Question

Figure below shows a conducting rod of negligible resistance that can slide on a smooth U-shaped rail made of a wire of resistance $1\Omega m^{-1}$. Position of the conducting rod at $t=0$ is shown. A time dependent magnetic field $B=2T\text{(tesla)}$ is switched ON at $t=0$.
At $t=0$, when the magnetic field is switched ON, the conducting rod is moved to the left at a constant speed of $5{\text{cms}}^{-1}$ by some external means. The rod moves perpendicular to the rail. At $t=2s$, induced emf has magnitude

• A. 0.12 V
• B. 0.08 V
• C. 0.04 V
• D. 0.02 V

Answer 1

The correct option is B 0.08 V

From Faraday's law induced emf, $E=-\frac{d\varphi }{dt}$
Since flux, $\varphi =B.A$
After $t$ sec if rod is moving with velocity $v$ area, $A=0.2\times (0.4-v\times t)$
Also magnetic field, $B=2t$
Therefore flux, $\varphi =B.A=2t\times 0.2\times (0.4-v\times t)$
$\Rightarrow \varphi =0.16t-0.4v{t}^2$
$\frac{d\varphi }{dt}=0.16-0.8vt$
At $t=2$ sec and $v=5$ cm/s = $0.05$ m/s
$\frac{d\varphi }{dt}=0.16-0.8\times 0.05\times 2=0.08$
$\Rightarrow E=-\frac{d\varphi }{dt}=-0.08V$
Since only magnitude is asked, therefore induced emf is $0.08V$