2. The current distribution of an infinite, wire can be described in cylindrical coordinates by J = J(p) 2. longitudinally-uniform, and axially-symmetric (a) Show that · J = 0. (b) Considerations of longitudinal and axial symmetry require that the mag- netic field can only depend upon p, i.e. that B(p) = B₂(p)ô + Bø(p)ô + B₂(p)2. Use Ampère's Law to determine B6 (p) in terms of I (p) = 2π J(p') p'dp'. (c) Use the Biot and Savart Law to show that Bp(p) = B₂ (p) = 0. Side note: it is pretty easy to show that B,(p) must be zero using Gauss's Law for B with a cylindrical volume. I am not aware of an "easy" way to see B₂ (p) = 0 - not that it is very difficult using the Biot and Savart Law... (d) Use the integral form for the vector potential [Jackson, Eq. (5.32)] to determine A for this current distribution. Hint: in order to deal with divergent integrals, you may want to limit the current distribution to -L ≤ z ≤ L, determine A, adjust A by a constant (as one is allowed to do), and then take L→ ∞. (e) Verify that the A in part (d) reproduces the B found in parts (b) and (c) when the curl is taken.

2. The current distribution of an infinite, wire can be described in cylindrical coordinates by J = J(p) 2. longitudinally-uniform, and axially-symmetric (a) Show that · J = 0. (b) Considerations of longitudinal and axial symmetry require that the mag- netic field can only depend upon p, i.e. that B(p) = B₂(p)ô + Bø(p)ô + B₂(p)2. Use Ampère's Law to determine B6 (p) in terms of I (p) = 2π J(p') p'dp'. (c) Use the Biot and Savart Law to show that Bp(p) = B₂ (p) = 0. Side note: it is pretty easy to show that B,(p) must be zero using Gauss's Law for B with a cylindrical volume. I am not aware of an "easy" way to see B₂ (p) = 0 - not that it is very difficult using the Biot and Savart Law... (d) Use the integral form for the vector potential [Jackson, Eq. (5.32)] to determine A for this current distribution. Hint: in order to deal with divergent integrals, you may want to limit the current distribution to -L ≤ z ≤ L, determine A, adjust A by a constant (as one is allowed to do), and then take L→ ∞. (e) Verify that the A in part (d) reproduces the B found in parts (b) and (c) when the curl is taken.