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materials called paramagnets and negative values for materials called diamagnets. For a linear medium the
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quantities B, M and H are related by
(2.6.69)
B i = µ 0 (H i + M i )= µ 0 H i + µ 0 χ m H i = µ 0 (1 + χ m )H i = µ 0 k m H i = µH i
where µ = µ 0 k m = µ 0 (1 + χ m ) is called the permeability of the material.
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Note: The auxiliary magnetic vector H for magnetostatics in materials plays a role similar to the
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displacement vector D for electrostatics in materials. Be careful in using electromagnetic equations from
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different texts as many authors interchange the roles of B and H. Some authors call H the magnetic field.
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However, the quantity B should be the fundamental quantity. 1
Electrodynamics
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In the nonstatic case of electrodynamics there is an additional quantity J p = ∂ ~ P called the polarization
∂t
current which satisfies
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∂P ∂ ∂ρ b
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∇· J p = ∇· = ∇· P = − (2.6.70)
∂t ∂t ∂t
and the current density has three parts
∂P ~
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J = J b + J f + J p = ∇× M + J f + (2.6.71)
∂t
consisting of bound, free and polarization currents.
Faraday’s law states that a changing magnetic field creates an electric field. In particular, the electro-
magnetic force induced in a closed loop circuit C is proportional to the rate of change of flux of the magnetic
field associated with any surface S connected with C. Faraday’s law states
Z ZZ
~ ∂ ~
r

E · d~ = − B · b e n dσ.
C ∂t S
Using the Stoke’s theorem, we find
ZZ ZZ ~
~ ∂B
(∇× E) · b e n dσ = − · b e n dσ.
S S ∂t
The above equation must hold for an arbitrary surface and loop. Equating like terms we obtain the differential
form of Faraday’s law
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∂B
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∇× E = − . (2.6.72)
∂t
This is the first electromagnetic field equation of Maxwell.
Ampere’s law, equation (2.6.65), written in terms of the total current from equation (2.6.71) , becomes

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∂P ∂E
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∇× B = µ 0 (∇× M + J f + )+ µ 0 0 (2.6.73)
∂t ∂t
which can also be written as
1 ∂
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∇× ( B − M)= J f + (P + 0 E)
µ 0 ∂t
1 D.J. Griffiths, Introduction to Electrodynamics, Prentice Hall, 1981. P.232.