Question

The value of time when the current reverse its sign for the first time will be

• A. T/2
• B. T/4
• C. T/6
• D. T/8

Answer 1

The correct option is B T/4

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

consider a concentric circular ring of radius x and radial thickness dx, The number turns in the ring
$N=ndx$
The flux linked with the ring $\varphi =\pi x^{2}B$

$=x^2{B}_0\sin \frac{2\pi t}{T}$

The emf induced in this ring $\mid e\mid =N^{\prime }\frac{d\phi }{dt}=\left(\frac{N}{(b-a)}dx\right)\pi x^2{B}_0\sin \frac{2\pi t}{T}$

$=\left(\frac{N}{(b-a)}\pi x^2{B}_{0}dx\right)\frac{2\pi }{T}.\cos \frac{2\pi t}{T}$

toatal e.m.f induced in the coil, E=${\int }_a^b\frac{2{\pi }^2}{T}.\frac{N{B}_0}{(b-a)}\cos \frac{2\pi t}{T}{x}^{2}dx$

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

$E=E_0\cos \frac{2\pi t}{T}$

$E_0=Maximumemf\frac{2}{3}\frac{{\pi }^{2}N{B}_0(a^2+ab+b^2)}{T}$

The current through the coil $i=\frac{E}{R}=\frac{E_0}{R}\cos \frac{2\pi t}{T}$

The current will be maximum when t=0 and t=T/2

Maximum current $i_{max}=\frac{E_0}{R}=\frac{2{\pi }^{2}N{B}_0(a^2+ab+b^2)}{3RT}$

The current will reverse its direction when i=0 it happen at t=T/4