EMF dependence on certain variables (radius, loops and velocity)

Question

So imagine we had a solenoid with N loops and radius r, then we dropped a magnet through the solenoid from a height of of h above the solenoid. Now when the magnet goes through the coil an emf is produced.

Now I am trying to work out the relationship between emf and r, N and h $$|\varepsilon| = N\frac{d\Phi_B }{dt} $$
Since we know the magnetic flux $\Phi_B=BA = B\pi r^2 $ $$|\varepsilon| =N \frac{d(B\pi r^2) }{dt} =N\pi r^2 \frac{dB }{dt} $$ Then using the chain rule $\frac{dB }{dt}=\frac{dB }{dx} \frac{dx }{dt}$. Hence $$|\varepsilon| =N\pi r^2 \frac{dB }{dx} \frac{dx }{dt}$$ But since the magnet was dropped from height h $\frac{dx }{dt}= v= \sqrt{2gh}$ Finally we can say $$|\varepsilon| =N\pi r^2 \sqrt{2gh} \frac{dB }{dx} $$

However I don't feel like this makes a lot of sense since it is kind of saying that the larger we make the radius of the solenoid the greater the emf but i feel like the opposite should happen (i.e. the bigger the radius the smaller the emf). Can some please explain what I have done wrong or explain how the emf is related to these variables?

Answer 1

The analysis you are attempting is much more complex because you have made some assumptions which are not realistic.

The first is that the magnet produces a magnetic field which is constant.
This is not true as by a simple experiment you can show the attractive power of a magnet for a piece of iron depends on the separation between the iron and the magnet.

So what happens is that the magnet is released and accelerated because of the gravitational attraction due to the Earth.

As the magnet gets closer to the solenoid the magnetic flux linked with the coil increases in a very non-linear way and thus an emf is induced in the solenoid.
The closer the magnet is to the solenoid the faster it is going and the greater is the magnetic field due to the magnet in the vicinity of the solenoid, so the emf increases.

AS the magnet enters the solenoid the rate of change flux linkage with the solenoid will start to decrease and so after passing through a maximum the induced emf will start to decrease even though the magnet is travelling faster.
Remember it is the rate of change of flux linkage which is important as the magnet passes though the solenoid less magnetic field lines are cut.

There will come a time when the magnet is at the centre of the solenoid when the rate of change of magnetic flux and hence the induced emf becomes zero.

Then the magnet start to "leave" the solenoid and the process described above is mirrored but with the emf now in the opposite direction becasue now the magnetic flux linked with the coil is decreasing.
The maximum emf during this phase will be larger that before because the magnet is travelling faster.

But that is not all you have to worry about in that the induced emf may induce a current if there is a complete conducting circuit.

So suppose the ends of the solenoid have a resistor connected across them and an induced current is produce.
Lenz tells you that that induced current will oppose the motion producing it and so the induced current will produce a magnetic field which will try and slow down the magnet.
This will mean that the acceleration of the magnet is no longer the acceleration of free fall but low and magnitude of the acceleration will depend on the position of the magnet within the solenoid.
A large induced emf will produce a large induced current and hence more opposition (force) to the downward fall of the magnet.

The slowing down effect will depend on the amount of resistance in the solenoid circuit.
If there ends of the solenoid did not have a resistor across them then there is still, albeit a very small, induced current in the copper windings of the solenoid which are called eddy currents.
However as these eddy currents are small the acceleration of the magnet as it falls though the magnet will be very close to the acceleration of free fall.

Graphically you might get something like this:

If eddy currents are significantly like this very strong neodymium magnet falling through a single turn "solenoid" the progress of the magnet through the solenoid is almost at constant speed as shown in this video.

So for a simple analysis you can probably only give an idea of how the emf changes as you variables, radius, loops and velocity, change rather that being able to evaluate the actual value of the induced emf.

Answer 2

The error you've made which led to the radius-squared in your answer is in assuming that the magnetic field from the falling magnet, $B$, is constant with $r$ in $\Phi_B=BA = B\pi r^2$. The magnetic field is not constant with $r$ but decreases with increasing $r$.

Lenz's law also tells us that the velocity is not determined by gravity alone, because the induced current in the solenoid produces a magnetic field which will slow the solenoid down.

There are some other things going wrong here too, but perhaps we can leave those for another opportunity.