Page 349 - Intro to Tensor Calculus

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or
~
∂D
~
~
∇× H = J f + . (2.6.74)
∂t
This is Maxwell’s second electromagnetic field equation.
To the equations (2.6.74) and (2.6.73) we add the Gauss’s law for magnetization, equation (2.6.61) and
Gauss’s law for electrostatics, equation (2.6.48). These four equations produce the Maxwell’s equations of
electrodynamics and are now summarized. The general form of Maxwell’s equations involve the quantities

E i , Electric force vector, [E i ]= Newton/coulomb
2
B i , Magnetic force vector, [B i ]= Weber/m
H i , Auxilary magnetic force vector, [H i ] = ampere/m
D i , Displacement vector, [D i ] = coulomb/m 2

J i , Free current density, [J i ] = ampere/m 2
P i , Polarization vector, [P i ] = coulomb/m 2

M i , Magnetization vector, [M i ] = ampere/m
for i =1, 2, 3. There are also the quantities

%, representing the free charge density, with units [%] = coulomb/m 3
2 2
0 , Permittivity of free space, [ 0 ] = farads/m or coulomb /Newton · m .
2
µ 0 , Permeability of free space, [µ 0 ] = henrys/morkg · m/coulomb
In addition, there arises the material parameters:

i
µ , magnetic permeability tensor, which is dimensionless
j
i
, dielectric tensor, which is dimensionless
j
i
α , electric susceptibility tensor, which is dimensionless
j
i
χ , magnetic susceptibility tensor, which is dimensionless
j
These parameters are used to express variations in the electric field E i and magnetic field B i when
acting in a material medium. In particular, P i ,D i ,M i and H i are defined from the equations
j i i j
D i = E j = 0 E i + P i = 0 δ + α i
i
j
j
i
i
i
j
B i =µ H j = µ 0 H i + µ 0 M i , µ = µ 0 (δ + χ )
i j j j
j
P i =α E j , and j for i =1, 2, 3.
i
i M i = χ H j
The above quantities obey the following laws:
Faraday’s Law This law states the line integral of the electromagnetic force around a loop is proportional
to the rate of flux of magnetic induction through the loop. This gives rise to the first electromagnetic field
equation:
~
∂B ∂B i
~ ijk
∇× E = − or E k,j = − . (2.6.75)
∂t ∂t