A cycle wheel is rotated with a speed of 240 rpm in a plane normal to horizontal component of earth's magnetic field of 0.36 G. Find the value of induced current between the rim and axle of the wheel given, there are 16 spokes, each 70 cm long. The resistance of each spoke is \( 2\,\Omega \).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 1.76 \times 10^{-3}\,\mathrm{A} \)
Each spoke rotating in the field is a conductor of length \( l \) rotating about one end, so a motional emf \( \varepsilon = \frac{1}{2} B \omega l^{2} \) appears between the axle and the rim.
\( B = 0.36\,\mathrm{G} = 0.36\times10^{-4}\,\mathrm{T} \)
\( l = 70\,\mathrm{cm} = 0.7\,\mathrm{m} \)
\( \displaystyle \omega = 240\,\mathrm{rpm} = \frac{240 \times 2\pi}{60}\,\mathrm{rad\,s^{-1}} = 8\pi\,\mathrm{rad\,s^{-1}} \approx 25.1\,\mathrm{rad\,s^{-1}} \)
\( \displaystyle \varepsilon = \frac{1}{2} \times 0.36\times10^{-4} \times 25.1 \times (0.7)^{2} \approx 2.2\times10^{-4}\,\mathrm{V} \)
All 16 spokes join the axle to the rim, so they act as 16 identical cells in parallel: the emf between the rim and the axle stays \( \varepsilon \), and the internal resistance is that of 16 spokes of \( 2\,\Omega \) in parallel.
\( \displaystyle R = \frac{2\,\Omega}{16} = 0.125\,\Omega \)
The current is taken for the rim and the axle joined externally by a connecting wire of negligible resistance, so only the spokes oppose it.
\( \displaystyle I = \frac{\varepsilon}{R} = \frac{2.2\times10^{-4}\,\mathrm{V}}{0.125\,\Omega} \approx 1.76\times10^{-3}\,\mathrm{A} \)