Why Is the Magnetic Field Uniform in a Non-Coaxial Cylindrical Cavity?

[c299792458]

Homework Statement

We are given an infinitely long cylinder of radius b with an empty cylinder (not coaxial) cut out of it, of radius a. The system carries a steady current (direction along the cylinders) of size I. I am trying to find the magnetic field at a point in the hollow. I am told that the answer is that the magnetic field is uniform throughout the cavity. and is proportional to $d\over b^2-a^2$ where $d$ is the distance between the centers of the cylinders.

The Attempt at a Solution

I have found by using Ampere's law that the magnetic field at a point at distance r from the axis in a cylinder of radius R carrying a steady current, I, is given by $\mu_0 I r\over 2\pi R^2$. So I thought I would use superposition. But what I get is ${\mu_0 I \sqrt{(x-d)^2+y^2}\over 2\pi b^2}-{\mu_0 I \sqrt{(x)^2+y^2}\over 2\pi a^2}$. However this is not the given answer!

 

[M Quack]

You are on the right track, but you have to superpose the magnetic field vectors.

 

[c299792458]

@M Quack: Thank you. I don't know how to change these into vectors, could you please kindly give me another nudge? Thanks again.

 

[M Quack]

The magnetic field generated by a long wire goes right around the wire. So it is perpendicular to the raidal vector.

If the wire is along (0,0,z) and your point at (x,y,z), you know that B_z=0 and that
B is perpendicular to (x,y,0). What vector has these properties?

 

[asteriatic]

I would first like to commend you for utilizing Ampere's law to approach this problem. Your attempt at using superposition is also a valid approach, but it seems that there may be a discrepancy in your calculations.

Upon further examination, I believe the given answer is indeed correct. The magnetic field within the cavity can be thought of as a combination of the magnetic fields from two separate sources: the outer cylinder and the inner cylinder. The field from the outer cylinder can be calculated using Ampere's law, as you have correctly done. However, for the inner cylinder, the distance from the axis to any point within the cavity will always be the same, regardless of the point's position. Therefore, the magnetic field from the inner cylinder will be constant throughout the cavity and proportional to 1/a^2.

To obtain the given answer, you can add the magnetic fields from the two sources using superposition. This will give you a uniform magnetic field throughout the cavity, proportional to d/(b^2-a^2).

In summary, your approach was correct, but there may have been a mistake in your calculations. The given answer is the result of considering the magnetic fields from both the outer and inner cylinders, and taking into account the distance between their centers. I hope this helps clarify the solution for you. Keep up the good work in your scientific endeavors!