Define the notions of magnetic flux, electromotive force and resistance, in the context of a single closed loop of wire. Use the Maxwell equation

$$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$

to derive Faraday's law of induction for the loop, assuming the loop is at rest.

Suppose now that the magnetic field is $\mathbf{B}=(0,0, B \tanh t)$ where $B$ is a constant, and that the loop of wire, with resistance $R$, is a circle of radius a lying in the $(x, y)$ plane. Calculate the current in the wire as a function of time.

Explain briefly why, even in a time-independent magnetic field, an electromotive force may be produced in a loop of wire that moves through the field, and state the law of induction in this situation.