Question

A thin conducting rod of length $l$ is moved such that its end $B$ moves along the X-axis while end $A$ moves along the Y-axis. A uniform magnetic field $B=B_0\hat{k}$ exists in the region. At some instant, the velocity of end $B$ is $v$ and the rod makes an angle of $\theta =60$ with the X-axis as shown in the figure. Then, at this instant

• A. angular speed of rod $AB$ is $\omega =\frac{2v}{\sqrt{3}l}$
• B. angular speed of rod $AB$ is $\omega =\frac{\sqrt{3}v}{2l}$
• C. e.m.f. induced in rod $AB$ is $Blv\sqrt{3}$
• D. e.m.f. induced in rod $AB$ is $Blv/2\sqrt{3}$

Answer 1

The correct option is A angular speed of rod $AB$ is $\omega =\frac{2v}{\sqrt{3}l}$

$\begin{array}{c}OB=x=l\cos \theta \\ \frac{dx}{dt}=-l\sin \theta \left(\frac{d\theta }{dt}\right)\\ \frac{d\theta }{dt}=-\frac{1}{l\sin \theta }\left(\frac{dx}{dt}\right)\\ \Rightarrow \omega =-\frac{1}{l\sin \theta }\times v_B\\ \Rightarrow \omega =-\frac{1}{l\sin 60}\times v\\ \Rightarrow \omega =-\frac{v}{l\times \sqrt{3}/2}=\frac{-2v}{\sqrt{3}l}\\ so,\omega =\frac{2v}{\sqrt{3}l}\end{array}$