Figure shows a situation similar to the previous problem. All par… - Vidyadip

Question

Figure shows a situation similar to the previous problem. All parameters are the same except that a battery of emf $\in$ and a variable resistance R are connected between O and C. The connecting wires have zero resistance. No external force is applied on the rod (except gravity, forces by the magnetic field and by the pivot). In what way should the resistance R be changed so that the rod may rotate with uniform angular velocity in the clockwise direction? Express your answer in terms of the given quantities and the angle $\theta$ made by the rod OA with the horizontal.

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Answer

$\text{emf}=\frac{1}{2}\text{B}\omega\text{a}^2$ [from previous problem]
Current $= \frac{\text{e}+\text{E}}{\text{R}}=\frac{\frac{1}{2}\times\text{B}\omega\text{a}^2+\text{E}}{\text{R}}$
$=\frac{\text{B}\omega\text{a}^2 +2\text{E}}{2\text{R}}$
$\Rightarrow \text{mg} \cos \theta =\text{ilB}$ [Net force acting on the rod is O]
$\Rightarrow \text{mg}\cos\theta =\frac{\text{B}\omega\text{a}^2+2\text{E}}{2\text{R}}\text{a}\times\text{B}$
$\Rightarrow \text{R}=\frac{\big(\text{B}\omega\text{a}^2+2\text{E}\big)\text{ab}}{2\text{mg}\cos\theta}$

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