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EAD 234A: E&M Homework #4 Due Thursday, February 4, 2010 1. A thin insulating rod, running from z = -a to z = +a, carries the following line charges: (a) 0 cos 2 z a π λλ ⎛⎞ = ⎜⎟ ⎝⎠ (b) 0 sin 2 z a = (c) 0 cos z a = In each case, find the leading term in the multipole expansion of the potential. 2. A charge Q is distributed uniformly along the z axis from z = -a to z =a. Find the first three non-vanishing terms in the multipole expansion of the electric potential V(r , θ ) for r > a. 3. ( a) Evaluate the exterior spherical multipole moments for a shell of radius R which carries a surface charge density σ ( θ , φ ) = σ 0 sin θ cos φ . (b) Write ϕ (r > R, θ , φ ) in the form ϕ (x, y, z, r). (c) Evaluate the interior spherical multipole moments for the shell of part (a). (d) Write ϕ (r < R, θ , φ ) in the form ϕ (x, y, z, r) (e) Check the matching conditions for ϕ and E at r = R. (f) Extract the dipole moment p of the shell from your answer to part (b). 4. (a). Calculate the multipole moments q lm of each of the two charge distributions shown in the figure below. Try to obtain results for the non-vanishing moments valid for all l, but in each case find the first

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