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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
2007-07-14