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Call A the vector magnetic potential. Let us now take eq. 2. We will find a characteristic solution E_{c} and a particular solution E_{p}. The total electric field will then be the sum E_{c} + E_{p}. a. Characteristic solution: x E_{c} = 0. By another theorem of vector calculus, this expression can be true if and only if there exists a scalar field such that E_{c} =  . 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)  ^{2}A
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 (^{2}A)
=
x (_{0}
_{0
}^{2}A/t^{2}
+ 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/c^{2} where c is the speed of light. We now introduce a new operator ^{2} defined by ^{2}
= ^{2}
 (1/c^{2}) ^{2}
/t^{2}
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/c^{2}, and using the operator ^{2} Eq. 13a is the other field equation that we sought. To summarize:


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