exam2sols.pdf - Exam II solutions 1 Calculate the potential...

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Exam II solutions 1) Calculate the potential everywhere, for a thin shell of charge, radius R and surface charge density σ ( θ ) = σ 0 cos 2 θ . Solution: First find how cos 2 θ projects onto the Legendre polynomial space. You should find that cos 2 θ = 2 3 P 2 + 1 3 P 0 . Including the fact that singularities at r = 0 , r = should be avoided, the solution is V ( r < R ) = A 0 + A 2 y 2 P 2 , V ( r > R ) = B 0 y + B 2 y 3 P 2 . where y = r/R has been used. From the continuity of the voltage at the boundary for every angle we have immediately A 0 = B 0 and A 2 = B 2 . From the discontinuity of the electric field we have ( V 0 = 0 / 0 ) 3 B 2 = - 2 A 2 + 2 3 V 0 , B 0 = 1 3 V 0 . The final expressions are V ( r < R ) = V 0 ( 1 3 + 2 15 y 2 P 2 ) , V ( r > R ) = V 0 ( 1 3 y + 2 15 y 3 P 2 ) . 2) A wire of linear charge density + λ has length d . After specifying your choice of axes, evaluate the first three non-zero moments of its multipole expansion at distances large compared to d . Solution: The z - axis is along the wire, with the origin at the mid-point. The charge density of the wire is ρ ( r 0 ) = λδ ( x 0 ) δ ( y 0 ) , - d/ 2 < z 0 < d/ 2 .
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  • Fall '15
  • Giovani Bonvicini
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