Question

An electron is moving in a circular orbit of radius R with an angular acceleration $\alpha$. At the centre of the orbit is kept a conducting loop of radius r, (r <<R). The e.m.f induced in the smaller loop due to the motion of the electron is

Answer 1

$\frac{\mu_0 e r^2 \alpha}{4R}$

Answer 2

The electron's circular motion with angular acceleration implies a changing angular velocity, which in turn means a changing current. This changing current produces a changing magnetic field at the center of the orbit. According to Faraday's law, a changing magnetic flux through the small conducting loop placed at the center induces an electromotive force (e.m.f.). The current due to the electron is $I = \frac{e\omega}{2\pi}$. The magnetic field at the center is $B = \frac{\mu_0 I}{2R} = \frac{\mu_0 e \omega}{4\pi R}$. The magnetic flux through the small loop of radius r is $\Phi_B = B \cdot (\pi r^2) = \frac{\mu_0 e \omega r^2}{4R}$. The induced e.m.f. is $\mathcal{E} = -\frac{d\Phi_B}{dt} = -\frac{\mu_0 e r^2}{4R} \frac{d\omega}{dt}$. Since $\alpha = \frac{d\omega}{dt}$, the induced e.m.f. is $\mathcal{E} = -\frac{\mu_0 e r^2 \alpha}{4R}$.