Subjective Type

Figure shows a circuit having a coil of resistance $$R=2.5 \Omega$$ and inductance 'L' connected to a conducting rod PQ which can slide on a perfectly conducting circular ring of radius 10 cm with its centre at 'P'. Assume that friction and gravity are absent and a constant uniform magnetic field of 5 T exists as shown in figure. At t =0,the circuit is switched on and simultaneously a time varying external torque is applied on the rod so that it rotates about 'P' with a constant angular velocity 40 rad/s.Find the magnitude of this torque ( in 'P') when current reaches half of its maximum value.Neglect the self inductance of the loop formed by the circuit.

Solution

Induced EMF =$$\dfrac{1}{2}B \omega l^2$$

Maximum current : $$i_0=\dfrac{B \omega l^2}{2R}$$

Torque about the hinge P is

$$ \tau =\int^l_0 i(dx)Bx$$

$$\Rightarrow \tau =\dfrac{1}{2}iBl^2$$

Putting $$i=i_0/2,$$

we get ; $$\tau =\dfrac{B^2 \omega l^4}{8R}=5 mNm$$

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