Electricity and Magnetism

Electric Energy and Capacitance

Electric potential

As stated earlier in our discussion of simple harmonic oscillator, there are two equivalent ways to describe an interaction. We can describe it either in terms of a force, or equivalently in terms of the corresponding potential energy. This is true for electric interactions as well.

Electric field (a vector) is a quantity that corresponds to the force-description of electric interaction. Basically, it is the force that would have been experienced by a unit test charge at a point in space:

E = F/q

Electric potential corresponds to the description of electric interaction in terms of a potential energy. Specifically, the potential energy of a charged particle at a point in space is related to the electric potential V at that point by

PE = qV

Electric potential is a scalar, as opposed to the electric field which is a vector.

The change of potential energy is thus related to the difference in electric potential by

DPE = q DV

The unit of electric potential is volt:

1 V = 1 J/C

Electric potential drops along the direction of the electric field.

The potential difference between two points in a constant electric field is

VB - VA = -Ed

where d is the distance between point A and point B along the direction of the electric field. This can be derived from the fact that the change in potential energy is equal to the negative of the work done by the electric field, i.e.

q(VB - VA)= Fd = -qEd

Example 16.1 The field between two parallel plates of opposite charge

The electric field near the middle is uniform.

Example 16.2 Motion of a proton in a uniform electric field

Conservation of energy.

Electron Volt (a unit of energy commonly used in atomic and nuclear physics)

1 eV = 1.6021773310-19 CV = 1.6021773310-19 J

Electric potential due to point charges

Electric potential due to a point charge q is given by

V = kq/r

where k the Coulomb constant and r is distance from the charge.

Electric potential due to multiple charges is an algebraic sum of the potentials due to the individual charges.

Example 16.3 Finding the electric potential

Equipotential surfaces and charged conductors

Equipotential surfaces: surfaces of equal potential

Electric field at a point in space is always perpendicular to the equipotential surface passing through that point.

Electric potential is the same everywhere inside a conductor and on its surface.

This is because the electric field is zero inside a conductor.




C = Q/V

is a measure of the capacity to store charge. The greater the capacitance, the greater amount of charge can be stored for the same applied voltage.

Capacitance is measured in units of farad (F)

1 F = 1 C/V

The Parallel-Plate Capacitor

C = e0(A/d)


and e0 = 8.8510-12 C2 / Nm2 is a constant called the permittivity of free space. It is related to the Coulomb constant by

k = 1/(4pe0)

It is important to note that capacitance depends only on the geometric factors A and d (and in fact the material we put between the plates, see Section 16.9). It is independent of the amount of charge we put on the plates.

Combinations of Capacitors

Parallel combination

Ceq = C1 + C2 + C3 + ...


Q = Q1 + Q2 + Q3 + ...

and V is the same for every capacitor.

Series combination

1/Ceq = 1/C1 + 1/C2 + 1/C3 + ...


V = V1 + V2 + V3 + ...

and Q is the same for every capacitor.

Energy in a charged capacitor

A charged capacitor contains an amount of energy:

E = (1/2)QV = (1/2)CV2 = Q2/2C

This relationship can be derived by considering the amount of work necessary to create the configuration of +Q and -Q separated by a potential V.

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Copyright 1997 by Bo Gao. All rights reserved.