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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.
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 By still another theorem of vector calculus, we have the identity
x (
x A) = (
. A) - 2A
so that eq. 8 becomes
or, after some simplification 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 2
= 2
- (1/c2) 2
/t2
so that eq. 10a becomes
Using eq. 10 in eq. 9, we obtain (
. A) = - (0
/t)
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 -
2
- 0
(-
0/t)/t
= /0
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:
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