   Next: Worked Examples Up: Electric Potential Previous: Electric Potential and Electric

## Electric Potential of a Point Charge

Let us calculate the electric potential generated by a point charge located at the origin. It is fairly obvious, by symmetry, and also by looking at Fig. 14, that is a function of only, where is the radial distance from the origin. Thus, without loss of generality, we can restrict our investigation to the potential generated along the positive -axis. The -component of the electric field generated along this axis takes the form (94)

Both the - and -components of the field are zero. According to Eq. (87), and are related via (95)

Thus, by integration, (96)

where is an arbitrary constant. Finally, making use of the fact that , we obtain (97)

Here, we have adopted the common convention that the potential at infinity is zero. A potential defined according to this convention is called an absolute potential.

Suppose that we have point charges distributed in space. Let the th charge be located at position vector . Since electric potential is superposable, and is also a scalar quantity, the absolute potential at position vector is simply the algebraic sum of the potentials generated by each charge taken in isolation: (98)

The work we would perform in taking a charge from infinity and slowly moving it to point is the same as the increase in electric potential energy of the charge during its journey [see Eq. (79)]. This, by definition, is equal to the product of the charge and the increase in the electric potential. This, finally, is the same as times the absolute potential at point : i.e., (99)   Next: Worked Examples Up: Electric Potential Previous: Electric Potential and Electric
Richard Fitzpatrick 2007-07-14