A long straight cable of length l is placed symmetrically

Question:

A long straight cable of length l is placed symmetrically along the z-axis and has radius a. The cable consists of a thin wire and a co-axial conducting tube. An

alternating current I(t) = I0 sin (2πvt) flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance s

from the wire inside the cable is E(s,t) = μ0I0v coz (2πvt). In

$\left(\frac{s}{a}\right) \hat{k}$a) calculate the displacement current density inside the cable

(b) integrate the displacement current density across the cross-section of the cable to find the total displacement current I

(c) compare the conduction current I0 with the displacement current I0d

Solution:

(a) The displacement current density is given as

$\vec{J}_{d}=\frac{2 \pi I_{0}}{\lambda^{2}} \ln \frac{a}{s} \sin 2 \pi v t \hat{k}$

(b) Total displacement current

$I^{d}=\int J_{d} 2 \pi s d s$

Id = (πa/λ)2I0 sin 2πvt

(c) The displacement current is I0d/I0 = (aπ/λ)2

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