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Homework4

# Homework4 - Homework 4 Answers 95.657 Electromagnetic...

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Homework 4 Answers, 95.657 Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Jackson 2.13 (a) Two halves of a long hollow conducting cylinder of inner radius b are separated by small lengthwise gaps on each side, and are kept at different potentials V 1 and V 2 . Show that the potential inside is given by  , = V 1 V 2 2 V 1 V 2 tan 1 2 b b 2 − 2 cos where is measured from a plane perpendicular to the plane through the gap. SOLUTION: Due to the symmetry of the problem, it is apparent that the solution will be best expressed in cylindrical coordinates. Additionally, because the solution will be independent of the z coordinate, the problem reduces to the two dimensions of polar coordinates  ,  . Because the problem contains no charge, the problem simplifies down to solving the Laplace equation 2 = 0 in polar coordinates and applying the boundary condition = b , = V  where: V = { V 1 if / 2  3 / 2 V 2 if / 2  3 / 2 } The Laplace equation in polar coordinates is: 1 ∂ ∂ ∂  1 2 2 ∂  2 = 0 Separation of variables leads to the general solution:  , = a 0 b 0 ln  A 0 B 0  , ≠ 0 a b − A e i  B e i  We desire a valid solution at the origin, which is only possible if b 0 =0 and b υ =0 so that the solution becomes:  , = A 0 B 0  , ≠ 0 A e i   B e i  We desire a single, valid solution over the full angular range, so the single-value requirement means  , = ,  2  . When we apply this, we get: A 0 B 0  , ≠ 0 A e i  B e i  = A 0 B 0  2  , ≠ 0 A e i  2  B e i  2  Which leads to B 0 =0 and υ = n where n =1, 2, ... We now have:

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 , = A 0 n = 1 n A n e in B n e in Now apply the last boundary condition = b , = V  V = A 0 n = 1 b n A n e in B n e in ( Eq. 1) Let us first find the A 0 term. Integrate both sides over the full angular sweep. 0 2 V  d = 0 2 A 0 d  n = 1 b n A n 0 2 e in d  B n 0 2 e in d −/ 2 / 2 V 1 d  / 2 3 / 2 V 2 d = A 0 2 V 1  V 2 = A 0 2 A 0 = V 1 V 2 2 Let us now find the A n coefficients. Multiply ( Eq. 1) on both sides by e in ' and integrate over all angles : 0 2 V  e in ' d = A 0 0 2 e in ' d  n = 1 b n A n 0 2 e i n n '  d  B n 0 2 e i n n '  d Use the orthonormality condition 0 2 e i k k ' x dx = 2   k , k ' 0 2 d
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