Work is change in potential energy: U B − U A = q′ Ed.
In general, the electrostatic potential difference, sometimes called the electric potential difference, is defined as the energy change per unit positive charge, or V B − V A = ( U B − U A )/ q′. For certain configurations of electric field, it may be necessary to use the integral definition of electrostatic potential:
where a test charge moves over a line integral from point A to point B along path s in an electric field ( E).
For the special case of parallel plates:
where V is the potential difference between the plates, measured in units of volts (V):
The electric potential due to a point charge (q) at a distance (r) from the point charge is
The following problem illustrates the calculations of electric field and potential due to point charges.
Example 3: Given two charges of +3 Q and – Q, a distance X apart, find the following: (1) At what point(s) along the line is the electric field zero? (2) At what point(s) is the electric potential zero? (See Figure 11.)
The arrangement of two point charges for the example.
The first task is to find the region(s) where the electric field is zero. The electric field is a vector, and its direction can be located by a test charge. Figure is divided into three regions. Between the opposite charges, the direction of the force on the test charge will be in the same direction from each charge; therefore, it is impossible to have a zero electric field in Region II. Even though the forces on the test charge from the two charges are in opposite directions in Region I, the force, and therefore the electric field, can never be zero in this region because the test charge is always closer to the largest given charge. Therefore, Region III is the only place where E can be zero. Select an arbitrary point (r) to the right of – Q and set the two electric fields equal. Because the fields are in opposite directions, the vector sum at this point will equal zero.
If X is given, solve for r.
Potential is not a vector, so the potential is zero wherever the following equation holds:
where r m is the distance from the test point to +3 Q and r 2 is the distance to– Q.
This example illustrates the difference in methods of analysis in finding the vector quantity ( E) and the scalar quantity (V). Note that if the charges were either both positive or both negative, it would be possible to find a point with zero electric field between the charges, but the potential would never be zero.
The electrical potential energy of a pair of point charges separated by a distance r is
Equipotential surfaces are surfaces where no work is required to move a charge from one point to another. The equipotential surfaces are always perpendicular to the electric field lines. Equipotential lines are two‐dimensional representations of the intersection of the surface with the plane of the diagram. In Figure , equipotential lines are shown for (a) a uniform field, (b) a point charge, and (c) two opposite charges.