Begin2.DVI

Example 9-14. The last Maxwell equation is hard to derive. Historically,

Ampere^14 showed that for straight line currents the curl of the magnetic field was

proportional to the volume current density J
so that one could write

∇× B
 =μ 0 J

Maxwell realized that this equation did not hold in general because it did not satisfy

the property that the divergence of the curl must be zero. Based upon theoretical

reasoning Maxwell came up with the modified equation

∇× B
 =μ 0 J
+μ 0

0

∂E


∂t (9 .160)

where the term μ 0

0 ∂

E


∂t

is known as Maxwell’s term for Ampere’s law. Taking the

divergence of equation (9.160) one can show

∇·(∇× B
) = μ 0 ∇·J
+μ 0

0

∂∇·E

∂t Use the first Maxwell equation and show

∇·(∇× B
) = μ 0 ∇·J
+μ 0

0 ∂(ρ/

^0 )

∂t

∇·(∇× B
) = μ 0

[

∇·J
+∂ρ

∂t

]

= 0

where the continuity equation from the previous example has been employed to show

the divergence of the curl is zero. The equation (9.160) is the last of the Maxwell

equations from (9.134).

Example 9-15. If there are no charges or currents in space, then the Maxwell

equations (9.134) simplify to the form

∇·E
 = 0

∇× E
=−

∂B


∂t

∇·B
 = 0

∇× B
 =μ 0

0

∂E


∂t

(9 .161)

Use the property of the del operator that

∇× (∇× A
) = ∇(∇·A
)−∇^2 A

(^14) Andr ́e Marie Amp`ere (1775-1836) A French physicist, chemist and mathematician.