A long straight cable of length l is placed symmetrically along z - axis and has radius `a(lt lt l)`. The cable consists of a thin wire and a co-axial conducting tube. An alternating current thin wire and returns along the coaxial conducting tube. The induced electric field at a distance s from the wire inside the cable is `vec(E )(s, t)=mu_(0)I_(0)v cos (2pi vt) ln ((s)/(a))hat(k)`
Calculate the displacement current density inside the cable.

Explanation

Induced electric field at distance s (which is less then radius of coaxial cable)
`vec(E )(s, t)=mu_(0)I_(0)v cos (2pi v t)ln ((s)/(a)) hat(k)`
Now displacement current density.
`J_(d) =in_(0)(dE)/(dt)`
`= in_(0)(d)/(dt)[mu_(0)I_(0)v cos (2pi vt)ln((s)/(a))hat(k)]`
`= in_(0)mu_(0)I_(0)v(d)/(dt)[cos (2pi v t) ln ((s)/(a))hat(k)]`
`=(1)/(c^(2))I_(0)v^(2)xx 2pi[-sin (2pi vt)ln((s)/(a))hat(k)] " " [because mu_(0)in_(0) =(1)/(c^(2))]`
`therefore J_(d)=+(n^(2))/(c^(2))xx 2pi_(0)sin(2 pi v t)ln ((a)/(s))hat(k) " " [because ln.(S)/(a)=-ln.(a)/(S)]`
`=(1)/(lambda^(2))xx 2pi_(0)ln((a)/(s))sin(2pi vt)hat(k) " " [because lambda = (c )/(v)]`
`therefore J_(d) = (2pi I_(0))/(lambda^(2))ln ((a)/(s))sin(2pi v t)hat(k) " "` .....(1)