# When a ‘J’ shaped conducting rod is rotating in its own plane with constant angular velocity ω about one of its ends P, in a uniform magnetic field $\vec{B}$ (directed normally into the plane of paper) then magnitude of emf induced across it will be Option 1) $Bw\sqrt{L^{2}+l^{2}}$ Option 2) $\frac{1}{2}BwL^{2}$ Option 3) $\frac{1}{2}Bw\left ( L^{2}+l^{2} \right )$ Option 4) $\frac{1}{2}Bwl^{2}$

As learnt in

Motional E.m.f due to rotational motion -

Conducting rod $\rightarrow$

$\varepsilon =\frac{1}{2}Bl^{2}\omega =Bl^{2}\pi\nu$

$\nu \rightarrow f\! r\! equency$

$T \rightarrow Time\; period$

- wherein

This system can be replaced by a single rod of length = $\sqrt{l^{2}+L^{2}}$

$\therefore$  E.m.f induced = $\frac{1}{2}B\omega \left ( l^{2}+L^{2} \right )$

Option 1)

$Bw\sqrt{L^{2}+l^{2}}$

This option is incorrect

Option 2)

$\frac{1}{2}BwL^{2}$

This option is incorrect

Option 3)

$\frac{1}{2}Bw\left ( L^{2}+l^{2} \right )$

This option is correct

Option 4)

$\frac{1}{2}Bwl^{2}$

This option is incorrect

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