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I have recently carried out an experiment to verify Faradays law for a falling magnet. My starting point was to keep both the area of the coil and the number of turns constant whilst changing the velocity (the different velocities were obtained by dropping from different heights).What would be good is if a graph of emf induced vs. $dB/dt$ could be plotted so that the gradient of the graph will be equal to the product of the area and number of turns. From $e = -nA(dB/dt)$

In short, is there an equation to change velocity to $dB/dt$?

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The problem is that the magnetic field around a bar magnet aren't uniform, so different parts of your coil will experience different values of $B$. If the magnetic is small enough, you can approximate it's field by a dipole,

$\mathbf{B}({\mathbf{r}})=\frac{\mu_{0}}{4\pi}\left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{r^{5}}-\frac{{\mathbf{m}}}{r^{3}}\right)$

(here $\mathbf{m}$ is a constant vector equal to the dipole moment).

Then, supposing your coil is very flat (not really a solenoid), you can calculate what $\frac{\partial\mathbf{B}}{\partial t}$ is when $\mathbf{m} \cdot \mathbf{r} = 0$ , i.e., at the moment the magnet passes through the coil like a basketball falling through a hoop. Of course, $\mathbf{m}\cdot \dot{ \mathbf{r}} \neq 0$. The peak EMF should then be somewhat close to what you would predict from experiment.

A lot of early experiments had to be really clever to isolate exactly what was going on. It really depends on what apparatuses are available to you.

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