A metallic rod of length $l$ and resistance R is free to rotate about one of its ends over a horizontal smooth, rigid circular metallic frame of radius $l$ in an inward magnetic field of induction B. The circuit is completed by means of a conducting wire connected between the centre O and the point on the right. If the required torque to maintain constant angular velocity $ω$ is $\frac{n B^{2} l^{4} ω}{R},$ find value of n.

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Detailed Solution

The torque applied by the magnetic force $F_{m}$ about O is given as $d τ_{m} = r (d F_{m}) = r {i (d r) B} \Rightarrow τ_{m} = \int d τ_{m} = \int r (i d r B);$

Putting $i = \frac{E}{R} = \frac{B l^{2} ω}{2 R}$

We obtain, $τ_{m} = (\frac{B^{2} l^{2} ω}{2 R}) \int_{0}^{l} r d r$

$τ_{m} = \frac{B^{2} l^{2} ω}{2 R} {\frac{r^{2}}{2} |}_{0}^{l} \Rightarrow τ_{m} = \frac{B^{2} l^{4} ω}{4 R}$

$∴$ The external torque must be equal and opposite to the magnetic torque $\Rightarrow τ_{e x t} = \frac{B^{2} l^{4} ω}{4 R}$

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