Discontinuities in the time derivative of the magnetic field

In summary, the conversation discusses an inductor and resistor arranged in parallel to a constant voltage source, with a switch that can create closed loops between the voltage source and resistor. The switch is left connected to the source and inductor for a long period of time, then flipped to only connect the resistor and inductor. It is questioned whether the magnetic field in the inductor can change instantaneously from 0 to B, with a mnemonic for the behavior of inductors and capacitors at DC and a discussion on the effects of adding a resistor to the circuit. The conversation also touches on the transient behavior of the circuit and the concept of a time constant.
  • #1
Nolan
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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.
 
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  • #2
Nolan said:
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.
 
  • #3
Dale said:
Can you sketch this circuit? I am guessing at what you mean with the switch creating closed loops, and I prefer not to guess.
IMG_4713.JPG
 
  • #4
Nolan said:
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?
 
  • #5
yes
 
  • #6
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.
 
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  • #7
Dale said:
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
 
  • #8
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?
 
  • #9
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?
 
  • #10
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".
 
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1. What are discontinuities in the time derivative of the magnetic field?

Discontinuities in the time derivative of the magnetic field refer to abrupt changes or breaks in the rate of change of the magnetic field over time.

2. What causes discontinuities in the time derivative of the magnetic field?

Discontinuities in the time derivative of the magnetic field can be caused by a variety of factors, including sudden changes in the magnetic field strength, variations in the direction of the magnetic field, and the presence of electric currents or charges.

3. What are the consequences of discontinuities in the time derivative of the magnetic field?

Discontinuities in the time derivative of the magnetic field can have significant consequences in various fields such as electromagnetism, astrophysics, and plasma physics. They can lead to disruptions in electronic systems, affect the behavior of charged particles, and influence the dynamics of magnetic fields in space.

4. How do scientists study and measure discontinuities in the time derivative of the magnetic field?

Scientists use various instruments and techniques to study and measure discontinuities in the time derivative of the magnetic field. These include magnetometers, which can measure changes in magnetic field strength, and spacecraft missions, which can provide data on magnetic fields in different regions of space.

5. How do discontinuities in the time derivative of the magnetic field relate to other physical phenomena?

Discontinuities in the time derivative of the magnetic field are often closely related to other physical phenomena, such as sudden changes in electric fields or the presence of shock waves. Understanding these connections can help scientists better understand the complex interactions between different physical processes in our universe.

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