Page 237 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 219

We now need the expansion in rectangular coordinates for ∇× ∇× A. Performing
the indicated partial differentiations and collecting the resulting terms, we may write
the result as

2
∇× ∇× A ≡∇(∇ · A) −∇ A (58)
where
2 2 2 2 (59)
∇ A ≡∇ A x a x +∇ A y a y +∇ A z a z
Equation (59) is the definition (in rectangular coordinates) of the Laplacian of a
vector.
Substituting (58) into (57), we have
1
2
∇× H = [∇(∇ · A) −∇ A] (60)
µ 0
and now require expressions for the divergence and the Laplacian of A.
We may find the divergence of A by applying the divergence operation to (52),

µ 0 J 1
∇ 2 · A 2 = ∇ 2 · dν 1 (61)
4π vol R 12
and using the vector identity (44) of Section 4.8,
∇ · (SV) ≡ V · (∇S) + S(∇ · V)
Thus,

µ 0 1 1
∇ 2 · A 2 = J 1 · ∇ 2 + (∇ 2 · J 1 ) dν 1 (62)
4π vol R 12 R 12
The second part of the integrand is zero because J 1 is not a function of x 2 , y 2 ,
and z 2 .
3
We have already used the result that ∇ 2 (1/R 12 ) =−R 12 /R , and it is just as
12
easily shown that
1 R 12
∇ 1 = 3
R 12 R 12
or that
1 1
∇ 1 =−∇ 2
R 12 R 12
Equation (62) can therefore be written as

µ 0 1
∇ 2 · A 2 = −J 1 · ∇ 1 dν 1
4π vol R 12
and the vector identity applied again,

µ 0 1 J 1
∇ 2 · A 2 = (∇ 1 · J 1 ) −∇ 1 · dν 1 (63)
4π vol R 12 R 12