19.1 Electric Potential Energy: Potential Difference

When a free positive charge qq is accelerated by an electric field, such as shown in Figure 19.2, it is given kinetic energy. The process is analogous to an object being accelerated by a gravitational field. It is as if the charge is going down an electrical hill where its electric potential energy is converted to kinetic energy. Let us explore the work done on a charge qq by the electric field in this process, so that we may develop a definition of electric potential energy.

The electrostatic or Coulomb force is conservative, which means that the work done on qq is independent of the path taken. This is exactly analogous to the gravitational force in the absence of dissipative forces such as friction. When a force is conservative, it is possible to define a potential energy associated with the force, and it is usually easier to deal with the potential energy (because it depends only on position) than to calculate the work directly.

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We use the letters PE to denote electric potential energy, which has units of joules (J). The change in potential energy, ΔPEΔPE, is crucial, since the work done by a conservative force is the negative of the change in potential energy; that is, W =-ΔPEW =-ΔPE. For example, work W W done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative ΔPEΔPE. There must be a minus sign in front of ΔPEΔPE to make W W positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point.

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Gravitational potential energy and electric potential energy are quite analogous. Potential energy accounts for work done by a conservative force and gives added insight regarding energy and energy transformation without the necessity of dealing with the force directly. It is much more common, for example, to use the concept of voltage (related to electric potential energy) than to deal with the Coulomb force directly.

Calculating the work directly is generally difficult, since W = FdcosθW = Fdcosθ and the direction and magnitude of F F can be complex for multiple charges, for odd-shaped objects, and along arbitrary paths. But we do know that, since F =qEF =qE, the work, and hence ΔPEΔPE, is proportional to the test charge q.q. To have a physical quantity that is independent of test charge, we define electric potential VV (or simply potential, since electric is understood) to be the potential energy per unit charge:

Since PE is proportional to qq , the dependence on qq cancels. Thus VV does not depend on qq. The change in potential energy ΔPEΔPE is crucial, and so we are concerned with the difference in potential or potential difference ΔVΔV between two points, where

The potential difference between points A and B, VB-VAVB-VA, is thus defined to be the change in potential energy of a charge q q moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta.

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The familiar term voltage is the common name for potential difference. Keep in mind that whenever a voltage is quoted, it is understood to be the potential difference between two points. For example, every battery has two terminals, and its voltage is the potential difference between them. More fundamentally, the point you choose to be zero volts is arbitrary. This is analogous to the fact that gravitational potential energy has an arbitrary zero, such as sea level or perhaps a lecture hall floor.

In summary, the relationship between potential difference (or voltage) and electrical potential energy is given by

Voltage is not the same as energy. Voltage is the energy per unit charge. Thus a motorcycle battery and a car battery can both have the same voltage (more precisely, the same potential difference between battery terminals), yet one stores much more energy than the other since ΔPE =qΔVΔPE =qΔV. The car battery can move more charge than the motorcycle battery, although both are 12 V batteries.

Note that the energies calculated in the previous example are absolute values. The change in potential energy for the battery is negative, since it loses energy. These batteries, like many electrical systems, actually move negative charge—electrons in particular. The batteries repel electrons from their negative terminals (A) through whatever circuitry is involved and attract them to their positive terminals (B) as shown in Figure 19.3. The change in potential is ΔV=VB-VA=+12 VΔV=VB-VA=+12 V and the charge qq is negative, so that ΔPE=qΔVΔPE=qΔV is negative, meaning the potential energy of the battery has decreased when qq has moved from A to B.

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