Power Generation from Axial and Transverse Emf

Question

We consider a flat rectangular plate moving horizontally in a vertical magnetic field,motion being in a direction perpendicular to the length of the plate. We have an emf=BLV between the tips,in the lenhgth wise direction[the axial emf]. During the formation of the axial emf a current flows along the length of the conductor. This current should get deflected in the lateral direction due to the existing magnetic field, producing a transverse emf between the lateral edges.If the tips are connected by a wire we have a closed circuit condition--we should simply have the Hall voltage between the lateral edges[in the direction of the motion].

Query: Would it be possible to use these voltages as supplementary power sourses in moving vehicles---in moving trains,cars ,aeroplanes etc? [We seem to have several ready-made parallel cells in these vehicles]

Answer 1

I'm assuming that the magnetic field you're referring to over here is the Earth's field, then we have the field strength of $6.5 * 10^{-5}$. If we assume the vehicle to be a bullet train, then we get the velocity of $300 km / h$ which is $83.33 m/s$ Assuming a scenario where a rectangular plate of $1m$ is being used and the current is being generated along its diagonal of length $\sqrt2 m$ Then as you stated,

$ \varepsilon = BLV $

$ \varepsilon = 6.5 * 10^{-5} * 83.33 * \sqrt2 $

Then this setup will generate $.00766$ $V$ of electricity, which is too low to be applied for most practical purposes.

If you notice over here then you already have an engine powering the vehicle, and in effect you're trying to capture some of its output. Wouldn't it be much better to capture it at the source?

Answer 2

They wouldn't be supplementary sources They have to get energy from somewhere, right? And the magnetic field isn't being changed, is it? Eventually, you'd have to supply the energy.

Otherwise, stuff like hydroelectric plants would just need an initial push and they'd give free energy forever.

Derivation

Lets consider a simple system: A stick in the y direction is moving along the x-axis. A uniform magnetic field exists in the -ve z direction (into the paper).

| x  x
| x  x
|----> v
| x  x
| x  x B
| x  x

y
^  (.)z
| 
-->x

An emf $Blv$ will be induced in the upwards direction. So far, so good.

Now, lets put it in a simple circuit with a bulb or something. The total resistance of the circuit is $R$. So we get an upwards current (+ve y direction)=$Blv/R$. The total obtainable power from this circuit is $\frac{(Blv)^2}{R}$.

Ok. So we managed to get some energy out of it. But, there's something else: There's a current in a magnetic field. That means that there's a force. Let's assume that you do whatever's necessary to keep velocity constant (Otherwise doing the problem completely correctly would require an infinite series and a differential equation). So, we have a current $Blv/R$, and a perpendicular magnetic field $B$. The force on it will be $ilB=B^2l^2v/R$. It's opposite to velocity (verify this if you want). To keep velocity constant, the car has to apply an equal and opposite force. Power that we must apply will be $Fv=F^2l^2v^2/R$.

Surprise, surprise! We have to input the same power as we get out. Effectively, it's useless for getting power. Practically, there are other losses, so we basically ended up losing energy. If we don't attempt to keep the velocity constant, the car will slow down. Also bad.

Note that there will be no energy lost if the circuit is not connected and if we just have a moving rod, as there is no current. So, the minute we try to obtain energy from a moving rod, we lose the same amount of energy in keeping it moving.

A better use

Of course, this can be used as a way to slow down the car and obtain extra energy. But, as @Anna has calculated, it's not much energy, so it would be a pretty ineffective way.

Conclusion

Trying to obtain energy in this manner will slow down your car, and you will have to burn extra gas to keep it moving. In a perfect world with no other resistive forces, we would lose exactly the amount of energy we obtain. In our not-so-perfect world, this makes us lose more energy. We can use it to slow the car down, but it's not effective.