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
`E(s,t)=mu_(0) I_(0)v "cos"(2pi vt) In ((s)/(a))hatk`
Now displacement current density ,
`J_(0) = epsilon_(0) (dE)/(dt) =epsilon_(0) (d)/(dt) [mu_(0) I_(0) v"cos" (2pi vt) In ((s)/(a)) hatk]`
`=epsilon_(0)mu_(0)I_(0)v(d)/(dt) ["cos"2pivt]In ((s)/(a))hatk`
`=(1)/(c^(2))I_(0)v^(2)pi[-"sin"2pivt]In ((s)/(a))hatk`
`=(V^(2))/(C^(2)) 2piI_(0)"sin"2pivtIn ((a)/(s))hatk" "[:. l_(4)(s)/(a) =- l_(4)(a)/(s)]`
`=(1)/(lambda^(2))2piI_(0) In ((a)/(s)) "sin"2pivthatk`
`=(2piI_(0))/(lambda_(2))In (a)/(s) "sin"2pivthatk`
`(ii)" "I_(d) = int J_(d) sdsd0 = underset(s=0)overset(a)(int) J_(d) sds underset(0)overset(2pi)(int)d0 =underset(s=0)overset(a)(int) J_(d) sds xx 2pi`
`=((2pi)/(lambda))^(2)`
`rArr" "=((2pi)/(lambda))^(2) I_(0) underset(s=0)overset(a)(int) In ((a)/(s)) 1/2d(s^(2))."sin" 2pivt`
`=(a^(2))/(2) ((2pi)/(lambda))^(2)I_(0) "sin"2pivt underset(s=0)overset(a)(int) In ((a)/(s)).d((s)/(a))^(2)`
`=(a^(2))/(4)((2pi)/(lambda))^(2) I_(0)"sin" 2pivt underset(s=0)overset(a)(int) In ((a)/(s))^(2).d((s)/(a))^(2)`
`=-(a^(2))/(4)((2pi)/(lambda))^(2)I_(0) "sin" 2pivt underset(s=0)overset(a)(int) In ((s)/(a))^(2).d((s)/(a))^(2)`
`=-(a^(2))/(4)((2pi)/(lambda))^(2) I_(0)"sin"2pivt xx(-1)" "[:.underset(s=0)overset(a)(int)In ((s)/(a))^(2)d((s)/(a))^(2)=-1]`
`:. I_(d) (a^(2))/(4)((2pi)/(lambda))^(2)I_(0)"sin" 2pivt`
`rArr" "=((2pia)/(2lambda))^(2)I_(0)"sin" 2pivt`
(iii) the displacement current
`I_(0) =((2pia)/(2lambda))^(2) I_(0)"sin" 2pivt =I_(0d)"sin" 2 pivt`
`"Here"" "I_(0d) =((2pia)/(2lambda))^(2)I_(0)=((api)/(lambda))^(2)I_(0)`
`(I_(0d))/(I_(0)) =((api)/(lambda))^(2)`