Page 235 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 217

7.7 DERIVATION OF THE
STEADY-MAGNETIC-FIELD LAWS
We will now supply the promised proofs of the several relationships between the
magnetic field quantities. All these relationships may be obtained from the definitions
of H,

IdL × a R
H = (3)
4πR 2
of B (in free space),
B = µ 0 H (32)
and of A,
B =∇ × A (46)

Let us first assume that we may express A by the last equation of Section 7.6,

µ 0 J dν

A = (51)
vol 4πR
and then demonstrate the correctness of (51) by showing that (3) follows. First, we
should add subscripts to indicate the point at which the current element is located
(x 1 , y 1 , z 1 ) and the point at which A is given (x 2 , y 2 , z 2 ). The differential volume
element dν is then written dν 1 and in rectangular coordinates would be dx 1 dy 1 dz 1 .
The variables of integration are x 1 , y 1 , and z 1 . Using these subscripts, then,

µ 0 J 1 dν 1
A 2 = (52)
vol 4πR 12
From (32) and (46) we have
B ∇× A
H = = (53)
µ 0 µ 0
To show that (3) follows from (52), it is necessary to substitute (52) into (53). This
step involves taking the curl of A 2 ,a quantity expressed in terms of the variables x 2 ,
y 2 , and z 2 , and the curl therefore involves partial derivatives with respect to x 2 , y 2 , and
z 2 .Wedo this, placing a subscript on the del operator to remind us of the variables
involved in the partial differentiation process,

∇ 2 × A 2 1 µ 0 J 1 dν 1
H 2 = = ∇ 2 ×
µ 0 µ 0 vol 4πR 12
The order of partial differentiation and integration is immaterial, and µ 0 /4π is
constant, allowing us to write
1 J 1 dν 1
H 2 = ∇ 2 ×
4π vol R 12
The curl operation within the integrand represents partial differentiation with
respect to x 2 , y 2 , and z 2 . The differential volume element dν 1 is a scalar and a function