25.6 The Electric Potential of a Point Charge

Another important electric potential is that of a point charge. FIGURE 25.25a shows charge q and a point in space at which we would like to know the electric potential. To do so, as shown in FIGURE 25.25b, we let a second charge $q'$ probe the electric potential of q. The potential energy of the two point charges is

U_{q' + q} = \frac{1}{4 π ϵ_{0}} \frac{q q'}{r}

(25.29)

Thus, by definition, the electric potential of charge q is

V_{point} = \frac{U_{q' + q}}{q'} = \frac{1}{4 π ϵ_{0}} \frac{q}{r} (electric potential of a point charge)

(25.30)

The figure consists of two parts labeled ("a") and (b) that illustrate the electric potential of charge q.

FIGURE 25.25 Full Alternative Text

The potential of Equation 25.30 extends through all of space, showing the influence of charge q, but it weakens with distance as $1 / r$ . This expression for V assumes that we have chosen $V = 0 V$ to be at $r = \infty$ . This is the most logical choice for a point charge because the influence of charge q ends at infinity.

The expression for the electric potential of charge q is similar to that for the electric field of charge q. The difference most quickly seen is that $V_{point}$ depends on $1 / r$ whereas $\vec{E}$ depends on $1 / r^{2}$ . But it is also important to notice that the potential is a scalar whereas the field is a vector. Thus the mathematics of using the potential are much easier than the vector mathematics using the electric field requires.

For example, the electric potential 1.0 cm from a $+ 1.0 nC$ charge is

V_{1 cm} = \frac{1}{4 π ϵ_{0}} \frac{q}{r} = (9.0 \times 10^{9} {N m}^{2} / C^{2}) \frac{1.0 \times 10^{- 9} C}{0.010 m} = 900 V

1 nC is typical of the electrostatic charge produced by rubbing, and you can see that such a charge creates a fairly large potential nearby. Why are we not shocked and injured when working with the “high voltages” of such charges? The sensation of being shocked is a result of current, not potential. Some high-potential sources simply do not have the ability to generate much current. We will look at this issue in Chapter 28.