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Consider a simple LCR ac circuit; generally I (might) understand that the charge exchange between capacitor and inductor would induce a harmonic current flow, but I remain confused for two reasons:

  • How does the magnetic field 'hold/store energy'? Or more particularly, how does it transfer it back to the wire? Is that akin to pulling a bar magnet our of the solenoid (inductor)?

  • How does the inductor hold energy without maintaining a change in current, resistivity, or back emf to ensure a continued change in flux, and thus a Magnetic field? Inherent is the assumption that the inductor would still have energy if you disconnected it from the rest of the circuit, which I what I've thus far understood.

I've looked at many similar questions, but they don't seem to address these questions specifically. More likely I'm just in the wrong direction.

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  • $\begingroup$ A current in the wire causes a magnetic field and changes in the magnetic field equal a change in flux, which cause an induced voltage on the wire. There is really not much more to it than that, unless you want a detailed calculation with Poynting vectors around the wires? $\endgroup$ – CuriousOne Jun 8 '15 at 17:37
  • $\begingroup$ I think I'm wondering how exactly that change in flux is maintained, especially in the absence of a an alternating/harmonic direct current $\endgroup$ – Pothoz Jun 9 '15 at 7:27
  • $\begingroup$ It's explained in the wikipedia article en.wikipedia.org/wiki/Inductor#Ideal_and_real_inductors: changes in I induce an EMF that acts against the direction of the current flow. $\endgroup$ – CuriousOne Jun 9 '15 at 8:30
  • $\begingroup$ I'm afraid I'm still confused, despite Wikipidea. Is Wikipedia suggesting that there is a resonance of the inductor when not attached to anything? (I'm speaking of a 'charged' inductor, that has thereafter been detached from a circuit) If so, how is the resonance maintained? $\endgroup$ – Pothoz Jun 9 '15 at 10:01
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    $\begingroup$ Real inductors have an inductance and a capacitance, together they form a resonant circuit. $\endgroup$ – CuriousOne Jun 9 '15 at 17:53
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A magnetic field is determined by the current and a changing electric field. And it has energy just for existing. It takes energy to make the magnetic field, for instance to increase the current, and you get energy back when magnetic fields decrease in strength.

For a common inductor the magnetic field and associated stored energy are due solely to the current through the wires at that moment and not due to anything else.

The capacitor is the same. An electric field is due to changing magnetic fields and from charge imbalances. And it has energy just for existing. It takes energy to make the electric field stronger, for instance to make larger charge imbalances, and you get energy back when electric fields decrease in strength. For a common capacitor the electric field and the associated stored energy are due solely to the charge imbalance on the capacitor.

For an LCR circuit it happens that to have a large current through the inductor the charge has to be flowing from one part of the capacitor to the other part of the capacitor with all the charge flowing through the inductor. This is because of how they are wired together and conservation of charge not because of a law of physics about the fields. But this means large currents through the inductor happen when the charge imbalance on the capacitor is changing a lot. So the magnetic field is strongest when the electric field is changing the most.

Similarly when the electric fields in the capacitor are strong then this requires a large charge imbalance, but charge imbalances between parts connected by a conductor require an external EMF to large EMF through the inductor to counter the push the charges would otherwise feel from the large charge imbalance. This requires a large EMF so a changing magnetic field. Since the magnetic field is associated with the current we require a changing current.

So large magnetic field requires a large current and a large current requires a changing charge imbalance so a changing amount of electric field inside the capacitor.

And a large electric field requires a strong charge imbalance and a large charge imbalance requires a large EMF from non electrostatic sources to build it strong so we need changing magnetic fields to make the EMF. Though if the circuit were moving or attached to a battery there would be other ways to get an opposing EMF. To get this large EMF we need changing magnetic fields to drive it.

So changing electric fields in the capacitor allows a large magnetic field in the inductor. And a large electric field requires a changing magnetic field. So the energy sloshes back and forth.

This is the harmonic current flow.

How does the magnetic field 'hold/store energy'? Or more particularly, how does it transfer it back to the wire?

So the magnetic field has energy just by existing, stronger field more energy. When the magnetic field changes there is circulating electric fields. Imagine electric fields going in circles in the entire space between the circular wires in the conductors similar to like as if water was flowing through the curves in a record just going around and around. And they get stronger spatially the farther from the center they get until they reach their peak at the location of the actual wire. So there are circularly pointing electric fields in between the wires. The overall strength is tied to the rate at which the magnetic field is changing. So even right inside the empty space inside the inductor there are electric fields. This not an accident that you have both electric and magnetic fields at the same point because this is mandatory whenever the energy changes in a region without charges. Over inside the capacitor the same thing happens, when the electric field is changing there are circulating magnetic fields going in circles and getting stronger as you closer to the edge of the capacitor and with a strength ties to the overall rate that the capacitor charge imbalance (hence electric field) changes.

In a region without charges there is energy stored in each little piece of space where there are electric or magnetic fields and the energy stored in a region without charges changes by that energy flowing towards other regions. In the case of the inductor the magnetic field is uniform and it decreases therefore there is a non uniform electric field. The nonuniformity represents the fact that total energy is flowing out of the center (smaller electric fields, same magnetic fields) towards the outside edge of the cylinder (larger electric fields, same magnetic fields).

Now when there are charges the energy of the fields can change and it changes by flowing from the fields to the charges and the rate of flow of energy is $\vec J\cdot \vec E$ so in our case the expanding rings of stronger electric field hit the wires and deliver energy to the moving charges there by doing work. In the case when the magnetic field is decreasing in strength the electric fields are pointing in the direction to give kinetic energy to the charges.

So changing magnetic fields inside the conductor have uneven circulating electric fields the unevenness represents the floe of energy and it keeps flowing until it gets to the wire where the electric field does work on the charges. And that's where and how the energy goes from the magnetic field inside the inductor to the wires that go around the inductor. If instead of empty space inside the inductor you have a magnetic material then there will be circulating electric fields inside the material in which case there might be some energy loss right away right there too. But you don't need that.

Now I answered your question about the changing magnetic field over inside the capacitor. But now you might be worried about how the electric field can change and where and how its energy goes. After all there aren't wires wrapped around the outside of it and even so when expanding magnetic fields get to wires they don't deliver energy by doing work. So let's see what happens.

When the electric field is changing the circulating magnetic fields over inside the capacitor will again be stronger near the edge of the capacitor and this does represent a flow of energy towards the edges and when it gets to the edge there is just more empty space. The electric field isn't really confined solely to the inside and so the energy does continue flowing out but curls up and around the capacitor and flows in the empty space outside the wires all along the circuit because remember how when the electric field changes there is a current brought the wires well there is an associated current hence magnetic field, those fields join up and the energy flows through the empty space outside the wires. Now the magnetic fields keep changing and therefore produce new electric fields, the kind that circulate as opposed to the kind that terminate on charge imbalances. But they circulate around the magnetic fields which themselves circulate around the wires so these new electric fields actually point in the direction of the current so they do do work on the charges and that's where the energy flows in the capacitor from the empty charge free space inside the capacitor between the plates to the outside because the magnetic fields get stronger there to the region outside and flowing around the wires along the wires and also flowing into every little region of current from the outside of the wires in. This also includes right inside the conductor straight into the plates (which is because the electric field doesn't actually point exactly orthogonal to the plate). And you might notice that if the electric field changes this is associated with a magnetic field and then if the electric field changes in a changing way then those magnetic fields are changing so there are new electric fields that circulate about the magnetic fields. So you have something that circles around something that circles around the original electric fields which actually just means there is a new electric field unrelated to the charge imbalance that is just from the changing magnetic field and it points in a direction so as to oppose the change in electric fields so every place where the electric field hits the current energy goes from the electromagnetic fields to the charges (or vice versa) and in the resistor this works goes into fighting the resistor that objects to current. In the rest if the wires charge imbalance form on the outside of the wires to help guide the current in the wire to flow in a steady way because if you had a larger current in one part if the wire then charge would pile up in the region were the current increased or decreased but they would pile up on the edges in a way so as to even out the current so as to not pike up anymore, so you just lay down some charge imbalance on the outside of the wire in proportion to the current. So they contribute to the electric field inside guiding the current and are another factor in the electric fields outside the wires. And the current (different than the charge imbalance on the outside of the wires) contributes to the magnetic field. And when you have both an electric field and a magnetic field (not pointing in the exact same direction or in the exact opposite direction) then you have a flow of field energy and where you have electric field and current (nit orthogonal to each other) then energy is flowing between the electromagnetic field and the charges.

How does the inductor hold energy without maintaining a change in current, resistivity, or back emf to ensure a continued change in flux, and thus a Magnetic field? Inherent is the assumption that the inductor would still have energy if you disconnected it from the rest of the circuit, which I what I've thus far understood.

Hopefully I already answered your question. However to emphasize that the magnetic field just has energy: if your inductor is made out of a perfect conductor and you disconnected it from the rest of the circuit and connected it to itself then it would maintain its current and hence the magnetic field. Having the current means having the magnetic field which means having the associated energy for magnetic field be stored in the field itself inside the inductor.

When there is no resistance in the inductor there is no cost to the current and there is also no cost to keeping the magnetic field. There is a cost to increase it and there is a gain from letting it get weaker. When you hook it up to the LCR then there is a cost because of the resistor, but that's different.

Just note that at any moment you have some energy in the electric field inside the capacitor and you have some energy in the magnetic field inside the inductor and as the current harmonically goes through the resistor you lose some in transit between the two.

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