Explanation Given: The length of carbon pipe is 1.0 m , its inner diameter is 3.0 cm and its outer diameter is 5.0 cm . Concept used: The Laplace equation for cylindrical coordinates is shown below. ∇ 2 V cylindrical = 1 ρ ∂ ∂ ρ ( ρ ∂ V ∂ ρ ) + 1 ρ 2 ( d 2 V d ϕ 2 ) + d 2 V d z 2 Calculation: Consider the inner radius be a and outer radius be b for the carbon pipe. The potential is a function of ρ so, Laplace equation can be written as, ∇ 2 V = 0 1 ρ ∂ ∂ ρ ( ρ ∂ V ∂ ρ ) = 0 ∂ ∂ ρ ( ρ ∂ V ∂ ρ ) = 0 Integrate the above equation with respect to ρ . ρ ∂ V ∂ ρ = A ∂ V ∂ ρ = A ρ Integrate the above equation with respect to ρ . V ( ρ ) = ∫ A ρ d ρ = A ln ρ + B Here, A and B are constants of integration and can be obtained from boundary conditions. The first boundary condition is V ( b ) = 0 , since the potential at outer surface is zero. So, the constant can be obtained as, 0 = A ln b + B B = − A ln b Substitute − A ln b for B in the expression for V ( ρ ) as, V ( ρ ) = A ln ρ − A ln b = A ( ln ρ − ln b ) = A ln ( ρ b ) The second boundary condition is V ( ρ ) = V ρ . So, substitute V ρ for V ( ρ ) in above equation. V ρ = A ln ( a b ) A = V ρ ln ( a b ) Now, the expression for V ( ρ ) can be written with the values of constants known as, V ( ρ ) = V ρ ln ( a b ) ln ( ρ b ) = V ρ ln ( ρ b ) ln ( a b ) The electric field intensity E is calculated as, E = − ∇ V ( ρ ) = − ∂ V ∂ ρ a ρ = − ∂ ∂ ρ ( V ρ ln ( ρ b ) ln ( a b ) ) a ρ = − V ln ( a b ) ∂ ∂ ρ [ ln ( ρ b ) ] a ρ Simplify the above equation. E = − V ρ ln ( a b ) 1 ρ b ⋅ 1 b a ρ = − V ρ ln ( a b ) ⋅ 1 ρ a ρ The relationship between current density and electric field can be written as, J = σ E The expression for current is shown below

Fundamentals of Electromagnetics with Engineering Applications

1st Edition

ISBN: 9780470105757

Author: Stuart M. Wentworth

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2, Problem 2.67P

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The resistance between the inner surface and the outer surface of a carbon pipe.

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Chapter 2, Problem 2.66P

Chapter 2, Problem 2.68P

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Fundamentals of Electromagnetics with Engineering Applications

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The resistance between the inner surface and the outer surface of a carbon pipe.

Solution Summary: The resistance between the inner surface and the outer surface of a carbon pipe is 5.4Omega .

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