Explanation Given: The long conducting cylinder is placed parallel to the z-axis in an uniform electric field in the negative x-direction. Calculation: Use Laplace’s equation V = ∑ k = 0 ∞ r k ( A k sin k θ + B k cos k θ ) + r − k ( A k ' sin k θ + B k ' cos k θ ) − − − ( i ) The coefficients multiplying r k must be zero because the potential is outside the cylinder. A k = 0 B k = 0 Substitute these values in equation ( i ) . V = ∑ k = 0 ∞ r − k ( A k ' sin k θ + B k ' cos k θ ) − − − ( i i ) The electric field ( on the x-axis) E = − E 0 i E = − ∇ V ' because the electric field is a negative gradient of the potential. − E 0 i = − ∇ V ' ⇒ E 0 i = ∇ V ' ⇒ E 0 i = ∂ V ' ∂ x i Integrate on both sides and find V ' . E 0 x = V ' ⇒ V ' = E 0 r cos θ − − − ( i i i ) From ( i i ) and ( i i i ) (find the solution for the Laplace’s equation) V = E 0 r cos θ + ∑ k = 0 ∞ r − k ( A k ' sin k θ + B k ' cos k θ ) − − − ( i v ) Find the coefficients values using boundary condition in the equation ( i v )

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Chapter 13.10, Problem 19MP

To determine

Find the potential in the region outside the cylinder.

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Chapter 13.10, Problem 18MP

Chapter 13.10, Problem 20MP

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Chapter 13 Solutions

Mathematical Methods in the Physical Sciences

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Find the potential in the region outside the cylinder.

Solution Summary: The author calculates the potential in the region outside the cylinder using Laplace's equation.

Find the potential in the region outside the cylinder.

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Chapter 13 Solutions