The Vector and Scalar Potentials
to the student:
This section is reserved for advanced students, with background
in electricity and magnetism, and vector differential equations.
Given Maxwell's four equations, demonstrate the existence of a vector
magnetic potential and a scalar electric potential. Derive field
equations for these potentials.
Maxwell's four equations read:
. H = 0
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,
x 10-7 H/m is the permeability of free space, and
is the scalar charge density (C/m3).
Of The Vector And Scalar Potentials
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
A the vector magnetic potential.
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.
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 = - .
the scalar electric potential.
x Ep = - 0
and H =
x A allows us to write
x Ep = - 0
x (- 0
may obtain a particular solution Ep by simply choosing
Ep = - 0
Between The Vector And Scalar Potentials
us now substitute the expressions derived above into eqs.
4 and 1. From eq. 4, we
from eq. 1, we obtain
x A) = 0
still another theorem of vector calculus, we have the identity
that eq. 8 becomes
after some simplification
. A) - 2A
= - (0
+ j 9.
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
which we may infer, without loss of generality, that
10a is one of the field equations we sought. We may simplify
eq. 10a somewhat if we recognize that 0
= 1/c2 where c is the speed of light. We now introduce
a new operator 2
that eq. 10a becomes
A = j
eq. 10 in eq. 9, we obtain
which we may infer, again without loss of generality, that
must next rewrite eq. 7 by distributing the operator
eq. 12 into eq. 7a gives
may be simplified:
recalling that 0
= 1/c2, and using the operator 2
13a is the other field equation that we sought.
- 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
= - /0
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
analysis on the above equations shows that A has units of electric
current (amperes) and
has units of electric potential (volts).