A coaxial cable has an inside conductor of radius 'a' carrying a current 'I'.The current returns as a uniform current through the outside conductor with a inside radius 'b' and outside radius 'c'.What is the magnetic field at a distance 'r' from the center, for the following cases.

r<a, a<r<b, b<r<c, c<r

You need to apply Ampere's law, which for a cylindrical geometry takes the form

B = uo I/(2 pi r)

Where uo is the permeability of free space and I is the total current inside radius r. Apply it in all regions. Assume the current is uniformly distributed in the inner and outer conductors.

To determine the magnetic field at different distances from the center of a coaxial cable, we can use Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space, μ₀.

1. r < a (Inside the inner conductor):
When the distance from the center is less than the radius of the inner conductor, the magnetic field is zero. Therefore, the magnetic field is B = 0.

2. a < r < b (Between the inner and outer conductors):
In this scenario, the closed loop is taken as a circle of radius r. The current enclosed by this loop is the current flowing through the inner conductor. Therefore, using Ampere's law:

∮ B * dl = μ₀ * I_enclosed

Since the current is uniformly distributed throughout the inner conductor, the current enclosed is simply I, the total current flowing through the cable. The perpendicular length dl is equal to the circumference of the circular loop, 2πr. Therefore:

B * 2πr = μ₀ * I

Rearranging the equation, we can find the magnetic field:

B = (μ₀ * I) / (2πr)

3. b < r < c (Between the outer conductor and the cable):
Similar to the previous case, we use Ampere's law to solve for the magnetic field. Now, the current enclosed by the loop is the total current flowing through the cable, since it returns uniformly through the outer conductor. Therefore:

B * 2πr = μ₀ * I

Rearranging the equation gives us:

B = (μ₀ * I) / (2πr)

The magnetic field between the outer conductor and the cable is the same as the magnetic field between the inner conductor and the cable.

4. r > c (Outside the outer conductor):
When the distance from the center is greater than the radius of the outer conductor, there is no current enclosed by the loop. Therefore, the magnetic field is again zero. B = 0.

In summary:
- Inside the inner conductor (r < a): B = 0
- Between the inner and outer conductors (a < r < b): B = (μ₀ * I) / (2πr)
- Between the outer conductor and the cable (b < r < c): B = (μ₀ * I) / (2πr)
- Outside the outer conductor (r > c): B = 0