Question

I am trying to the max voltage produced from a magnetic being dropped through a wire with a relationship on the height the magnet is dropped from. I know that emf=-V= N(partial derivative of magnetic flux). But I am not sure how to relate this to the height the magnet is dropped through a solenoid (wire coil). It would be amazing if anyone could help me with this formula. I tried finding the velocity with Ki, Ko and Ui, and Uo, but i don't know how to join them. I am dropping a magnet through a wire from varing heights and measureing the max voltage that is measured. I am trying to derive the ideal equation for this procress.

Answer 1

The magnet is dropped through a wire coil.

I assume a heavy cylindrical magnet falling through a round coil with both their axes alligned.

I am going to assume that the resistive force that the wire loop exerts on the magnet as per Lenz's law is negligible as the magnet is quite heavy.

Now, the velocity of an object falling from a height h is:

The voltage induced in a coil is given by:

Now. see the term  

This refers to the change of flux as the magnet is at different heights with respect to the coil (not the dropping height!). I am talking about the variation of the flux as the magnet is falling through the coil.

If the magnet's length is L, when it falls, as the magnet's gemetrical center lies in the plane of the coil, the flux has increased from zero to max flux. As it falls through, the flux again reduces to zero.

This easily gives the term

during the first half of the motion.(the second half gives the same term but with sign changed)

should be established experimentally.

Substituting the obtained quantity:

So, in the end, the required result is:

where h is the dropping height, the only variable quantity.

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Extra:

Now, for the more interested mind, this is what happens if you have a stack of such wire loops:

I again assume a cylindrical magnet falling through a solenoidal coil.

Consider the scenario that the wire coil is infinite in length and the magnet keeps falling through it. Forget dropping it from different heights on top of the coil.

Due to acceleration by gravity, the magnet gets faster and faster. This means that the rate of change of flux through any loop of coil gets greater as time increases. This makes the coil resist the magnet's motion more as time increases.

There will be a certain point of time where the magnet will have attained a terminal velocity where the magnetic force has increased so much that it completely cancels out the gravity force. From then on, that velocity will be maintained forever through the infinite pipe and the magnetic force will also be the same and so will be the induced voltage. ( I have absurdly taken the gravitational force to be constant along infinite pipe , but in practice, it is probably a few meters before the terminal velocity is achieved and a few meters is a good enough approximation).

Now, to work out the voltage: This is better done as a simulation using Finite Element Analysis softwares such as COMSOL Multiphysics.

We know that:

for a ring of wire

I am going to assume that the solenoid is a whole bunch of rings of wire stacked on top of each other. This eliminates an axial current flow and makes the problem simpler.

Furthermore, we have to find the voltage only on the wire loop that the magnet is centered at.

The flux linked with the coil is given by:

where A is the area of the coil.

Now, for some mathematics:

or,

Now, the term

can be found out easily by experimenting with the magnet and a wire loop. Just stop the magnet at various positions with respect to the loop's plane and note down the values of the flux phi for various points and then make it as

Now, you can also find the derivative easily and it is fixed for the given system.

(obviously, the axes of the magnet and the loop should coincide to make the problem simpler).