Question #43076

A long straight wire carries an alternating current $I(t) = I_0 \sin(\omega t)$, where $I_0$ is the current amplitude in Amperes and $\omega$ is the angular frequency in rad/s. A rigid rectangular conducting loop of side lengths $a$ and $b$ (in meters) is placed in the same plane as the wire. The side of length $b$ is parallel to the wire and is at a distance $d$ (in meters) from it. The loop has a constant self-inductance $L$ (in Henries). Due to high-frequency skin effects, the effective resistance of the loop is strictly proportional to the angular frequency, given by $R = \alpha \omega$, where $\alpha$ is a positive constant in Henries. Let $M = \frac{\mu_0 b}{2\pi} \ln\left(1 + \frac{a}{d}\right)$ be the mutual inductance (in Henries) between the wire and the loop. What is the total electrical energy $E$ dissipated as heat in the loop during exactly one complete cycle of the alternating current?