Question

The maximum emf induced in the coil will be

• A. $\frac{2}{3}\frac{{\pi }^{2}N{B}_0(a^2+ab+b^2)}{T}$
• B. $\frac{{\pi }^{2}N{B}_0(a^2+ab+b^2)}{T}$
• C. $\frac{1}{3}\frac{{\pi }^{2}N{B}_0(a^2+ab+b^2)}{T}$
• D. $\frac{2}{3}\frac{{\pi }^{2}N{B}_0(a^2+b^2)}{T}$

Answer 1

The correct option is A $\frac{2}{3}\frac{{\pi }^{2}N{B}_0(a^2+ab+b^2)}{T}$

The number of turms in he spiral coil per unit radial width $n=\frac{N}{b-a}$

The emf induced in an 'almost circular' part of spiral at radial distance $r$ is

$dV=\frac{d\varphi }{dt}$

$=\frac{d(nB.\pi r^2)}{dt}$

$=n{B}_0\frac{2\pi }{T}cos\left(\frac{2\pi }{T}t\right)\pi r^2$

Since these are almost circular, but actually end to end connected, each of these rings at radial distances varying from $a$ to $b$ would add up to given total emf across the loop.

Hence $V={\int }_a^{b}n{B}_0\frac{2\pi }{T}cos\left(\frac{2\pi }{T}t\right)\pi r^{2}dr$

$=\frac{2}{3}\frac{{\pi }^{2}N{B}_0(a^2+ab+b^2)}{T}cos\left(\frac{2\pi }{T}t\right)$ [Since $(b^3-a^3)=(b-a)(a^2+ab+b^2)$]

Thus the maximum value of the emf is

$=\frac{2}{3}\frac{{\pi }^{2}N{B}_0(a^2+ab+b^2)}{T}$