We now want to figure out what the magnetic field will be for a coaxial cable. T

Question

We now want to figure out what the magnetic field will be for a coaxial cable. This cable has an inner wire with radius R carries a total current I into the page, the current density is uniform throughout the wire. There is an cylindrical conducting shell with inner radius R_in and outer radius R_out. The same total current as the inner wire is going in the opposite direction. What is the magnetic field in all regions of space (inside the wire, between the wire and shell, inside of the conducting shell and outside of the cable.) There are different ways to do this problem, any technique may be used.

Explanation / Answer

for 0<r<R:

current through the cylindrical surface of radius r=I*(r^2/R^2)

then using ampere's law:

if magnetic field intensity is H

H*2*pi*r=current enclosed=I*(r^2/R^2)

==>H=I*r/(2*pi*R^2)

magnetic field=mu*H=mu*I*r/(2*pi*R^2)

for R<r<Rin:

current enclosed=I

if magnetic field intensity is H,

H*2*pi*r=I

=>H=I/(2*pi*r)

magnetic field=mu*H=mu*I/(2*pi*r)

for Rin<r<Rout:

current enclosed =I-(I*(r^2-Rin^2)/(Rout^2-Rin^2))

if magnetic field intensity is H,

using ampere's law:

H*2*pi*r=I-(I*(r^2-Rin^2)/(Rout^2-Rin^2))=I*(Rout^2-r^2)/(Rout^2-Rin^2)

==>H=I*(Rout^2-r^2)/(2*pi*r*(Rout^2-Rin^2))

magnetic field=B=mu*H

=mu*I*(Rout^2-r^2)/(2*pi*r*(Rout^2-Rin^2))

for r<Rout:

current enclosed=0

so magnetic field=0

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