators. There are virtually no free electrons in non-metallic materials. Semiconductors, which are the widely used in modern electronics, have physical properties of conductors and insulators.

© Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Force

Coulomb’s Law Static charges experience an electric force. Charles Coulomb 1785: the electric force between two point charges is proportional to the product of the charges, and inversely proportional to the distance squared between them. Coulomb’s Law The electric force between two point charges q1 and q2 at vacuum is 1 q1 q2 q1 q2 F= =k 2 2 4πϵ0 r r

where ϵ0 = 8.85 × 10−12 C2 N−1 m−2 is the permittivity of free space, k = 8.99 × 109 Nm2 C−2 is the Coulomb constant, and r is the distance between the charges. Electric force may be either repulsive or attractive. Like charges repel and unlike charges attract each other. © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

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Topic 5.1 Electric Force and Field Electric Force

Coulomb’s Law in Medium The Coulomb’s Law on the previous page is valid as such in air. For other media the law has to be modified. For that we introduce a new quantity, the relative permittivity of the material. The relative permittivity of the material, ϵr , expresses how much the electric force between the two interacting charges is reduced in the medium. Coulomb’s Law The electric force between two point charges q1 and q2 in a medium is 1 q1 q2 F= 4πϵrϵ0 r 2

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where ϵr is the relative permittivity of the medium, ϵ0 = 8.85 × 10−12 C2 N−1 m−2 is the permittivity of free space, and r is the distance between the charges. © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Field

Electric Field Electric charge creates an electric field around a charged object. The simplest type of a charged object is a point charge, which is very small electrically charged object. In a field line representation, the density of the lines represents the field strength, and arrowheads the direction. 3

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Figure: A positive point charge creates an outward radial electric field. © Kari Eloranta 2016

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Figure: A negative point charge creates an inward radial electric field. Topic 5.1 Electric Force and Field

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Topic 5.1 Electric Force and Field Electric Field

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Electric Field Strength E

When we place a small positive test charge q to a point in an electric field, the field exerts an electric force on the charge in the direction of the field. The electric field strength is a quantity that gives the magnitude and direction of the electric field at a certain point in the field. Electric Field Strength The electric field strength is the electric force per small positive unit charge. Electric Field Strength The electric field strength is #» #»

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F E= q

where F is the electric force on a small positive test charge q at a point in the electric field. © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

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Topic 5.1 Electric Force and Field Electric Field

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Electric Field Strength E (cont.) The SI derived unit of the electric field strength is NC−1 (newton per coulomb)(or V m−1 (volt per metre) as we study later). Electric field strength is a vector quantity. Direction of Electric Field The direction of the electric field at a point is the direction of the electric force on a small positive test charge at that point. Charge q should be as small as possible, so that it exerts essentially no force on the charges which created the field. That way the test charge does not change the original charge distribution that created the field.

© Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Field

Electric Field Strength of Point Charge Electric Field Strength of Point Charge The magnitude of the electric field strength at a distance r from a point charge q is q qt F k r2 q E= = =k 2 qt qt r

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where k = 8.99 × 109 Nm2 C−2 is the Coulomb constant, and qt a small positive test charge. A point charge creates a conservative radial field around itself. In principle, the field extends to infinity, but decreases rapidly as distance to the charge increases.

© Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Field

Electric Potential Difference An electric field is a conservative field: the work done by the electric field on the charge, as the charge moves from point A to point B in the field, depends only on the points A and B, not on the path along which the charge travelled. As the result, we can attach electric potential to all points in an electric field. Only differences in electric potential have physical meaning, not the value of a potential on a point which may be artificially normalised. Electric Potential Difference V When charge q travels from point A to point B in an electric field, the electric potential difference between the points is W V= q

where W is the work done by the electric field on the charge. © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

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Topic 5.1 Electric Force and Field Electric Field

Electric Potential

Electric potential is a scalar quantity. The unit of electric potential is [E ]/[Q] = JC−1 = 1V (joule/coulomb=volt), according to italian physicist Alessandro Volta (1745 – 1827). An alternative unit for electric potential is the electronvolt: 1eV = 1.60 × 10−19 J

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which equals the kinetic energy of an electron when the electron is accelerated by an electric potential difference of 1 V.

© Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Battery and Electric Potential Difference

Electric Potential Difference Across Battery Terminals A battery is an electrical device that converts chemical energy into electric energy. A battery has two metallic connectors called terminals. There is an electric potential difference between the terminals. The terminals are called positive and negative based on their electrical properties. The positive (+) terminal is at higher electric potential with respect to the negative (−) terminal. For example, in a 9 V battery, the electric potential difference across the unplugged terminals is 9 V. © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Battery and Electric Potential Difference

Electric Potential Difference V and Electric Current I When the terminals of a battery are connected to an electric circuit, an electric field is established in the wires, and other circuit elements of the circuit (resistors, lamps, ...). The field exerts an electric force on the free electrons, while the interactions between the electrons and the metallic lattice oppose to motion. As a result, a slow steady electric current flows in the wire. The free electrons move slowly towards the positive terminal, while electrons move from the negative terminal into the circuit. The electric current flows through the battery and the circuit, because the battery sustains an electric potential difference across the terminals. A battery produces a direct current (DC) in the circuit. An alternating voltage source, such as a wall outlet, produces an alternating electric current (AC). © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Current

Electric Current I Electric Current I Electric current is the time rate of change of electric charge flowing through a cross-sectional area in a conductor. Electric Current I For a steady direct current, the electric current is defined as ∆q I= ∆t

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where ∆q is the electric charge that flows trough a cross-sectional area in a conductor in time ∆t . The unit of electric current in SI system is [q] C = = Cs−1 = 1A (ampere). (8) [I ] = [t ] s By definition, the direction of electric current is from the higher potential to the lower potential (the direction of movement of a positive charge in an electric field). © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Current

Drift Speed There is a vast number of free electrons in a metal. For example, copper has about 8 × 1022 free electrons per cubic centimetre. As a potential difference is applied across a wire, an electric field is established in the wire. The free electrons start moving slowly against the field from the lower potential towards the higher potential. The speed of the free electrons due to electric field is called the drift speed. Next we derive an equation for the electric current I as the function of the drift speed v . Suppose that there are n charge carries (electrons) per unit volume in a wire of cross-sectional area A . Then, in time ∆t , the total charge ∆Q flowing through a cross-sectional area is ∆Q = nV q = n Av∆t q . The electric current in the wire is thus ∆Q n Av∆t q I= = = n Av q. ∆t ∆t © Kari Eloranta 2016

Topic 5.1 Electric Force and Field

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Topic 5.1 Electric Force and Field Electric Current

Electric Current I as a Function of Drift Speed v Electric Current I as a Function of Drift Speed v If the drift speed of the charge carries (charge q ) is v , the electric current I flowing through a cross-sectional area A of a wire is I = n Av q

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where n is the number of charge carries per unit volume (charge density) in the wire. Thermal speed of electrons is tens of thousands of metres per second. Electric field disturbances travel by the speed of light. A typical drift speed is of the order of magnitude of a few fractions of millimetres per second! That is really slooooooooow.

© Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Current

Example on Electric current Example A student used an ammeter to measure the electric current of 178 mA in a metallic wire. How many electrons passes through a cross sectional area in the wire each second?

© Kari Eloranta 2016

Topic 5.1 Electric Force and Field

Topic 5.1 Electric Force and Field Electric Current

Example on Electric current Example A student used an ammeter to measure the electric current of 178 mA in a metallic wire. How many electrons passes through a cross sectional area in the wire each second? Answer. The electric current in the wire is ∆q I= ∆t

from where it follows that the electric charge going through a cross sectional area in the wire in time ∆t is ∆q = I ∆t .

Since the charge of an electron is the negative elementary charge q e = −e = −1.602 × 10−19 C, the number of electrons passing through a cross sectional area in a wire each second is ∆q I ∆t 178 × 10−3 A × 1s 18 N= = = ≈ 1 . 11 × 10 . − 19 |q e | |q e | 1.602 × 10 C © Kari Eloranta 2016

Topic 5.1 Electric Force and Field