A long cylindrical wire kept along Z-axis carries a current of density $\bar{J} = J_{0} r \hat{k}$ where $J_{0}$ is a constant and r is the radial distance from the axis of the cylinder. The magnetic induction B inside the conductor at a distance d from the axis of the cylinder is

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$μ_{0} J_{0}$

$- \frac{μ_{0} J_{0} d^{3}}{4}$

$- \frac{μ_{0} J_{0} d}{2}$

$\frac{μ_{0} J_{0} d^{2}}{3}$

answer is C.

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Detailed Solution

Current density $\bar{J} = j_{0} r \hat{k}$
Current within a distance d $I = \int_{r = 0}^{d} j . d s = \int_{0}^{d} J_{0} r .2 π r d r = 2 π J_{0} \frac{d^{3}}{3}$
From ampere’s Law $\int_{C} \bar{B} . \vec{d l} = μ_{0} 2 π J_{0} \frac{d^{3}}{3}$
Here loop is a circle of radius, r, B is magnetic field at r
$2 π r . B = μ_{0} 2 π J \frac{d^{3}}{3}$
$\Rightarrow B_{r (= d)} = \frac{μ_{0} J_{0} d^{2}}{3}$