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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

enter image description here

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.

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  • $\begingroup$ Please show some effort to solve your problem. Also, see this meta.physics.stackexchange.com/q/714/81224 $\endgroup$ Mar 11 '17 at 9:27
  • $\begingroup$ I did make some progress on it. $\endgroup$
    – Curiosity
    Mar 11 '17 at 9:30
  • $\begingroup$ I used Ampere's circuital law to find the field at a point P when the hole didn't exist. $\endgroup$
    – Curiosity
    Mar 11 '17 at 9:31
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    $\begingroup$ Perhaps you can save yourself some time and think about the magnetic field before the hole was cut out and the magnetic field which the cut out conductor would have produced alone before it was cut out? $\endgroup$
    – Farcher
    Mar 11 '17 at 10:39
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    $\begingroup$ @Curiosity heard of superposition principle? Just like negative mass in centre of mass you can do it here too. $\endgroup$
    – user147979
    Mar 11 '17 at 11:06
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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.

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