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A coaxial cable contains an insulating material of conductivity σ .If the radius of the central wire is a and that of the sheet is b, then the conductance per unit length is

A
ln(ba)2πσV0
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B
4πσln(ba)
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C
2πσln(ab)
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D
2πσln(ba)
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Solution

The correct option is D 2πσln(ba)

Consider the length of the coaxial cable as L

Let V0 be the potential difference between the inner and outer conductors so that,

V(ρ=a)=0

and V(ρ=b)=V0

J=σE=σV0ρlnba^ap

dS=ρdϕdz^ap

I=J.dS=2πΦ=0Lz=0V0σρlnbaρdϕdZ=2πLσV0lnba

Resistance per unit lengh , RL=V0I=ln(ba)2πσ

The conductance per unit length is,

G=1R=2πσln(ba)


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