The diagram shows a circuit having a coil of resistance $R = 2.5 Ω$ 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 magnitude of this torque (in milli Nm) when current reaches half of its maximum value. Neglect the self-inductance of the loop formed by the circuit.

Question

The diagram shows a circuit having a coil of resistance $R = 2.5 Ω$ 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 magnitude of this torque (in milli Nm) when current reaches half of its maximum value. Neglect the self-inductance of the loop formed by the circuit.

Infinity Learn · Accepted Answer

Induced emf $$=12B$$ω$$l2Maximum$$ current: $$i0=B$$ω$$l22RTorque$$ about the hinge P isτ$$=$$∫$$0li(dx)Bxd$$⇒τ$$=12iBl2Put$$ i $$=$$ $$i0/2$$, we get : τ$$=B2$$ω$$l48R=5$$ mNm

Magnetic flux and induced emf

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Induced emf $= \frac{1}{2} B ω l^{2}$
Maximum current: $i_{0} = \frac{B ω l^{2}}{2 R}$
Torque about the hinge P is
$τ = \int_{0}^{l} i (d x) B x d \Rightarrow τ = \frac{1}{2} i B l^{2}$
Put i = i₀/2, we get : $τ = \frac{B^{2} ω l^{4}}{8 R} = 5 m N m$

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Magnetic flux and induced emf

Remember concepts with our Masterclasses.

Induced emf =12Bωl2Maximum current: i0=Bωl22RTorque about the hinge P isτ=∫0li(dx)Bxd⇒τ=12iBl2Put i = i0/2, we get : τ=B2ωl48R=5 mNm

Ready to Test Your Skills?

free study material

Induced emf $= \frac{1}{2} B ω l^{2}$
Maximum current: $i_{0} = \frac{B ω l^{2}}{2 R}$
Torque about the hinge P is
$τ = \int_{0}^{l} i (d x) B x d \Rightarrow τ = \frac{1}{2} i B l^{2}$
Put i = i₀/2, we get : $τ = \frac{B^{2} ω l^{4}}{8 R} = 5 m N m$