Jackson 6.9 Homework Problem Solution
Dr. Christopher S. Baird
University of Massachusetts Lowell
PROBLEM:
Discuss the conservation of energy and linear momentum for a macroscopic system of sources and
electromagnetic fields in a uniform, isotropic medium described by a permittivity ε and a permeability
μ. Show that in a straightforward calculation the energy density, Poynting vector, field-momentum
density, and Maxwell stress tensor are given by the Minkowski expressions:
u=1
2(ϵ E2+μ H2)
S=E×H
g=μ ϵE×H
Tij=[ ϵ EiEj+μ HiHj−1
2δij( ϵ E2+μ H2)]
What modifications are made if ε and μ are functions of position?
SOLUTION:
As derived in class, the energy density and energy flux density in linear, low-dispersion, low-loss
materials are given by:
u=1
2
(
H⋅B+E⋅D
)
and
S=E×H
For linear material, B = μH and D = εE so these become:
u=1
2
(
ϵE2+μ H2
)
and
S=E×H
The field-momentum density is given by:
g=S
v2
In linear materials, the speed of the waves are given by
v=1
√
ϵμ
so that
g=μ ϵ E×H
The Maxwell stress tensor describes the electromagnetic momentum flux and is given by:
