Answer to Question #216369 in Electricity and Magnetism for Sandeep

Question

Consider a square loop of wire, of side length l, lying in the xy−plane as shown in
figure. Suppose a particle of charge q is moving with velocity v, where v << c, in the
xz−plane at a constant distance z0 from xy−plane (Assume the particle is moving
in the positive x−direction). Suppose the particle crosses the z−axis at t=0. Give
the induced emf in the loop as a function of time.

Answer 1

Gives

Square loop current =I

Length=l

Plane=x-yplane

$\phi=B.A=BAcos\theta\\\smallint d\phi=\smallint_{z_0}^{z_0+l}\frac{\mu_0 2I}{4 \pi x}adx \\\phi=[\frac{\mu_0 2Il}{4\pi }]_{z_0}^{z_0+l}\\\phi =\frac{\mu_0 Il}{2 \pi}ln \frac{z_{0+l}}{z_0}$

Induced emf

$e=-\frac{d\phi}{dt}=-\frac{\mu_0Il}{2\pi}\frac{l}{l+z_0}\frac{dl}{dt}$