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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.) 1