3.3 Calculations of Electric Potential
LEARNING OBJECTIVES
- Calculate the potential due to a point charge
- Calculate the potential of a system of multiple point charges
- Describe an electric dipole
- Define dipole moment
- Calculate the potential of a continuous charge distribution
Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (such as charge on a metal sphere) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider.
We can use calculus to find the work needed to move a test charge from a large distance away to a distance of from a point charge . Noting the connection between work and potential , as in the last section, we can obtain the following result.
ELECTRIC POTENTIAL V OF A POINT CHARGE
The potential at infinity is chosen to be zero. Thus, for a point charge decreases with distance, whereas for a point charge decreases with distance squared:
Recall that the electric potential is a scalar and has no direction, whereas the electric field is a vector. To find the voltage due to a combination of point charges, you add the individual voltages as numbers. To find the total electric field, you must add the individual fields as vectors, taking magnitude and direction into account. This is consistent with the fact that is closely associated with energy, a scalar, whereas is closely associated with force, a vector.
EXAMPLE 3.3.1
What Voltage Is Produced by a Small Charge on a Metal Sphere?
Charges in static electricity are typically in the nanocoulomb () to microcoulomb () range. What is the voltage away from the center of a diameter solid metal sphere that has a static charge?
Strategy
As we discussed in Electric Charges and Fields, charge on a metal sphere spreads out uniformly and produces a field like that of a point charge located at its centre. Thus, we can find the voltage using the equation .
Solution
Entering known values into the expression for the potential of a point charge, we obtain
Significance
The negative value for voltage means a positive charge would be attracted from a larger distance, since the potential is lower (more negative) than at larger distances. Conversely, a negative charge would be repelled, as expected.
EXAMPLE 3.3.2
What Is the Excess Charge on a Van de Graaff Generator?
A demonstration Van de Graaff generator has a diameter metal sphere that produces a voltage of near its surface (Figure 3.3.1). What excess charge resides on the sphere? (Assume that each numerical value here is shown with three significant figures.)
(Figure 3.3.1)
Strategy
The potential on the surface is the same as that of a point charge at the center of the sphere, away. (The radius of the sphere is .) We can thus determine the excess charge using the equation
Solution
Solving for and entering known values gives
Significance
This is a relatively small charge, but it produces