Question

$N$ turns are tightly wound to form a spiral plane of outer radius $a$. If the magnetic induction $B=B_0\sin \omega t$ varies perpendicular to the plane of spiral then find the emf induced in the spiral.

• A. $\frac{\pi a^2{B}_0\omega N\cos \omega t}{2}$
• B. $\frac{\pi a^2{B}_{0}N\omega \cos \omega t}{4}$
• C. $\pi a^2{B}_{0}N\omega \cos \omega t$
• D. $\frac{\pi a^2{B}_{0}N\omega \cos \omega t}{3}$

Answer 1

The correct option is D $\frac{\pi a^2{B}_{0}N\omega \cos \omega t}{3}$

Let the hypothetical elementary loop have radius $x$. Then emf induced in this loop $d\epsilon =\frac{AdB}{dt}=\pi x^2\frac{d\left(B_0\sin \omega t\right)}{dt}dN=\pi x^2{B}_0\omega \cos \omega t\left(\frac{N}{a}\right)dx$

or $\epsilon ={\int }_0^a{B}_0\pi \omega \frac{N}{a}\cos \omega t{x}^{2}dx=\frac{\pi a^2{B}_0\omega N\cos \omega t}{3}$