Maxwell's
Equations:
The Vector and Scalar Potentials
Note
to the student:
This section is reserved for advanced students, with background in electricity
and magnetism, and vector differential equations.
Problem:
Given Maxwell's four equations, demonstrate the existence of a vector
magnetic potential and a scalar electric potential. Derive field equations
for these potentials.
Solution:
Maxwell's four equations read:
. H = 0 |
|
where
H is the magnetic field (A/m), E is the electric field (V/m), j is the
vector current density (A/m2), 0
= 8.8542 x 10-12 F/m is the permittivity of free space, 0
= 4
x 10-7 H/m is the permeability of free space, and
is the scalar charge density (C/m3).
Existence
Of The Vector And Scalar Potentials
Let
us begin with eq. 3. By a theorem of vector calculus,
.
H = 0 if and only if there exists a vector field A such that
H
=
x A |
5. |
Call
A the vector magnetic potential.
Let
us now take eq. 2. We will find a characteristic solution
Ec and a particular solution Ep. The total electric
field will then be the sum Ec + Ep.
a.
Characteristic
solution:
x Ec = 0. By another theorem of vector calculus, this expression
can be true if and only if there exists a scalar field
such that Ec = - .
Call
the scalar electric potential.
b.
Particular
solution:
x Ep = - 0
H/t,
and H =
x A allows us to write
x Ep = - 0
(
x A)/t
=
x (- 0
A/t)
We
may obtain a particular solution Ep by simply choosing
Ep = - 0
A/t.
c.
Total
electric field:
Relationships
Between The Vector And Scalar Potentials
Let
us now substitute the expressions derived above into eqs.
4 and 1. From eq. 4, we obtain
and
from eq. 1, we obtain
x (
x A) = 0
(-
- 0
A/t)/t
+ j |
|
By
still another theorem of vector calculus, we have the identity
so
that eq. 8 becomes
or,
after some simplification
(
. A) - 2A
= - (0
/t)
- 0
0
2A/t2
+ j 9.
Since
x grad = 0, eq. 9 may be manipulated by taking the
curl of both sides. The terms involving the gradient then vanish leaving
us with the identity
x (2A)
=
x (0
0
2A/t2
+ j)
from
which we may infer, without loss of generality, that
Eq.
10a is one of the field equations we sought. We may simplify eq.
10a somewhat if we recognize that 0
0
= 1/c2 where c is the speed of light. We now introduce a
new operator 2
defined by
so
that eq. 10a becomes
2
A = j |
11. |
Using
eq. 10 in eq. 9, we obtain
from
which we may infer, again without loss of generality, that
We
must next rewrite eq. 7 by distributing the operator
.
Substituting
eq. 12 into eq. 7a gives
which
may be simplified:
or,
recalling that 0
0
= 1/c2, and using the operator 2
Eq.
13a is the other field equation that we sought.
To
summarize:
-
The
existence of a vector magnetic potential A and a scalar electric
potential
was demonstrated. The respective field equations for A and
were found to be 2
A = j
and 2
= - /0
- where
j is the vector current density (A/m2),
is the scalar charge potential (C/m3), and 0
= 8.8542 x 10-12 F/m is the permittivity of free space.
-
Dimensional
analysis on the above equations shows that A has units of electric
current (amperes) and
has units of electric potential (volts).
|