An infinitely long current carrying thick cylindrical conductor, has a cylindrical cavity of radius $\frac{R}{2}$ . The axis of the cylindrical cavity is at a distance $\frac{R}{2}$ from the axis of the cylindrical conductor. A current with current density J flows along the length of the cylindrical conductor(without cavity). The magnetic field at a point ‘P’at a distance 2R from the axis of the cylinder is (point ‘P’ is on the plane containing both the axes)

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Magnetic field on the axis of the cavity is non-zero

Magnetic field at P is $\frac{μ_{0} J R}{12}$

Magnetic field on the axis of the conductor is non-zero

Magnetic field at P is $\frac{μ_{0} J R}{6}$

answer is B, C, D.

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Detailed Solution

When we fill the cavity with same current density, then magnetic field at
$P = B_{1} = \frac{μ_{0}}{2 π} \times \frac{I}{2 R}$ Direction is into the plane
$= \frac{μ_{0}}{4 π} \frac{J \times π R^{2}}{2 R} J = \frac{I}{π R^{2}} B_{1} = \frac{μ_{0} J R}{4} - - - - (1)$
Take cavity portion alone.
Question Image

Dir. into the plane

$∴ B = B_{1} - B_{2} = \frac{μ_{0} J R}{4} [1 - \frac{1}{3}]$

$B = \frac{μ_{0} J R}{6} T e s l a$

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