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)`.

Explanation

(i) Given, the induced electric field at a distance r from the wire inside the cable is
`vecE(s,t)=mu_0I_0 v cos (2pivt)log_e(s/a)hatk`
Displacement current density,
`vecJ_D=in_0 (vec(dE))/(dt)=in_0 d/(dt)[mu_0I_0 v cos (2pivt)log_e(s/a)hatk]`
`=in_0 mu_0 I_0 v d/(dt)[cos 2pivt]log_e (s/a)hatk=1/(c^2) I_0 v^2 2pi[-sin 2pivt]loe_e (s/a)hatk`
`=(v^2)/(c^2)2piI_0 sin 2pivt log_e (a/s)hatk [ :' log_e(s/a)=-log_e(a/s)]`
`=1/(lambda^2) 2pi I_0 log_e(a/s) sin 2pivthatk=(2piI_0)/(lambda^2) log_e a/s sin 2pivt hatk`
(ii) `I_D=int J_D s ds d theta=int_(s=0)^a int_0^(2pi) d theta=int_(s=0)^a J_Dsdsxx2pi`
`=int_0^a[(2pi)/(lambda^2)I_0log_e(a/s)s ds sin 2pivt]xx2pi=((2pi)/lambda)^2I_0 int_(s=0)^alog_e(a/s)s ds sin 2pivt`
`((2pi)/lambda)^2I_0 int_(s=0)^alog_e(a/s) 1/2d(s^2).sin 2pivt=(a^2)/2((2pi)/lambda)^2I_0 sin 2pivt int_(s=0)^a log_e(a/s). d(s/a)^2`
` (a^2)/4((2pi)/lambda)^2I_0 sin 2pivt int_(s=0)^a log_e(a/s). d(s/a)^2=-(a^2)/4 ((2pi)/lambda)^2I_0 sin 2pivt int_(s=0)^a log_e(a/s). d(s/a)^2`
`=-(a^2)/4 ((2pi)/lambda)^2I_0 sin 2pivtxx(-1) [ :' int_(s=0)^a log_e(a/s). d(s/a)^2=-1]`
`I_D=(a^2)/4 ((2pi)/lambda)^2I_0 sin 2pivt=((2pia)/(2lambda))^2I_0sin 2pivt`
(iii) The diplacement current, `I_D=((2pia)/(2lambda))^2I_0sin 2pivt=I_(0D)sin 2pivt`
where `I_(0D)=((2pia)/(2lambda))^2I_0=((pia)/(lambda))^2I_0 :. (I_0D)/(I_0)=((api)/(lambda))^2`