Time varying magnetic field, yielding two effects of induction?

Question

$B_s$ is nonuniform, and it's generated from a movable source(e.g magnet or electromagnet).

A rectangular loop of area $A$ is stationary.

The variation of flux for this case is caused from the following:

1. Spatial movement of $B_s$.
2. Strength variation $B_s$ of due to (1).

How can I quantitatively formulate $\varepsilon_{induced}$ here?

I could assume that:

$$\varepsilon_{induced} =\oint \vec{E} \cdot \vec{dl} = -\frac{\delta \Phi}{\delta t}$$

to:

$$\varepsilon_{induced} =\oint \vec{E} \cdot \vec{dl} = -\int \frac{\delta B_s}{\delta t} \cdot \vec{da}$$

But how can I factor in the effects of 1&2 to $\delta B_s$?

From my textbook,the examples brought up had the effects(1&2) mutually exclusive, but for this case both add to $\varepsilon_{induced}$

Answer 1

Faraday-Lenz's law says that

$$\varepsilon_{ind}=-\frac{\partial\Phi_B}{\partial t}$$

In your case, the flux is variying due to the change in two factors. Recall that $\Phi_B=B\cdot S\cdot\cos(\alpha)$, so the derivative would be

$$\frac{\partial\Phi_B}{\partial t}=\frac{\partial B}{\partial t}\cdot S\cdot\cos(\alpha)+B\frac{\partial S}{\partial t}\cos(\alpha)$$