Introduction
There is a scalar quantity
, called the electric potential, which satisfies
where
is the electric field.
It is often easier to handle scalar fields rather than vector fields. It is therefore convenient to work with
and then derive
from it.
Problem in words
Given the electric potential, find the electric field.
Mathematical statement of problem
For a point charge,
, the potential
is given by
Verify, using spherical polar coordinates, that
is indeed
Mathematical analysis
In spherical polar coordinates:
Interpretation
So
as required.
This is a form of Coulomb’s Law. A positive charge will experience a positive repulsion radially
outwards
in the field of another positive charge.
Using spherical polar coordinates, find
for the following vector functions.
-
-
-
-
Note :- in Cartesian coordinates, the corresponding vector is
with
(hence consistency).
-
-
Find
for the following vector fields
.
-
, where
is a constant
-
-
-
Using spherical polar coordinates, find
for
-
-
-
-
,
-
,
-
-
For
, find
and
.
-
For
, find
and
.
-
For
find
.
-
,
-
,
-