Potential from a Point Charge

The expression for electric potential due to a point charge is:

(10)

where K is a physical constant which depends on the system of units we use, Q is the charge which creates the field, and r is the distance between the charge and the point of observation. In the SI system,

where is the electrical permitivity of free space. Its value is 8.85 x 10-12 C2 / N m2. It is a physical constant meaning its value never changes. It can also be helpful to remember that K is equal to 8.99 x 109 N m2 / C2.

The electric field is then

(11)

It can be seen from eq. 10 and 11 (above) that:

  1. The equipotential surfaces are a series of concentric spherical surfaces, and
  2. The electric field is directed along the radial vector r, and is perpendicular to the equipotential surfaces.

If we project the equipotential surfaces onto the xy-plane, they become equipotential lines. The electric field vectors still point radially outward, perpendicular to these equipotential lines. This is shown in the following figure:


If you were to plot this potential function in three-dimensional space such that the potential V(x,y) were plotted on the z-axis, what would be the resulting shape?

Answer


The electric potential produced by a more complex charge distribution can be obtained by adding or integrating the point charge potential given by (10). Thus,

(12)

The first term gives the contribution from a group of discrete point charges qi (i = 1, 2, ..., n) located at a radial distance ri, and the second term gives the contribution from a continuous charge distribution. The integral is to be taken over the entire such charge distribution.

It is important to note that the electric potential given in Equations (10) and (12) is scalar. Therefore, it is generally much easier to calculate the electric potential and from that to obtain the electric field, E, using (6).

Now go on to get an intuitive feel for a single point charge, using our charge applet.


Copyright 1998-2000 Rensselaer Polytechnic Institute. All Rights Reserved.