Question

Consider a cylindrical coaxial cable. The center conductor is solid with radius R, while the outer conductor is hollow with inner radius a and outer radius b, Both conductors have the same total current I (of uniform current density), but moving in the opposite direction of each other. Find the magnitude of the magnetic field as a function of r (where r=0 at the central axis) in the interval a is less than or equal to r is less than b.

Note: your final equation should be in terms of the given parameters (r, a, I, etc.) and any other known constants.

Answer 1

The Magnetic field can be found ou using Ampere's law.

Where B is Magnetic field vector (T)

dl is a length element (m)

is total current enclosed in the region.

Let the current flowing in the core is represented in the positive direction(flowing out of paper) and current through the hollow part in negative direction (into the paper)

The magnetic field lines will be in concentric circles around the cylinder.

We have two co-axial cylinders.

The solid cylinder has radius R and carries a current I which flows out of paper.

The hollow cylinder has inner radius a and outer radius b.

They are insulated from each other.

Let Curve C be the region of our interest.

This region has radius r from the axis of the cylinder.

This region also encloses the core cylinder and inner part of the hollow cylinder.

The total current enclosed by the curve c (blue region) is current I flowing through the core cylinder plus the current flowing in the small part across the hollow cylinder which we have to find using current density.

To get that current, we find the current density in the hollow cylinder which is

and multiply this by area bounded by our curve C to get the required current.

Using Ampers's Law,

The B vector acts tangentially to the curve such that the dot product between the lenth element and magnetic field is 1 or we can say that angle between the magnetic field and length element is 0 degrees.