# Intuitive way of "visualizing" how the energy is stored in an inductor?

I've been trying to more or less understand intuitively how energy is stored in an inductor, but I don't seem to get anywhere decent.

In a capacitor I understand, I believe: an external battery pushes electrons and holes (going with the electron/hole theory, even though it's only electrons) to opposite sides and they remain on the 2 plates of the capacitor, forced to be there by the battery. Where we disconnect the battery and let the capacitor discharge through a resistor, there is already a difference in potential in the 2 terminals of the capacitor, created by many +s in one side and many -s in the other side. So as soon as there is a path for the current to go, it will go at full speed and start decreasing, since the difference in potential is also decreasing as the current goes (the number of +s in one side is starting to equal the number of -s in the other side), until Nature goes back to the equilibrium state of same potential in both terminals and we are in current of 0 A.

I'd imagine it as something like this:

But how can I "visualize" this in an inductor? Maybe I don't have the necessary concepts clean enough in my head, so I can't see it clearly as maybe I'm supposed to?

Any help is appreciated!

EDIT: Just to say, the idea here is to understand intuitively, so I don't need to memorize the formulas to know what's gonna happen in some part of a circuit.

• Does visualizing the electrons as flowing past and pushing a very heavy paddle wheel with inertia satisfy you? The paddle wheel with inertia is the magnetic field. It's not perfect since it sort of implies something about speed of flow, but in my experience, you can only really begin "visualize" something based on something you are already familiar with. But after being around something long enough, you develop a different mental image of something based on its own merits. Commented Jan 16, 2021 at 20:11
• analogy to a hydraulic system (ignores actual E&M physics): charge is volume of fluid, current is flow, voltage is pressure. Effect of capacitor is like elasticity in the system. Put more fluid in, volume increases, elasticity creates pressure. It resists sudden changes in pressure, as it can just expand/compress a little to take up the change. Effect of inductor is like the momentum of the fluid. Resists flow being suddenly accelerated. If you want to produce sudden acceleration or deceleration, you must momentarily supply more pressure than you need to keep the flow going at a steady rate. Commented Jan 16, 2021 at 20:21
• I visualize inductors as simply giving inertia to current flow. For whatever reason, I think most people (myself included) find capacitors more intuitive than inductors in terms of visualizing what is happening with charges and whatnot. But after all these years, I do have an intuitive feel for how inductors work. It just doesn't explicitly visualize fields or charges. Commented Jan 16, 2021 at 23:03
• @DADi590 First off, I'd recommend that you get and read the most recent edition of Chabay & Sherwood's "Matter & Interactions." It's truly unique in teaching physics and it covers capacitors. There, you will get a unique and wonderful view about capacitors and a useful explanation that is as simple as it can get without being so simple that it's no longer useful. But there are some simplifying ideas I use that haven't been written in any book I've read, but which do provide a useful viewpoint that can be stretched and applied in different cases without breaking down. I may write.
– jonk
Commented Jan 17, 2021 at 0:29
• @jonk Right I still have to get that book. I keep getting other books but keep forgetting about that one. Not that I ever find the time to read them. Pricey though. Commented Jan 17, 2021 at 1:37

I'm going to start by discussing capacitors then move to inductors. I'll avoid complex equations (to save myself as well as this discussion.) I may even discuss duals between capacitance and inductance and how that relates also to physical ideas such as mass, force, velocity, and momentum. (That will treat them in isolation with the purpose of providing some intuition, only.) But when I get some time for that. I think it is important to start with the area where we each may feel more comfortable.

The following will be more qualitative and will discuss ideas that aren't found in textbooks. I'm doing that because it may help you visualize, despite the risks of taking a non-standard approach.

## Capacitors

For capacitors, you probably know already that the plate area (assuming both plates have the same area), the distance separating the plates, and the physical matter (inserted to replace a vacuum or air) used in between the plates relate directly to its capacitance value. These ideas of plate area, plate separation, and inserted medium (if any) aren't hard to grasp. The formula is kind of easy, too:

$$C=\epsilon_{_0}\frac{A}{d}$$

It's important that you think of the above as implying a vacuum in between the plates. It's possible, of course, to stick bits of matter in between them, too. And who knows? Something different might happen. It turns out that for certain types of matter, the above equation does appear to require an added factor, the relative permittivity. This is just a fudge-factor that must either be 1 or larger than 1 and tells you something about the inserted material's ability to respond to a charge difference between the plates.

The new equation with its fancy new fudge-factor is:

$$C=\epsilon_{_r}\epsilon_{_0}\frac{A}{d}$$

And it works great, if you know $$\\epsilon_{_r}\$$. If you fill in the entire volume in between the plates (no gaps, etc) then you can just use the material's $$\\epsilon_{_r}\$$. Of course, if you do something screwy like filling oddball portions of the volume then matters aren't so simple. (But people mostly don't do things like that. They try and keep it simple and cheap to make.)

What does the fudge factor do? What does it mean? What does it relate to, physically?

Well, the charge is separated by a gap that keeps them from crossing over and neutralizing themselves. They want to, but can't. This sets up a field in-between the plates, with the field lines perpendicular to the plate surface and passing straight through from one plate to the other. There's also a field elsewhere too, surrounding the capacitor in the air around it and those field lines point all over the place, depending on where you are (and/or the time you are there for changing situations), as rotating vectors as you move through that space. But the round-trip path integral requires that the lines in between the plates point directly from one plate to the other and that is the important point that matters to a dielectric medium that may be inserted into that volume.

When something is inserted there, the matter within the material may be able to form dipoles. Molecules are a little more complicated athan a dipole. I have to admit that using the idea of an idealized dipole is a bit of a simplification. But the simplification is pretty close and good enough for most uses.

Now, before I go further, let's go back to the case where there's only a vacuum or air in between the plates. When a capacitor is charged, there is a certain amount of usable energy stored somewhere. Where? Well, in the vacuum in between the plates. I want you to firmly fix that in your mind. It's not stored in the plates, where there is no electric field at all (if there were, electrons would be moving around and we know that does not happen.) Instead, it can be reasonably visualized as being stored in the vacuum in between the plates.

Now let's return to the dipole idea, where we've inserted some kind of magic material that can form dipoles when exposed to the electric field lines between the plates. Some molecules or groups of molecules in a dielectric can rotate and align themselves with the electric field lines. It's a response to the field. In rotating into alignment, these dipoles can be thought of as forming a kind of "short-circuit" or "shortcut" that "bridge-over" (if you like) some of the vacuum. Vacuum, where the energy is stored, also has resistance let's say. And the molecules form a short circuit that bypasses this vacuum resistance and reduces the effective gap distance.

Now, according to the equations above, if the distance between the plates is less then the capacitance value is increased, other things being the same. So if you accept this way of thinking about it, it can explain why a dielectric increases capacitance.

Some materials won't polarize and form dipoles, though. So those are not considered to be dielectrics. And nothing I know of forms dipoles oppositely directed against the electric field line direction, so unless someone knows something I'm missing we cannot reduce the capacitance of a vacuum-gap capacitor by inserting an anti-dielectric.

From this, it's not so much about the little charges themselves but about fields. The fields that exist when there is a charge difference between the plates. Also, it's useful to imagine here that energy cannot be stored in matter but instead only in vacuum. And that if you fill that vacuum up with something that can form dipoles, then there is less vacuum within which energy can be stored -- or, in effect, a smaller effective gap.

So there is a gap you can measure with a tape measure. That's one number. But there's also a gap you cannot measure directly, which is the effective gap within which energy is stored. The value of $$\\epsilon_{_r}\$$ is just a quantity telling you by how much the measurable gap is shortened in order to get the remaining true vacuum gap, which is the only gap a capacitor cares about.

Note that this isn't strictly true, either. It's a simplification. Matter that rotates into alignment can rotate back out of alignment, too. And it takes some energy to achieve rotation and releases some when the field lines are diminishing and the molecules can "unwind" a bit. For capacitors, this effect can result in some heating as molecules moving about is pretty much the definition of heat energy. And if you knock them around by oscillating the field then some of that will wind up as extra vibrational energy in the capacitor. Also, there's a question about how much "memory" these molecules might have -- perhaps they don't completely go back to where they were before the capacitor was charged, once discharged. They may "remember" something. But with most capacitors this effect is pretty minor and so capacitors usually are not considered to have hysteresis.

(This hysteresis detail is not true for core materials used in inductors where hysteresis is an important detail you often cannot ignore.)

There are more effects (minor for the most part), such as electric field fringing at the edges of a capacitor's physical design. But they aren't usually important.

## Inductors

For inductors, you probably know already that the coil of wire and the physical matter (again inserted to replace a vacuum or air) that the wire coil surrounds relate directly to its inductance value. Here, things are a little bit more complicated to grasp. The equivalent to plate separation, but for an inductor, is the magnetic path length.

The problem with inductors is that there aren't any monopole magnetic charges. So you cannot just stick magnetic charges on some surface. So, here, you cannot have these magnetic field lines between two plates.

With the capacitor's case, I mentioned that there is a path around the outside of the capacitor, too, but that for all intents we can just focus on the field lines between the plates and ignore the rest. But with an inductor, we do have to focus on the entire magnetic path and not just a small segment of it.

This means that the simplest inductor is a toroid. That's because the entire magnetic path is a simple circle -- the average circumference around the toroid. (That $$\2\pi\,r\$$ thing.)

(Okay, let me take that back. The simplest inductor is a solenoid that extends to infinity. But since infinitely long inductors are hard to come by and short ones add some complexity I don't want to deal with here, it's back to that toroid.)

The toroid formula here is pretty simple, too:

$$L=\mu_{_0}N^2\frac{A}{d}$$

Well, $$\d=2\pi\,r\$$ so I really should have written this:

$$L=\mu_{_0}N^2\frac{A}{2\pi\,r}$$

That looks worse, but conceptually it is not. I've replaced $$\d\$$ with the circumference which is $$\2\pi\,r\$$, but that's not a conceptual problem of any kind. $$\A\$$ is just the cross-sectional area of the toroid. Then there is the added $$\N^2\$$ factor. You can probably imagine why the number of loops would have an effect. However, the reason the value is squared is a little beyond the scope of where I want to go. So let's just call that a slight added complexity. But it's not that much.

Once again, though, keep in mind that the energy of an inductor is not stored in the wire. It's stored in the vacuum volume within which the magnetic field resides. Here, that's the interior of the toroid.

So, what happens when a material with permeability is inserted into the vacuum volume? Well, in certain cases things "line up" again, just like those dipoles before. In this case, it's more complicated. So the term domain is used to represent collections of small bits of matter that can align themselves with a magnetic field. Electrons orbiting individual atoms also can line up (and they do to some degree.) But suffice it that the basic idea remains. Some magnetic domains line up with the magnetic field lines and form "shortcuts" through the vacuum along the lines of the magnetic field that causes them to align. This shortens the effective magnetic path length and therefore increases the inductance.

Modify the above equation to:

$$L=\mu_{_r}\mu_{_0}N^2\frac{A}{2\pi\,r}$$

Here, $$\\mu_{_r}\$$ is just a number that tells you by how much the vacuum path length is shortened. That can be by a factor of 1000 or more.

So, once again there is a magnetic path you can measure with a tape measure. That's one number. But there's also a magnetic path length you cannot measure directly, which is the effective length within which energy is stored. (The total vacuum volume that stores the energy will be this shortened length times the toroid's cross-section area.)

With materials we can use to increase the inductance, with domains that align, there is hysteresis. If you start with such a material with domains aligned in random directions, then apply a magnetic field that forces them into some alignment, and then remove that magnetic field, they will partially return to their earlier state (or something close enough for now) but they will not entirely return to their earlier randomized condition. Some "memory" of the prior aligning field will still hang around.

If you reverse the field, you'll first have to overcome that memory and then re-align them oppositely. Removing the field will leave some memory of that field, too, which was opposite. This process continues over and over when AC is applied to an inductor.

Just as with any material, too, rotating the domains does leave some vibrational energy in the material -- heat.

A changing magnetic field induces a non-Coulomb electric field. That means currents can flow, under AC conditions. Since some materials we use (iron) are conductive, these currents do in fact flow and will also generate heat. The first order non-Coulomb effect is called an Eddy current. This isn't something you have to worry about in a capacitor because, by definition, a dielectric is an insulator and currents just don't flow well in those. But for inductors, and at higher frequencies, it's a problem. Ferrites help with this because conductive particles are mixed with non-conductive bits that help block these currents.

Finally, there are also practical limits on the strength of the magnetic field (the number of Teslas) that can be supported. Different materials will place differing limits here, as well. Once this is exhausted (i.e., all of the magnetic domains have done all of the "lining up" they can do and there's nothing left over, anymore), then the inductor starts behaving as though there was no core material, at all. More like an air-core inductor. The process is often gradual so that the effect isn't sudden. Some oscillating circuits actually depend upon this "feature" in order to work.

## Simplification

Before I go on, I want to return to the capacitor and dig one level below my qualitative hand-sweeping above. The reason is that I want to point out a new concept that you should burn into your every thought about the world around you -- the idea of emergent phenomena.

Sometimes, theories describing well one level of experience and that are well verified, experimentally, have no meaning whatsoever at a deeper level. Instead, these ideas are emergent -- usually as a result of large number population statistics.

The concepts of temperature and entropy from statistical thermodynamics are well established and extremely important. They just work. But the ideas of temperature and entropy have no meaning whatsoever at the atomic level. They just don't exist. There is energy. And energy is an important concept both at the atomic level and in thermodynamics. So there is some common ground. It's just that individual particles don't have the idea of temperature. They may have velocity, momentum, energy, etc. But not temperature or entropy. Those two arise out of the statistics of quadrillions (and more) particles that interact with something else that also includes similar large numbers of particles.

For example, a cup of water and a thermometer. These both have huge, inconceivable numbers of individual particles in them. And when you insert a thermometer into a cup of water to measure its temperature, you are asking about "what happens with the quintillions of particles in the thermometer when immersed into septillions of water particles that will all be randomly bouncing around and affecting each other?" That has meaning because of the large numbers of possible starting states and the myriad results that are possible, almost all of which will produce a "reading" on the thermometer that you expect to see within error bounds. There are a few outlier initial states that would yield different results. But the likelihood of that happening is so low that the entire universe's lifetime would have expired before it occurs. So it is valid to talk about temperature and entropy as if they exist, at this emergent level. (Because outlier behaviors are extremely rare.)

(Note: The 0th law of thermodynamics is worth a nod at this point, too.)

Just remember this general rule: no matter what level you think you understand something, there's a deeper level at which you don't and out of which what you think you know emerges!

The capacitor is just such an example. I hand-waved above about dielectric dipoles and so on. But deeper down, it is of course at lot more interesting and complex. And further down yet, even still more exciting and complex. Let me give you a taste of just the next level down, drawn mostly from ideas expressed in Chabay & Sherwood's excellent book for learning physics called "Matter & Interactions."

Here's my rendition of a combo of what they write about:

I've tried to show a cloud of electrons in the two metal plates, but where the left side of both plates has just a few extra electrons near the surface (and just a few less electrons near the surfaces on the right side.) In between, I'm showing the polarized dielectric dipoles. And around the outside around the bottom I'm showing the e-field direction vectors as you move around the outside of the capacitor along some curve.

(Keep in mind that out of the very, very large number of conduction band electrons present on both plates, there are slightly fewer of them on the left plate than on the right plate.)

I've also added a green and an orange dot at two interesting locations within the dielectric. The green dot is right in the middle of one of those dipoles. From its perspective the vector points as indicated above. The orange dot is between two dipoles and from its perspective the vector points the other way, also as indicated above. As you move around within the dielectric, there are some very complex e-field directions, as you can imagine. I've only illustrated two simpler ones.

The question is, what's the net direction of the e-field within the dielectric? It seems very hard to compute that. But if you look at the direction vectors and imagine summing them as you move yourself along a closed path circling around the outside and then through the middle of the capacitor to finish up where you started, we know this sum must be zero. (It's impossible for it to be non-zero.)

But you can see that two vectors on the outside (furthest left and furthest right) both point in the same direction. So the sum around the exterior and ignoring the dielectric for a moment, must be non-zero and the net pointing to the right. Therefore, the net direction of the dielectric itself must be pointing to the left by just exactly the right amount needed to cancel the exterior path sum.

I told you that capacitors are interesting. Note that my earlier hand-waving didn't get close to this level of detail. And there's still deeper levels, yet, too.

## Momentum, Mass, Energy, Etc.

You may have heard that the energy on a capacitor is $$\\frac12 C V^2\$$ and that for an inductor it is $$\\frac12LI^2\$$. You may also know that the kinetic energy of a particle is $$\\frac12 mv^2\$$. It seems interesting that there is some similarities, just the same.

\begin{align*} \begin{array}[t]{r} {\text{momentum:}}\vphantom{q=C\:V}\\\\ {\text{momentum:}}\vphantom{\text{d}\,q=C\:\text{d}\,V}\\\\ {\text{force:}}\vphantom{\frac{\text{d}q}{\text{d}t}=C\:\frac{\text{d}\,V}{\text{d}t}}\\\\ {\text{mass:}}\vphantom{C=\frac{\text{d}q}{\text{d}V}}\\\\ {\text{velocity:}}\vphantom{q=C\:V}\\\\ {\text{acceleration:}}\vphantom{q=C\:V}\\\\ {\text{energy:}}\vphantom{\frac12 C \,V^2} \end{array} && \overbrace{ \begin{array}[t]{c} q=C\:V\\\\ \text{d}q=C\:\text{d}V\\\\ I=\frac{\text{d}q}{\text{d}t}=C\:\frac{\text{d}\,V}{\text{d}t}\\\\ C=\frac{\text{d}q}{\text{d}V}\\\\ V\\\\ \frac{\text{d}V}{\text{d}t}\\\\ \frac12 C \,V^2 \end{array} }^{\text{capacitor}} && \overbrace{ \begin{array}[t]{c} p=m\:v\vphantom{q=C\:V}\\\\ \text{d}p=m\:\text{d}v\vphantom{q=C\:V}\\\\ F=\frac{\text{d}p}{\text{d}t}=m\:\frac{\text{d}\,v}{\text{d}t}=m\:a\\\\ m=\frac{\text{d}p}{\text{d}v}\vphantom{C=\frac{\text{d}q}{\text{d}V}}\\\\ v\\\\ a=\frac{\text{d}v}{\text{d}t}\\\\ \frac12 m\,v^2\vphantom{\frac12 C \,V^2} \end{array} }^{\text{particle}} && \overbrace{ \begin{array}[t]{c} \phi=L\:I\vphantom{q=C\:V}\\\\ \text{d}\phi=L\:\text{d}I\\\\ V=\frac{\text{d}\phi}{\text{d}t}=L\:\frac{\text{d}\,I}{\text{d}t}\\\\ L=\frac{\text{d}\phi}{\text{d}I}\vphantom{C=\frac{\text{d}q}{\text{d}V}}\\\\ I\\\\ \frac{\text{d}\,I}{\text{d}t}\\\\ \frac12 L\,I^2\vphantom{\frac12 C \,V^2} \end{array} }^{\text{inductor}} \end{align*}

Also, look up Lagrangian mechanics and the principle of least action. (I started with momentum for a reason.)

I'm sure you know that the capacitor's charge is conserved. You can kind of think about the capacitor's charge as being related to potential energy in a physical system and think about the inductor's Webers as being related to the kinetic energy in a physical system. (You could choose the other way, if you like.)

Regardless, I like to think of the Webers (or volt-seconds) in an inductor as the magnetic dual of the capacitor's electric charge. It's a little weird at first, because charge is countable (in our minds) but volt-seconds seems to depend upon time (and it does) and isn't countable in the same way. But for all intents and purposes, $$\L I\$$ is magnetic charge and $$\C V\$$ is electric charge. These things are conserved just as momentum in a physical system is also conserved.

Similarly, the current into a capacitor is force. When you apply that force to a capacitor, it accelerates, changing the voltage. The voltage across an inductor is a force. When you apply a voltage across an inductor, it also accelerates, changing the current.

It's also useful (especially when you consider an LC tank) to think of the inductor's energy as kinetic and the capacitor's energy as potential. The LC tank is a simple system that converts one to another and then back again. This is perhaps very similar to the idea of a comet in a very highly elliptical orbit. At apoapsis, almost all of the kinetic energy has been converted to potential energy. Then as the comet accelerates back towards the sun, this potential energy is converted into kinetic energy. At periapsis, then, almost all of the potential energy has been converted to kinetic energy. And the comet then continues, over and over. An LC tank is kind of like that, except that the polarities flip such that there are four states instead of two if you include polarity. If just energy, then two states just like the comet.

A really good paper to read is Introduction to Quantum Electromagnetic Circuits, Vool and Devoret where the authors discuss a recipe of sorts for finding Lagrangians without having to guess about it. Lots of classical physics there, so don't worry too much about that word in the paper's title that suggests otherwise.

Somewhere in that paper, they choose differently than I did above, saying that the inductor's energy represents potential, and not kinetic, energy. But like I said, you get to choose. So I prefer to think of the Coulombs on a capacitor as position coordinates and the Webers of an inductor as momentum coordinates, which swaps those roles. (But the authors are a lot smarter than I am. So perhaps you should listen to them, instead of me.)

• Well, as a start, thank you so much for what you wrote! Gave me a better perspective on this! Just need to go see more about magnetic and electric fields when I have more time again (not now) to complete it in my head. But yes, please continue. Curious about the rest haha. Commented Jan 18, 2021 at 0:02
– jonk
Commented Jan 18, 2021 at 5:06
• Yep. Amazing answer. Both this and kruemi's answer helped! Though, you explain more things and from various ways. And loved the table there comparing a particle to capacitors and inductors. I noticed that but didn't went looking for more about it (forgot to do that). Seems you answered that too haha. So will be this one. Thank you! Commented Jan 19, 2021 at 1:12

It seems to me that if you can "visualize" electric charges being separated in a capacitor as energy storage you should be able to do a similar thing with an inductor.

The energy in an inductor is stored in the MAGNETIC field that is created by the electric current in the coil windings. The inductor opposes current flow when there is no magnetic field and that energy goes into building that field. Then when the current is stopped, the field begins to collapse which creates a current that wants to continue what was going on before the interruption.

The capacitor and inductor are two sides of the same coin. One is based on electric field storage and the other on magnetic field storage.

• The key point to seeing this duality is to visualize the energy in the capacitor as being stored in the field rather than in the charges. Commented Jan 16, 2021 at 19:20
• Quite so, the energy is stored in the magnetic field in the core, and this energy can turn back into electrical energy by pushing electrons along against a resistance. Conceptually there’s something is a difference in that a capacitor can be left charged for many seconds with little leakage, while an inductor is not generally able to do this unless it’s superconducting.
– Frog
Commented Jan 16, 2021 at 20:14
• So getting the answer and The Photon's comment together, the issue seems to be just that: both are stored in fields and it's that that must be visualized, not as charge storage in the capacitor. So it seems that I must go and see good ways to visualize electric and magnetic fields in these 2 cases, I guess (capacitor and inductor). I was thinking the same analogy with charges could be done on the inductor but with some other thing, instead of using the magnetic field. But I guess that might be the problem here. So thank you, and everyone else, for describing what happens in various ways. Commented Jan 16, 2021 at 22:33

If you are looking for mechanical analogues (that's what I do when I try to explain those things) I describe the capacitor as a spring and the inductor as a flywheel (you might know those toy cars you did not pull back but had to push forward several times to "charge" the flywheel inside and when you let them go they would drive quite some distance). If you apply force (apply voltage) on a flywheel it starts to accelerate slowly (current trough the inductor rises slowly). The more force you apply (higher voltage) the faster the acceleration (current rises faster). If you stop applying force, the flywheel will keep it's speed (current continues to flow) and if you try to abruptly stop it (open the circuit), it takes a lot of force (a high voltage results).

• This also helped! Thank you! Commented Jan 19, 2021 at 1:12

Well, imagine your electrons as marbles in a tube. Now the tube consists of little segments that will rotate when a marble passes through. Now in a solenoid, we have lots of those rotating segments and they can propel water on the outside (of course, they should all be rotating the same direction to make a real difference). If we make a nice circular water course (the magnetic circuit of an inductor) passing in and out of the solenoid, the water will flow in sync with the electrons running through the tube. To get electrons through the tube, you need to add the resistance of the water to your equation: you'll need to put a lot of pressure on the electrons to get them through while starting to get the water to circle, and the more turns of the tube are around your water course, the tighter the electron and water movement will be coupled. If the water is circling, stopping pressure on the electrons will not make the water stop: it will keep pumping electrons through the tube until the water rushing around its water course will exhaust itself and come to rest again.

Like any analogy, this is of course a tradeoff between being nice to visualize and having a rigorous equivalence to actual physical behavior. Basically you have to take this with an imperial pound of salt.