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## 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:

 x H = 0 E/t + j 1
 x E = - 0 H/t 2
 . H = 0 3
 . E = /0 4

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:

 E = Ec + Ep = - - 0 A/t 6
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

 . (- - 0 A/t) = /0 7

and from eq. 1, we obtain

 x ( x A) = 0 (- - 0 A/t)/t + j 8

By still another theorem of vector calculus, we have the identity

x ( x A) = ( . A) - 2A

so that eq. 8 becomes

 ( . A) - 2A = 0 (- - 0 A/t)/t + j 8a.

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

 2A = 0 0 2A/t2 + j 10
 or 2A - 0 0 2A/t2 = j 10a.

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

2 = 2 - (1/c2) 2 /t2

so that eq. 10a becomes

 2 A = j 11

Using eq. 10 in eq. 9, we obtain

( . A) = - (0 /t)

from which we may infer, again without loss of generality, that

 . A = - 0 /t 12

We must next rewrite eq. 7 by distributing the operator .

 . (- - 0 A/t) = /0 7
 becomes - 2 - 0 ( .A)/t = /0 7a.

Substituting eq. 12 into eq. 7a gives

- 2 - 0 (- 0/t)/t = /0

which may be simplified:

 2 - 0 0 2/t2 = - /0 13

or, recalling that 0 0 = 1/c2, and using the operator 2

 2 = - /0 13a.

Eq. 13a is the other field equation that we sought.

To summarize:

1. 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
1. 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.

2. Dimensional analysis on the above equations shows that A has units of electric current (amperes) and has units of electric potential (volts).