Homework 10 Answers, 95.658 Spring 2011, Electromagnetic Theory II
Dr. Christopher S. Baird, UMass Lowell
Problem 1
Show whether the following laws obey Galilean Relativity:
(a) The law of conservation of electromagnetic energy
(b) The chargecurrent continuity equation
SOLUTION:
If we have a frame
K
' moving with velocity v relative to frame
K
, then the coordinates transform using
Galilean Relativity according to:
x
'
=
x
−
v
t
,
t
'
=
t
Expanding out partial derivatives, this means:
∂
∂
x
'
=
∂
∂
x
and
∂
∂
t
'
=
v
⋅∇+
∂
∂
t
which leads to:
∇
'
2
=∇
2
∂
2
∂
t
'
2
=
v
⋅∇
v
⋅∇
2
v
⋅∇
∂
∂
t
∂
2
∂
t
2
The law of conservation of electromagnetic energy states if energy is lost in a small volume, it must
either have flowed out of the volume, or have accelerated charges:
−
∂
u
'
∂
t
'
=∇
'
⋅
S
'
+
J
⋅
E
'
Applying a Galilean relativity transformation:
−
v
⋅∇
u
−
∂
u
∂
t
=∇⋅
S
+
J
⋅
E
This equation obviously does not have the same for in the
K
frame as it did in frame
K
'. The law of
conservation of electromagnetic energy does not therefore obey Galilean relativity. This should be
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 Spring '11
 Staff
 Charge, Current, Energy, Mass, Work, Special Relativity, galilean relativity

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