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Question

A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magnitude of the magnetic field at the point P is given by N12μ0aJ, then the value of N is

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Solution

The magnetic field for an infinitely long cylinder is given by, Bsurface=μoJR2
Bout=μoJR22r
r= distance from the axis of the cylinder.
R= Radius of the cylinder.


Assuming the bigger cylinder to carry a positive current density and the smaller cylinder carry a negative current density of magnitude J each.


Net Magnetic field
B=B1(2a diameter)+B2(a diameter)
Therefore, magnitude of magnetic field at point P
B=μ0(J×a)2μ0(J×a24)2×3a2=512μ0Ja
Thus, on comparing with given relation
N=5

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