Discontinuities in the time derivative of the magnetic field

[Nolan]

An inductor and resistor are arranged in parallel to a constant voltage source. There is a switch connected to a terminal on the inductor that can create a closed loop that includes either the voltage source, or the resistor. The switch is left connecting the source and inductor for a long period of time, then abruptly flipped to only connect the resistor and inductor.

The magnetic field produced by the inductor goes from being zero, to some value that induces a current that will ensure a zero net current at the instant the switch is flipped. Is it possible for the magnetic field in the inductor to change instantaneously from 0 to B? No way! I feel there is an argument to be made which takes into account the propagation speed of the magnetic field (c), which means there is a finite amount of time that is required for the field to increase. I know it would be negligible, but I am just curious if there is some way to understand this seemingly "discontinuous" behavior.

Any explanations or comments would be much appreciated.

 

[Dale]

An inductor and resistor are arranged in parallel to a constant voltage source. There is a switch connected to a terminal on the inductor that can create a closed loop that includes either the voltage source, or the resistor. The switch is left connecting the source and inductor for a long period of time, then abruptly flipped to only connect the resistor and inductor.

Can you sketch this circuit? I am guessing at what you mean with the switch creating closed loops, and I prefer not to guess.

 

[Nolan]

Can you sketch this circuit? I am guessing at what you mean with the switch creating closed loops, and I prefer not to guess.

 

[Dale]

The switch is left connecting the source and inductor for a long period of time, then abruptly flipped to only connect the resistor and inductor.

Thanks for the sketch, that helps. So the switch is in the "A" position for all t<0, and is flipped to "B" at t=0, correct?

 

[Nolan]

yes

 

[Dale]

A good menmonic is to remember that at DC an inductor acts like a short circuit and a capacitor acts like an open circuit. So while the switch is set to "A" for a long time you are essentially shorting the inductor. With ideal components the current would increase without bound. So your initial condition at t=0 would be an infinite current and an infinite magnetic field.

 

[Nolan]

A good menmonic is to remember that at DC an inductor acts like a short circuit and a capacitor acts like an open circuit. So while the switch is set to "A" for a long time you are essentially shorting the inductor. With ideal components the current would increase without bound. So your initial condition at t=0 would be an infinite current and an infinite magnetic field.

Thank you Dale, I'm sorry for the silly question. I was being completely oblivious to the fact that the short circuit current through the inductor was even creating a magnetic field... which is embarrassing

 

[Dale]

No reason to be embarrassed. That is what PF is all about!

Do you want to discuss the case where the "A" switch has been closed for a finite time before t=0, or do you understand that case also now?

 

[Nolan]

Sure! But maybe it would be beneficial to add a resistor in the "A" configuration so that the current reaches a fixed value? From what I could find, there is a transient period before t=0 where the voltages across the resistor and inductor are changing as the circuit moves from an open circuit to configuration "A" (inductive time constant = L/R). Back when there wasn't a resistor in configuration "A" the time constant would have been undefined, which I would guess means that it takes an infinite amount of time for the inductor to reach steady state given an infinitely large current?

 

[Dale]

Yes. If you add a resistor in the A configuration then things get easier. In each state you can find the steady-state current simply by considering the inductor to be a short circuit and finding the current through the resistors. Then, the transient behavior is an exponential decay having a time constant as you described above. A good rule of thumb is that after about 5 time constants the circuit is "at steady state".