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Question

(B) The motional electromotive force $(EMF)$ induced in a rod of length $r$ rotating with angular velocity $\omega$ in a uniform magnetic field $B$ is

Vedclass · Accepted Answer

$|V_O - V_C| = \frac{7}{2} B \omega l^2$

Vedclass · Answer

(B) The motional electromotive force $(EMF)$ induced in a rod of length $r$ rotating with angular velocity $\omega$ in a uniform magnetic field $B$ is given by $\varepsilon = \frac{1}{2} B \omega r^2$.For segment $AO$ with length $r = l$:$|V_A - V_O| = \frac{1}{2} B \omega l^2$.For segment $OC$ with length $r = 3l$:$|V_O - V_C| = \frac{1}{2} B \omega (3l)^2 = \frac{1}{2} B \omega (9l^2) = \frac{9}{2} B \omega l^2$.Since the rod is rotating,the induced EMFs in segments $AO$ and $OC$ act in opposite directions relative to point $O$. Thus,the potential difference between $A$ and $C$ is:$|V_A - V_C| = |(V_A - V_O) + (V_O - V_C)| = |\frac{1}{2} B \omega l^2 - \frac{9}{2} B \omega l^2| = |-4 B \omega l^2| = 4 B \omega l^2$.Comparing these results with the given options:Option $A$ is correct.Option $B$ is incorrect because $|V_O - V_C| = \frac{9}{2} B \omega l^2$,not $\frac{7}{2} B \omega l^2$.Option $C$ is correct.Option $D$ is correct.

$A$ conducting rod $AC$ of length $4l$ is rotated about a point $O$ in a uniform magnetic field $\vec{B}$ directed into the paper. $AO = l$ and $OC = 3l$. Then which of the following is incorrect?

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