Question #43295

A rectangular conducting loop of length $L$ and width $w = \frac{\pi}{2k}$ is placed in the $x-y$ plane, bounded by the lines $x = 0$, $x = \frac{\pi}{2k}$, $y = 0$, and $y = L$. A spatially varying magnetic field $\vec{B}(x,t) = B_0 \cos(kx - \omega t) \hat{k}$ is applied across the region, where $B_0, k, \omega$ are positive constants. The induced EMF $\mathcal{E}(t)$ in the loop is measured with the counter-clockwise direction (viewed from the $+z$ axis) taken as positive. If the induced EMF is expressed in the form $\mathcal{E}(t) = \mathcal{E}_0 \cos(\omega t + \phi)$ with $\mathcal{E}_0 > 0$ and $\phi \in (-\pi, \pi]$, find the ordered pair $(\mathcal{E}_0, \phi)$.