Question

An elliptical loop having resistance R, of semi major axis a, and semi minor axis b is placed in magnetic field as shown in the figure. If the loop is rotated about the x-axis with angular frequency \(\omega\), the average power loss in the loop due to Joule heating is :

• A. \(\frac{\pi ab B\omega}{R}\)
• B. \(\frac{\pi^2 a^2 b^2 B^2 \omega^2}{R}\)
• C. \(\frac{\pi^2 a^2 b^2 B^2 \omega^2}{2R}\)
• D. Zero

Hint:

Flux through the rotating loop is \(B\pi ab\cos\omega t\); find the induced emf and average its square over a cycle.

Answer 1

The Correct Option is C

The field B is along z and the loop (area \(\pi ab\)) lies initially in the xy plane. On rotating about the x-axis, the flux is \(\phi = B\pi ab\cos\omega t\).
Induced emf: \(\varepsilon = -\frac{d\phi}{dt} = \pi abB\omega\sin\omega t\).
Instantaneous power: \(P = \frac{\varepsilon^2}{R} = \frac{\pi^2a^2b^2B^2\omega^2\sin^2\omega t}{R}\).
Average of \(\sin^2\omega t\) over a cycle is \(\frac12\), so \(\langle P \rangle = \frac{\pi^2a^2b^2B^2\omega^2}{2R}\).
Answer: (3) \(\frac{\pi^2 a^2 b^2 B^2 \omega^2}{2R}\)