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Recall that earlier we defined electric field to be a quantity independent of the test charge in a given system, which would nonetheless allow us to calculate the force that would result on an arbitrary test charge. (The default assumption in the absence of other information is that the test charge is positive.) We briefly defined a field for gravity, but gravity is always attractive, whereas the electric force can be either attractive or repulsive. Therefore, although potential energy is perfectly adequate in a gravitational system, it is convenient to define a quantity that allows us to calculate the work on a charge independent of the magnitude of the charge. Calculating the work directly may be difficult, since $W=\overrightarrow{\text{F}}\xb7\overrightarrow{\text{d}}$ and the direction and magnitude of $\overrightarrow{\text{F}}$ can be complex for multiple charges, for odd-shaped objects, and along arbitrary paths. But we do know that because $\overrightarrow{\text{F}}=q\overrightarrow{\text{E}}$ , the work, and hence $\text{\Delta}U,$ is proportional to the test charge q . To have a physical quantity that is independent of test charge, we define electric potential V (or simply potential, since electric is understood) to be the potential energy per unit charge:
The electric potential energy per unit charge is
Since U is proportional to q , the dependence on q cancels. Thus, V does not depend on q . The change in potential energy $\text{\Delta}U$ is crucial, so we are concerned with the difference in potential or potential difference $\text{\Delta}V$ between two points, where
The electric potential difference between points A and B , ${V}_{B}-{V}_{A},$ is defined to be the change in potential energy of a charge 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 .
The familiar term voltage is the common name for electric 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. It is worthwhile to emphasize the distinction between potential difference and electrical potential energy.
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 because $\text{\Delta}U=q\text{\Delta}V.$ The car battery can move more charge than the motorcycle battery, although both are 12-V batteries.
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