Page 350 - Intro to Tensor Calculus

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Ampere’s Law This law states the line integral of the magnetic force vector around a closed loop is
proportional to the sum of the current through the loop and the rate of flux of the displacement vector
through the loop. This produces the second electromagnetic field equation:

~
∂D ijk i ∂D i
~
~
∇× H = J f + or H k,j = J + . (2.6.76)
f
∂t ∂t
Gauss’s Law for Electricity This law states that the flux of the electric force vector through a closed
surface is proportional to the total charge enclosed by the surface. This results in the third electromagnetic
field equation:
1 ∂ √
~ i i
∇· D = ρ f or D = ρ f or √ gD = ρ f . (2.6.77)
,i i
g ∂x
Gauss’s Law for Magnetism This law states the magnetic flux through any closed volume is zero. This
produces the fourth electromagnetic field equation:
1 ∂ √ i
~
i
∇· B =0 or B =0 or √ gB =0. (2.6.78)
,i i
g ∂x
When no confusion arises it is convenient to drop the subscript f from the above Maxwell equations.
Special expanded forms of the above Maxwell equations are given on the pages 176 to 179.

Electromagnetic Stress and Energy
Let V denote the volume of some simple closed surface S. Let us calculate the rate at which electro-
magnetic energy is lost from this volume. This represents the energy flow per unit volume. Begin with the
first two Maxwell’s equations in Cartesian form

∂B i
ijk E k,j = − (2.6.79)
∂t
∂D i
ijk H k,j =J i + . (2.6.80)
∂t
Now multiply equation (2.6.79) by H i and equation (2.6.80) by E i . This gives two terms with dimensions of
energy per unit volume per unit of time which we write

∂B i
ijk E k,j H i = − H i (2.6.81)
∂t
∂D i
ijk H k,j E i =J i E i + E i . (2.6.82)
∂t
Subtracting equation (2.6.82) from equation (2.6.81) we find

∂D i ∂B i
ijk (E k,j H i − H k,j E i )= − J i E i − E i − H i
∂t ∂t
∂D i ∂B i
ijk [(E k H i ) ,j − E k H i,j + H i,j E k ]= − J i E i − E i − H i
∂t ∂t
Observe that jki (E k H i ) ,j is the same as ijk (E j H k ) ,i so that the above simplifies to

∂D i ∂B i
ijk (E j H k ) ,i + J i E i = − E i − H i . (2.6.83)
∂t ∂t