ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 48

Substituting (2-86) into (2-85) gives

1
R

µ 0
4 π

B =

C − Idl ×∇

(2-87)

The del operator ( ) does not operate on source variables and therefore does not

affect Idl . Therefore, it is acceptable to rewrite (2-87) in the form of a curl:

∇

l
R

µ 0 I
4 π

d

B =∇×

(2-88)

C

Also note that the negative sign has been eliminated because it indicates current
direction, which is comprehended by the vector Idl . Comparing (2-88) to the
definition of the magnetic vector potential in (2-83), we can deduce the form
of

A :

l
R

µ 0 I
4 π

d

A =

(2-89)

C

2.5.2 Inductance

Suppose that two current loops are in close proximity to each other as shown
in Figure 2-19. If a steady-state current ( I 1 ) is flowing in loop 1, it produces
a magnetic field B 1 , as predicted by (2-85). If some of the magnetic field lines
pass though loop 2, the flux passing through loop 2 is calculated using the form
of equation (2-17):

B 1

= · s 2

ψ 2 d (2-90)

Note that the flux in loop 2 is proportional to B 1 and is therefore also proportional
to I 1 . This allows us to define a constant of proportionality, more commonly
known as the mutual inductance :

ψ 2
I 1

L 21 ≡ (2-91)

Substituting the (2-83) into (2-90) gives the flux passing through loop 2 in terms
of the magnetic vector potential, where s 2 is the surface enclosed by loop 2:

ψ 2 = ( ∇× A 1 ) · d s 2 (2-92)

A [Jackson,

Now we can simplify using Stokes' theorem and substitute (2-89) for

1999]:

µ 0 I 1
4 π

dl
R

l 1
R

d µ 0 I 1

4 π

dl 2 dl 2 ψ 2 = ( ∇× A 1 ) · d s 2 = · = ·

C

(2-93)