In a dynamo, a coil of 200 turns and area 0.1 m2 rotates in a magnetic field of 0.1 Wb m−2 with an angular velocity of 100 rad s−1. The output terminals are connected through a resistance of 100 Ω. What will be the expression for torque τ as a function of time t required to keep the coil rotating?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4 sin2 100 t
The induced current in the coil feels a magnetic torque that opposes the rotation (Lenz's law). To keep the coil rotating uniformly, an external torque equal to this opposing torque must be applied.
The emf of a coil rotating with angular velocity \( \omega \), measuring \( \omega t \) from the position of maximum flux, is
\( \displaystyle \varepsilon = NBA\omega\sin\omega t \)
\( \displaystyle \varepsilon_0 = 200 \times 0.1\,\mathrm{T} \times 0.1\,\mathrm{m^{2}} \times 100\,\mathrm{rad\,s^{-1}} = 200\,\mathrm{V} \)
\( \displaystyle i = \frac{\varepsilon}{R} = \frac{200\sin 100t}{100}\,\mathrm{A} = 2\sin 100t\,\mathrm{A} \)
The torque on a coil carrying current i, whose normal makes angle \( \omega t \) with B, is
\( \displaystyle \tau = NiAB\sin\omega t \)
\( \displaystyle \tau = 200 \times 2\sin 100t \times 0.1 \times 0.1 \times \sin 100t\,\mathrm{N\,m} \)
\( \displaystyle \tau = 4\sin^{2}100t\,\mathrm{N\,m} \)
The same result follows from energy conservation: the mechanical power \( \tau\omega \) equals the electrical power \( \varepsilon i \).
\( \displaystyle \tau = \frac{\varepsilon i}{\omega} = \frac{200\sin 100t \times 2\sin 100t}{100} = 4\sin^{2}100t\,\mathrm{N\,m} \)
Note: NTA dropped this question in the Final Answer Key, probably because the correct torque, \( 4\sin^{2}100t \) N m, was not among the options; Option (2) originally read “4 sin 100 t”. Option (2) has been changed to “4 sin2 100 t” so that the question has one clear answer.