An elliptical loop having resistance $$R$$, of semi major axis $$a$$, and semi minor axis $$b$$ is placed in a 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:

Area of the elliptical loop: $$A = \pi a b$$

Magnetic flux through the loop at any time $$t$$: $$\phi = \vec{B} \cdot \vec{A} = B A \cos(\omega t) = B \pi a b \cos(\omega t)$$

Induced electromotive force by Faraday's law: $$\varepsilon = -\frac{d\phi}{dt} = B \pi a b \omega \sin(\omega t)$$

Peak induced electromotive force: $$\varepsilon_0 = \pi a b B \omega$$

Average power loss over a complete cycle: $$P_{\text{avg}} = \frac{\varepsilon_0^2}{2R}$$

$$P_{\text{avg}} = \frac{(\pi a b B \omega)^2}{2R} = \frac{\pi^2 a^2 b^2 B^2 \omega^2}{2R}$$