Question

4. A steady current I flows down a long cylindrical wire of radius a. 

(a) Find the magnetic field, both inside and outside the wire, if the current is uniformly dis- tributed over the outside surface of the wire. 

(b) Find the magnetic field, both inside and outside the wire, if the current is distributed in such a way that the current density J is proportional to s2, where s is the distance from the axis. 

(c) Show that your answers to (a) and (b) are consistent with the magnetostatic boundary conditions (Griffiths Eqn (5.76)) at the outside surface s=a. 

Answer 1

(a) According to Ampere's Law

\(\oint B \cdot d l=\mu I(\) enclosed \()\)

let \(s\) be the distance from the axis of wire to the point where we want to find out the magnetic field

For \(s ; \underline{I}_{\text {enclosed }}=\) zero

\(\oint B \cdot d l=\mu I(\) enclosed \() = \) zero

For s>a;

\(I_{\text {enclosed }}=1\)

\(\oint B . d l=B(2 \pi s)=\mu I\)

\(B=\mu I / 2 \pi s\) along \(\phi\)

(b) \(J=\mathrm{ks}^{2}\)

\(I=\int_{0}^{a} J d a=\int_{0}^{a} k s^{2} s d s d \phi=2 \pi k \int_{0}^{a} s^{3} d s=2 \pi k a^{4} / 4\)

now compute \(I_{\text {enclosed }}\)

Fors;

\(I(\) enclosed \()=\int_{0}^{s} k s^{2} s d s d \phi=2 \pi k s^{4} / 4=I s^{4} / 4\)

\(B=\left(\mu I s^{3}\right) /\left(2 \pi a^{4}\right)\) along \(\phi\)

For s>a;

\(I_{\text {enclosed }}=I\)

\(B=(\mu I) /(2 \pi s)\) along \(\phi\)