An infinitely long uniform solid wire of radius a carries a uniform DC current of density J.
A hole of radius b (b < a) is now drilled along the length of the wire at a distance d from the center of the wire.
The magnetic field inside the hole is(A) uniform and depends only on d
(B) uniform and depends only on b
(C) uniform and depends on both b and d
(D) non uniform
Figure source: https://qph.ec.quoracdn.net/main-qimg-951b32724c2858cd9823dcf535fbd49a
As far as my approach goes, the progress I made is:
a) I used Ampere's circuital law to find field B at a point P when the hole didn't exist.
By using it, $H 2\pi R = J\pi R^2$, where $R$ is the distance from the origin to a point P.
b) Now, once the hole is drilled, a part of the cross section becomes hollow. I wonder how to approach the problem from hereon.
Will the hollow portion be able to have any current flowing through it ? If no, there won't be any magnetic field inside it.
But, the options clearly don't suggest such a possibility.
Can a hollow portion of a wire support any current? The possibility doesn't seem logical, but if there's no current in the hollow portion, how can there be a magnetic field at a point within it? Maybe it is due to the current in the rest of the solid cross section. I would like a clarification in this regard.
Best Answer
First, consider that current I flows inside the cylinder without hole in it. After the drill, its like the current is reduced for the whole cylinder. The reduction is equivalent of saying the current I' is flowing in the opposite direction to I.Now the whole setup is reduced to two infinite wires carrying current I and I' separated by distance d. Hence we can calculate magnetic field between these wires. The resulting magnetic field will be the field inside the hole.