A clever use of superposition should can make seemingly complicated situations e

Question

A clever use of superposition should can make seemingly complicated situations easier to solve. A long (infinite) wire (cylindrical conductor of radius R, whose axis coincides with the z axis) carries a uniformly distributed current I_0 In the + z direction. A cylindrical hole is drilled out of the conductor, parallel to the z axis, (see figure above for geometry). The center of the hole is at x = b, and the radius is a. Determine the magnetic field in the hole region. If this is an ordinary wire carrying ordinary household currents, and the drilled hole has dimensions roughly shown to scale in the figure above, make an order of magnitude estimate for the strength of the B field in that region. How does it compare to the earth's field? You should find that the B field in the hole is uniform - that was just a little surprising to me!

Explanation / Answer

1) its similar to a case when there are two cylinders. We can consider hole as carrying current in negative direction.

B= uoi enclosed/ 2 pib + B due to hole(i.e. negative current)

   = uo* io*[R^2/(R^2-a^2)]/(2 pi b) + 0 as negative current produces no field at its center

= uo* io*[R^2/(R^2-a^2)]/(2 pi b)

2)it is not possible to scale as size of image differs on jooming, just scale it and put the values in above equation.

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