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Question

A long straight cable of lenght l is placed symmetrically along z - axis has radius a(<<l). The cable consists of a thin wire and a coaxial conducting tube. An alternating current I(t)=I0sin(2πυt)flows down the central thin wire and return along the coaxial conducting tube. The induced electric field at a distance s from the wire inside the cable is E(s,t)=μoIoυcos(2πυt) ln(sa)^k. The displacement current density inside the cable is

A
2πλ2I0 ln(as)sin(2πυt)^k
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B
Iλ2I0ln(as)sin(2πυt)^k
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C
πλ2I0ln(as)sin(2πυt)^k
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D
Zero
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Solution

The correct option is B 2πλ2I0 ln(as)sin(2πυt)^k
Given: The induced electric field at a distance from wire is E(s,t)=μ0I0υ cos(2πυt)ln(sa)^k

Displacement current density is given by:
Jd=ϵ0d¯Edt

ϵ0μ0I0v(t)(cos2πvt)(sa)^k

Substitute c for 1μ0ϵ0.

1c2I02πv2(sin(2πvt) ln(sa)^k

(vc)22πI0sin(2πvt) ln(as)^k

2πλ2I0 ln(as)sin(2πvt)^k

Option (A) is correct.

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