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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
20070714