Classical Electrodynamics
Physics 6561
Homework 3. Due: In class on Tuesday, 9/23
1. Force of charged particle with constant velocity (5 points)
Two equal charges
q
move with equal velocity
~v
. What is the force acting between the
two charges? The distance between the two charges is
R
and the angle between
~v
and
~
R
is
θ
.
2. Gauge invariance of interacting Lagrangian (5 points)
We have made the gauge invariance one of the basic principles for writing down the
Maxwell Lagrangian. Argue, that the interaction described by the usual Lagrangian
L
int
=

1
c
2
A
μ
j
μ
is also gauge invariant.
3 Scalar ﬁeld theory in 2D (10 points)
Consider a real scalar ﬁeld
φ
in 2 dimensions, described by the Lagrangian density
L
=
1
2
∂
μ
φ∂
μ
φ

λ
(
φ
2

φ
2
0
)
2
.
(a) Derive the EulerLagrange equation of motion for this ﬁeld, and the symmetric energy
momentum tensor. The energy momentum tensor is deﬁned as
T
μν
=
∂
L
∂∂
μ
φ
∂
ν
φ

g
μν
L
.
(b) Find an explicit STATIC solution (eg.
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 Fall '08
 LIPSON/POLLOCK
 Fundamental physics concepts, Energy density, Noether's theorem, Lagrangian density, Proca Lagrangian density, usual Lagrangian Lint

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