Question

A wire loop enclosing a semicircle of radius $R$ is located on the boundary of uniform magnetic field $B$. At time $t=0$, the loop is set into motion with constant angular accleration $\alpha$ about an axis passing through $O$ and is perpendicular to semicircle. The clockwise emf direction is taken to be positive. The modulus of variation of emf as a function of time is ($0<\theta <\pi$)

• A. $\frac{B{R}^2\alpha t}{2}$
• B. $\frac{3B{R}^2\alpha t}{2}$
• C. $\sqrt{3}B{R}^2\alpha t$
• D. $\frac{B{R}^2\alpha t}{\sqrt{2}}$

Answer 1

The correct option is A $\frac{B{R}^2\alpha t}{2}$


Let the loop is at some arbitrary angle $\theta$.
As area vector and magnetic field is parallel, therefore flux through loop at this instant, $\varphi =B\frac{\theta R^2}{2}\text{cos}0$
$\varphi =B\frac{\theta R^2}{2}$
$ϵ=-\frac{d\varphi }{dt}=-\frac{B{R}^2}{2}\frac{d\theta }{dt}$
$ϵ=-\frac{B{R}^2\omega }{2}$.........(1)
As the loop is rotating with an angular acceleration $\alpha$
$\omega ={\omega }_0+\alpha t$
But ${\omega }_0=0$
$\omega =\alpha t$
Substituting value of $\omega$ in (1), we get
$ϵ=-\frac{B{R}^2\alpha t}{2}$