homework_5_solution - x = a 2 x= a 2 a 0(x < 2 = ex a < x <...

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x = a 2 x = a 2 0( x < a 2 ) E = σ 0 e x ( a 2 < x < a 2 ) 0( x > a 2 ) 0( x < a 2 ) Φ = σ 0 x ( a 2 < x < a 2 ) 0( x > a 2 ) ∆Φ = σ a 2 0 2 Φ ∂ρ 2 + 1 ρ Φ ∂ρ + 1 ρ 2 2 Φ ∂φ 2 + 2 Φ z 2 = 0 Φ ( ρ , φ , z ) = R ( ρ ) Q ( φ ) Z ( z )
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d 2 Z dz 2 k 2 z = 0 d 2 Q d φ 2 + ν 2 Q = 0 d 2 R d ρ 2 + 1 ρ dR d ρ + ( k 2 ν 2 ρ 2 ) R = 0 Φ ( z ) | z =0 = 0 Q ( φ ) = A Z ( z ) = sinh kz R ( ρ ) = CJ 0 ( k ρ ) + DN 0 ( k ρ ) D = 0 ρ = 0 ρ = a k k n = x n a x n J 0 ( x ) Φ ( ρ , φ , z ) = + n =1 C n J 0 ( k n ρ ) sinh k n z z = L V = + n =1 C n J 0 ( k n ρ ) sinh k n L C n = csc k n L π a 2 J 2 1 ( k n a ) · V · 2 π ˆ a 0 d ρρ J 0 ( k n ρ )
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