motion of a conductor in a varying magnetic field, is motional emf still valid?

Question

If a conductor moves within a uniform constant magnetic field, the magnetic force would separate the charges leading to an induced emf. And if the conductor was stationary, and the magnetic field were to change,from Maxwell's equations I understood that there would be a nonelectrostatic electric field that would yield the to the induced emf.

What if the conductor moved, and the exterior magnetic field was varying with time(increasing/decreasing). Is motional emf in it's basic form valid?

$$ \varepsilon = -Bl\nu $$

Answer 1

The motional emf will not remain same.

Emf is defined as

$$\xi=\oint \vec{f}_{mag}.\vec{dl}$$

where $f_{mag}$ is the total force per unit charge which drives the current around the circuit. This can be due to the battery or can be due to a non electrostatic electric field or a magnetic field. The source cannot be an electrostatic field since the line integral of an electrostatic field over the entire circuit is $0$ since it is a conservative field.

For the simple case of a conductor moving in a magnetic field, $\vec{f}_{mag}=\vec{v} \times\vec{B}$.

Now for the case of a conductor moving in a time varying magnetic field, $\vec{f'}_{mag}=\vec{v} \times\vec{B(t)}+\vec{E}$, where $\vec{\nabla}\times\vec{E}=-\frac{\partial \vec{B(t)}}{\partial t}$.

Hence the total induced emf will be,

$$\xi=\oint \vec{f'}_{mag}.\vec{dl}$$.

If you still have doubts regarding induced emf, you can refer to the book by David J. Griffiths.