Begin2.DVI

~r′=x′ˆe 1 +y′ˆe 2 +z′ˆe 3 to the point(x′, y′, z′)inside the volumeV. The vector~r−~r′

then points from the point(x′, y′, z′)to the point(x, y, z). The vector ˆe= ~r−~r

′
|~r−~r′| is a
unit vector in this direction as illustrated in the accompanying figure.

The magnetic fieldB~ =B~(x, y, z)at the point(x, y, z)due to a current density

J~=J~(x′, y′, z′)insideV is given by the Biot^12 -Savart^13 law

B~=B~(x, y, z) =μ^0

4 π

∫ ∫ ∫

V

∇·

[

J~(x′, y′, z′)×(~r−~r′)

|~r−~r′|^3

]

dx′dy′dz′ (9.152)

The divergence of this magnetic field is determined by calculating

∇·B~=μ 4 π^0

∫ ∫ ∫

V

∇·

[

J~×(~r−~r′)

|~r−~r′|^3

]

dV (9.153)

In order to evaluate the divergence as given by equation (9.153) one can employ the

vector identities
∇×(fA~) =(∇f)×A~+f(∇×A~)
∇·(A~×B~) =B~·(∇×A~)−A~·(∇×B~)

(9.154)

LetB~ = ~r−~r

′

|~r−~r′|^3

andA~=J~along with the second of the equations (9.154) to

show

∇·(J~×B~) =B~·(∇×J~)−J~·(∇×B~) =−J~·(∇×B~) (9.155)

This holds becauseJ~is a function of the primed coordinates and∇involves differ-

entiation with respect to the unprimed coordinates so that∇×J~is zero. Using the

first equation in (9.154) withf=^1
|~r−~r′|^3

one can write

∇×B~=∇×(~r−~r

′)

|~r−~r′|^3

=^1

|~r−~r′|^3

∇×(~r−~r′)−(~r−~r′)×∇

(

1

|~r−~r′|^3

)

(9.156)

One can verify that

∇×(~r−~r′) =

∣∣

∣∣

∣∣

ˆe 1 ˆe 2 ˆe 3

∂

∂x

∂

∂y

∂

∂z

(x−x′) (y−y′) (z−z′)

∣∣

∣∣

∣∣=~^0

and iff=|~r−~r′|−^3 , then∇f=∂f∂xˆe 1 +∂f∂yˆe 2 +∂f∂zˆe 3 where

∂f

∂x=−^3 |~r−~r

′|− 4 ∂

∂x

√

(x−x′)^2 + (y−y′)^2 + (z−z′)^2 =

−3(x−x′)

|~r−~r′|^5

(^12) Jean Baptiste Biot (1774-1862) A French mathematician.
(^13) F elix Savart (1791-1841) A French physician who studied physics.