There exists a uniform magnetic field perpendicular to the plane of the figure in a cylindrical region of radius $a$ . The magnetic field is increasing at a constant rate of $α T s^{- 1}$ . A particle having charge $q$ is at a point P outside the field region. The particle is slowly moved to infinity. $α > 0$ .The work done by external agent has a magnitude $k (q a^{2} α)$ along PQ, then $k$ is_______

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Detailed Solution

$E (2 π \sqrt{x^{2} + d^{2}}) = π a^{2} (α)$

$\Rightarrow E = \frac{a^{2} (α)}{2 \sqrt{x^{2} + d^{2}}}$

$W_{e x t} = \int E q \cos θ d x = \int_{0}^{\infty} \frac{a^{2} (α)}{2 \sqrt{x^{2} + d^{2}}} q \times \frac{d}{\sqrt{x^{2} + d^{2}}} \times d x$

$\Rightarrow W_{e x t} = \frac{a^{2} (α) q d}{2} \int_{0}^{\infty} \frac{d x}{x^{2} + d^{2}} = \frac{a^{2} (α) q d}{2} \int_{0}^{\frac{π}{2}} d θ = \frac{a^{2} (α) q π}{4}$

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