A thought experiment involving inductors

[bitrex]

I have a theoretical question involving the properties of inductors, I hope someone is able to help! I know that the voltage across an inductor is proportional to the time rate of change of the current through the inductor, and that if the current is changed the inductor will generate a "back-emf" proportional to the rate of change of current through the inductor in an attempt to keep the current constant. Of course, the back-emf can't stay constant forever, because if it did there would be no change in current to produce the back-emf! So the changing current eventually "wins" over the back emf and the voltage spike generated decays away as the current rises or falls to its new value.

My question relates to the following situation: In a circuit like a relay, or a vacuum cleaner motor which is suddenly unplugged, as I think about it it seems the current is changed from whatever it was before to zero in "infinitesimal" time, i.e. there's a moment where the circuit is connected, and then there's a moment an infinitesimal amount of time later when it's not. Perhaps that's not an accurate way to look at it from a quantum-mechanical point of view as there may be a more broad interval when the wavefunctions of the electrons in the contacts of the circuits are still interacting. In any case, in the real world the potential across the inductor doesn't rise to infinity if the contacts are switched in "zero" time, what happens is, in the running vacuum cleaner motor for example, the potential exceeds the breakdown voltage of the air and current continues to flow through a spark from the plug to the outlet. What would happen, though, if you had a circuit in vacuum with no other conductors except the wire completing the inductor circuit (in the thought experiment universe) and were able to take some kind of material with a large dielectric constant, make it ridiculously thick (say 100 miles) and break the connection between the inductor and the wire? What would the back-emf of the inductor do, if the current suddenly dropped to zero like that but there was just no way for the back-emf to keep current flowing? My intuition doesn't seem to be much help here so any ideas would be appreciated!

 

[TurtleMeister]

My guess is that the inductor would short internally. In other words the potential across the windings of the inductor would exceed the breakdown voltage of the insulation between the windings. And if the breakdown voltage were not exceeded, the current would oscillate back and forth. Either way the energy stored in the magnetic field would be converted to heat and/or radiated. I may be wrong, but that's what my intuition tells me. Maybe someone more in the know can give you a better answer.

 

[Dadface]

The collapsing flux tends to maintain a current and the energy stored in the field is converted to other forms of energy and my guess is that the main changes would be to electrical energy which eventually is dissipated as heat etc.You don't need a complete circuit for the current to flow the connection break gap acting like a capacitor, albeit very small.Like Turtle Meister I imagine that there will be oscillation of the current and that the current reduces to zero because of resistance losses.Also, with a real experiment there will be a finite time to break the circuit and breakdown will occur...I cannot see a way of constructing the thought experiment to avoid this.These are just my first impression thoughts .

 

[berkeman]

Remember, you can get an arc in a vacuum. Wiki Paschen Curve

Also, remember that a real inductor will have a parasitic capacitance, and it's that parasitic capacitance that limits the kickback voltage (if there are no other components connected when the inductor is open-circuited). Then, as already mentioned, you get an RLC oscillation with decaying voltages and currents until you get zero in the end, with all the initial energy converted to heat.

 

[Bob S]

Your inductance carrying a current in series with a switch is very similar to the old ignition systems in cars. There was an inductance (the primary of the ignition coil) in series with a switch (breaker points), and in addition there was a capacitor (condenser) in parallel (shunting) with the switch. During the time the switch was closed, the current in the coil was about 2 amps. When the switch opened and interrupted the current, the voltage (L dI/dt) did not instantly rise, but the flowing current charged the shunt capacitor to a voltage of the order of 300 volts. The LC resonance was about 40 kHz, and with a 100:1 turns ratio, the secondary voltage was about 30 kilovolts.

 

[Count Iblis]

You could also let the coil explode. What will happen is that with the current gone, you now have the consider the displacement current (the time derivative of the electric field) that is present in the same Maxwell equation relating the current to the magnetic field.


You can then treat the problem as an intitial value problem: the initial field configuration right at the time of the explosion is known. The time derivatives of the fields are given in terms of the field configuration, so you can solve for the fields at some arbitrary time and position.

 

[berkeman]

Explode?

 

[Count Iblis]

Explode?

Yes, suppose you want to make an EM-pulse generator. There are different ways to do that, but the simplest way is to destroy the coil. You need to make sure the energy in the coil does not get dissipated to heat when large EMFs are generated that could drive currents.

 

[berkeman]

Interesting. But I guess we shouldn't talk any more about EMP generators, or the Mentors may get mad at us...

 

[Dadface]

In the circuits described by bitrex the energy transferred at break will be "very small" and the em pulse will have a very small effect.About the worst you could expect is something like a momentary interference crackle on a nearby radio.

 

[bitrex]

Thank you for all your responses! I had forgotten about the effect of parasitic capacitance between the windings of the inductor, I see how in a a case where the inductor is stymied in making an external connection to keep current flowing it will pump current into its own parasitic capacitance, making an RLC tank circuit that will dissipate the energy.

There's an apocryphal story I read about a manufacturer of some kind of microprocessor controlled motorized wheeled toy; to cut costs (you may be able to guess the nationality of the manufacturer ) they removed some of the interfacing circuitry between the motor and the microprocessor designed to protect it from voltage spikes. Supposedly there were a lot of disappointed kids one Christmas morning when they manually rolled their toy along the floor, and then brought it to a quick stop!