A long straight cable of length l is placed symmetrically along z-axis and has radius `a(ltlt l).` The cable consists of a thin wire and a co- axial conducting tube. An alternating current `I(t) = I_(0) " sin " (2pi vt)`. Flows down the central thin wire and returns along the co-axial conducting tube. the induced electric at a distance s from the wire inside the cable is
`E(s ,t) =mu_(0) I_(0) v " cos "(2pivt) In ((s)/(a)) hatk`.
(i) Calculate the displacement current density inside the cable.
(ii) Integrate the displacement current density across the cross- section of the cable to find the total displacement current `I^(d)`.
(iii) compare the conduction current `I_(0)` with the displacement current `I_(0)^(d)`.

Options

A. `(2pi)/(lamda^(2))I_(0)ln((a)/(s))sin(2pi upsilont)hat(k)`
B. `(1)/(lamda^(2))I_(0)ln((a)/(s))sin(2pi upsilont)hat(k)`
C. `(pi)/(lamda^(2))I_(0)ln((a)/(s))sin(2pi upsilon t)hat(k)`
D. Zero

Answer

A

Explanation

Displacement current density, `vecJ_(d)=epsilon_(0)(dvecE)/(dt)`
`=epsilon_(0)mu_(0)I_(0)upsilon(del)/(delt)upsiloncos(2piupsilont)ln.((s)/(a))hatk`
`=(1)/(c^(2))I_(0)2piupsilon^(2)(-sin(2piupsilont))ln.((s)/(a))hatk`
`=((upsilon)/(c))^(2)2piI_(0)sin(2piupsilont)ln.((a)/(s))hatk=(2pi)/(lambda^(2))I_(0)ln.((a)/(s))sin(2piupsilont)hatk`