PHGN462: Advanced E&M
Quiz 5 – October 26, 2009
NAME: KEY
y
I(t)
R
x
1. Consider a semi-circular segment of wire in the upper half x-y plane centered on the origin with carries a clockwise current
~
which varies as: I(t) = I0 t/τ . Find the retarded vector potential, A(t),
at the origin. Show all work.
R 3 0
R
0
~ thus the retarded vector potential at the origin is:
~ r , t) → I(t)dl,
Solution: For a current-carrying wire, one has d r J(~
~ = µ0
A(t)
4π
Z
~
I(tR )dl
R
~ = Rdθ(sin θx̂ − cos θŷ). Using I(t) = I0 t/τ , we get
where tR = t − R/c. The vector line element is dl
Z π
(t − R/c)Rdθ(sin θx̂ − cos θŷ)
µ0 I0
~ = µ0 I0
A(t)
=
(t − R/c)x̂
4πτ 0
R
2πτ
(1)
(2)
(Comment: One could decide to work in cylindrical coordinates which would give a φ̂ unit vector in the integral. However,
the final answer cannot have φ̂ because it is θ-dependent and therefore does not factor out of the integral.)
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