(a) Explanation The following figure shows the diagram of the wire and the cylinder. Figure-(1) Consider a Gaussian cylinder having length l and radius r . Write the equation for the electric field by using Gauss’ law. ∮ E → ⋅ d a → = Q ε 0 (I) Here, E is the electric field, d a is the area of an infinitesimal element, ε 0 is the permittivity of the free space and Q is the electric charge. Write the equation for the difference in the electric potential between the cylinder and the wire. V w − V C = ∫ C W E ⋅ d r Δ V = ∫ C W E ⋅ d r (II) Here, V W is the electric potential in the wire and V C is the electric potential in the cylinder. Conclusion: Substitute E ⋅ 2 π r l for ∮ E ⋅ d a and λ l for Q in the equation (I) to calculate the electric field (b) To determine The proof for E = Δ V ln ( r a r b ) ( 1 r ) .

Physics for Scientists and Engineers with Modern Physics, Technology Update

9th Edition

ISBN: 9781305401969

Author: SERWAY, Raymond A.; Jewett, John W.

Publisher: Cengage Learning

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Chapter 25, Problem 68AP

(a)

To determine

The proof for ΔV=2keλln(rarb).

(b)

To determine

The proof for E=ΔVln(rarb)(1r).

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Chapter 25, Problem 67AP

Chapter 25, Problem 69AP

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Physics for Scientists and Engineers with Modern Physics, Technology Update

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The proof for Δ V = 2 k e λ ln ( r a r b ) .

Solution Summary: The following figure shows the diagram of the wire and the cylinder.