Write down Faraday's law of electromagnetic induction for a moving circuit $C(t)$ in a magnetic field $\mathbf{B}(\mathbf{x}, t)$. Explain carefully the meaning of each term in the equation.

A thin wire is bent into a circular loop of radius $a$. The loop lies in the $(x, z)$-plane at time $t=0$. It spins steadily with angular velocity $\Omega \mathbf{k}$, where $\Omega$ is a constant and $\mathbf{k}$ is a unit vector in the $z$-direction. A spatially uniform magnetic field $\mathbf{B}=B_{0}(\cos \omega t, \sin \omega t, 0)$ is applied, with $B_{0}$ and $\omega$ both constant. If the resistance of the wire is $R$, find the current in the wire at time $t$.