The Imaging of a Magnetic Source - Biological Imaging and Sensing - page 150

S 1

G 1

σ 1

G 2

S 3

σ 2

G 3

S 2

σ 3

Fig. 3.27. An inhomogeneous conductor

where means the gradient operator of the scalar potential. Using this equa-

tion, (3.59) is transformed into

∇

J = J i − σ ∇ V. (3.62)

If G is divided by surface S j ,j =1 ,,n , into subregions G j ,j =1 ,,n ,sothat
σ = σ j in each G j (see Fig. 3.27), (3.60) is transformed into

r

( r )= μ 0
4 π

r −

J i ( r ) r ) r )]

3 d v

B [ − σ ( ∇ V ( ×

| r − r | G

G j ∇

n

r − r
| r −

μ 0
4 π

r )

3 d v ,

= B 0 ( r ) − σ j V ( × (3.63)

r |

j =1

where

μ 0
4 π

r − r

r )

3 d v .

B 0 = J i ( × (3.64)

r |

| r − G

B 0 is the magnetic field caused by the applied current only. We can derive
the identity

∇ V ×∇ g = ∇× ( V ∇ g ) , (3.65)

with

r
| r − r |

r −

∇ g =

3 .

(3.66)

Using the above identity and Stokes' theorem,
G ∇× A

d v =

∂G n ×

A d s, (3.67)