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Question

A long straight cable of length l is placed symmetrically along z-axis and has radius a(<<l). the cable consists of a thin wire and co-axial conducting tube. An alternating current I(t) = I0 sin (2πνt)flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance s from the wire inside the cable is E(s, t) = μ0I0cos(2πνt) ln(sa)^k. The displacement current density inside the cable is

A
2πλ2I0ln(as)sin(2πνt)^k
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B
1λ2I0ln(as)sin(2πνt)^k
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C
πλ2I0ln(as)sin(2πνt)^l
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D
Zero
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Solution

The correct option is B 2πλ2I0ln(as)sin(2πνt)^k
Given : The induced electric field at a distance from wire is E(s,t)=μ0I0vcos(2πvt)ln(sa)^k
Displacement current density is given by :
Jd=ϵ0dEdt

ϵ0μ0I0v(t)(cos2πvt)(sa)^k

Substitute for 1μ0ϵ0

1c2I02πv2(sin(2πvt))ln(sa)^k

(vc)22πI0sin(2πvt))ln(as)^k

2πλ2I0ln(as)sin(2πvt)^k

Option (A) is correct.

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