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Question

Find the electrostatic energy stored in a cylindrical shell of length l, inner radius a and outer radius b, coaxial with a uniformly charged wire with linear charge density λ.

A
U=λ2l4πε0log(ba)
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B
U=λ2l8πε0log(ba)
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C
U=λ2l2πε0log(ba)
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D
U=λ2l4πε0log(ab)
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Solution

The correct option is A U=λ2l4πε0log(ba)
For this we consider an elemental shell of radius x and width dx. The volume of this shell dV can be given as dV=(2πxl)dx

The electric field due to the wire at the shell is E=λ2πε0x

The electrostatic field energy stored in the volume of this shell is dU=[12ε0E2]dV

dU=12ε0(λ2πε0x)2(2πxl)dx

The total electrostatic energy stored in the above mentioned volume can be obtained by integrating the above expression within limits from a to b as

U=dU=ba12ε0(λ2πε0x)2(2πxl)dx

U=λ2l4πε0ba1xdx

U=λ2l4πε0(logxba)

U=λ2l4πε0log(ba)

Hence, option (a) is the correct answer.

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