Motional electrodynamics

One broad class of problems in electrodynamics involves bodies/particles with a time-varying position. From the Lorentz force law, we have:

$${\vec{F}}_m=q(\vec{v}\times \vec{B})$$

The magnetic force exerted on the particle is equivalent to the electric force, such that the motion of the particle/body produces a motional electric field.

$${\vec{E}}_m=\frac{{\vec{F}}_m}{q}=\vec{v}\times \vec{B}$$

This in turn induces a voltage difference, called the motional emf. This is defined as the line integral between two ends of the wire:

$$V_{\text{emf}}^m=V_{12}={\int }_2^1(\vec{v}\times \vec{B})\cdot d\vec{l}$$

For a closed circuit (i.e., a closed contour), we have a closed contour integral that describes the induced motional emf. Only segments of the circuit that cross magnetic field lines contribute to the motional emf.

Computations

We usually want to figure out the motional emf. In this case,