# Energy of magnetic field

## Main Question or Discussion Point

If a current is put trough a solenoid, it produces a magnetic field which has a certain energy. This energy is not 'stored' in the kinetic energy of the moving electrons, because that is multiple order of magnitudes smaller than the energy of the magnetic field.

So, where is the energy of the magnetic field stored?

Related Classical Physics News on Phys.org
in the magnetic field.

if you reduce the current in the solenoid you would get back the same amount of energy (due to inductance) you once used (against inductance) for creating the field. being able to get back E is as good as saying 'the field stores E'

If a current is put trough a solenoid, it produces a magnetic field which has a certain energy. This energy is not 'stored' in the kinetic energy of the moving electrons, because that is multiple order of magnitudes smaller than the energy of the magnetic field.

So, where is the energy of the magnetic field stored?
We say the energy is "stored in the field". What does that mean? You are correct that it's not stored in the kinetic energy of the electrons. Maxwell himself commented upon this idea which can be called they "hose" analogy. If you have a garden hose holding a certain amount of water flowing through it that represents kinetic energy you can store or get back. There might be a temptation (as you noted) to make an analogy with electricity in a wire. But Maxwell observed that this is wrong. And the reason it it wrong is that the energy stored in the hose does not depend upon the physical shape of the hose. If you coil it up you do not store anymore energy than if it is straight! The energy stored by current in a wire is related to a parameter termed "inductance". The inductance of a piece of wire very much depends upon it's shape. A coiled wire has much more inductance than the same piece straightened out! Making Maxwell's point.

As to where exactly the energy is stored "in a field" or how it does this is a mystery. It is known that one can find the energy flow from fields in space by calculating a thing called the "Poynting vector". Thus if we have a certain surface in space, if we integrate the Poynting vector over that surface we can find how much energy is flowing through that surface. Hence if we have a solenoid, and we put a surface around it we can determine the amount of energy going in and out of the solenoid (and presumably in and out of the fields)

People are often tempted to use the Poynting vector as an indication of the energy density in space, mathematically, this interpretation is not valid and only the INTEGRAL over the surface is a valid calculation.

So basically this all leaves us with a conundrum. Energy clearly seems to be stored somehow and somewhere in space due to the fields. But space is in fact empty and believed to consist of "nothing at all" since the 19th century ether theory was abandoned. The end result is sort of a logical impasse where the energy from a current-carrying solenoid is somehow stored somewhere in nothing at all. Indeed there is no credible model even for what electromagnetic fields might be! All that is known is that the forces that arise from these fields are quite real and the mathematics of field theory provides excellent agreement with all experiments. On thing is certain however, the water in a hose model is not valid for electricity.

Dale
Mentor
If a current is put trough a solenoid, it produces a magnetic field which has a certain energy. This energy is not 'stored' in the kinetic energy of the moving electrons, because that is multiple order of magnitudes smaller than the energy of the magnetic field.

So, where is the energy of the magnetic field stored?
Here is my favorite page on the subject:
http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

It goes through a detailed derivation of the energy conservation laws for Maxwell's equations. Basically, the square of the electric and magnetic fields are proportional to the energy density, so the fields themselves store energy which can later be retrieved as described by Maxwell's equations.

Dale
Mentor
So basically this all leaves us with a conundrum. Energy clearly seems to be stored somehow and somewhere in space due to the fields. But space is in fact empty and believed to consist of "nothing at all" since the 19th century ether theory was abandoned. The end result is sort of a logical impasse where the energy from a current-carrying solenoid is somehow stored somewhere in nothing at all.
The fields are certainly not "nothing at all" and a portion of space containing a field is hardly empty. Your mentioned conundrum simply doesn't exist, neither in classical physics nor in quantum physics.

The fields are certainly not "nothing at all" and a portion of space containing a field is hardly empty. Your mentioned conundrum simply doesn't exist, neither in classical physics nor in quantum physics.
Obviously the fields are not "nothing at all" as I already noted that the forces the fields produce are very real. But these fields, whatever they are, are known to exist in a perfect vacuum. In the old days it was alleged that a perfect vacuum was filled with a substance termed the "aether" which was assumed to be the media in which light waves propagated and the like. But Einstein and the Michelson-Morley experiment led to the abandonment of the old aether theory. So if a perfect vacuum is not empty, then the question you have to answer is what exactly fills that vacuum space that allows the fields to transmit forces there? I sure don't know for sure what it might be. I think there is some speculation of "strings" or "dark energy" or "zero point energy" or "neutrino sea" or the like but I've heard of no proof that any of this speculation has been established as true. So to me that's a conundrum. What do you say is the material that fills a perfect vacuum that gives rise to the forces of fields? Which I might mention raises the question that if "something" fills a "perfect vacuum" then that vacuum isn't exactly "perfect", is it?

I don't get it.
Who ever said energy is stored into something at all?
it's not a material.
it's just like asking where the velocity is stored..
velocity is not stored - location (if at all) is stored,
and the velocity is a mathematical reference of the change of it.
specially given that velocity is something that relative to other bodies.
(same thing with acceleration only a body can accelerate relative to the universe itself)

anyway, energy is a mathematical concept as well, as far as i get it.
excluding modern physics which i don't know at all,
i can't see what value it has to look for something that stores the energy..

Obviously the fields are not "nothing at all" as I already noted that the forces the fields produce are very real. But these fields, whatever they are, are known to exist in a perfect vacuum. In the old days it was alleged that a perfect vacuum was filled with a substance termed the "aether" which was assumed to be the media in which light waves propagated and the like. But Einstein and the Michelson-Morley experiment led to the abandonment of the old aether theory. So if a perfect vacuum is not empty, then the question you have to answer is what exactly fills that vacuum space that allows the fields to transmit forces there? I sure don't know for sure what it might be. I think there is some speculation of "strings" or "dark energy" or "zero point energy" or "neutrino sea" or the like but I've heard of no proof that any of this speculation has been established as true. So to me that's a conundrum. What do you say is the material that fills a perfect vacuum that gives rise to the forces of fields? Which I might mention raises the question that if "something" fills a "perfect vacuum" then that vacuum isn't exactly "perfect", is it?
but why do you think there must be a material to transfer a force, or field?
a field is probably something that just can transfer forces, and exist in a vacuum - that what's fields are for.
if perfect vacuum is a space with no material at all, what the problem of a field exist in it, which is definitely not material?

as a material creature we might have a hard time to understand fields, but fields do exist, and they are not materials, it just other kind of existence..

tiny-tim
Homework Helper
everything's … gotta be … somewhere!

I don't get it.
Who ever said energy is stored into something at all?
it's not a material.
it's just like asking where the velocity is stored..
Hi Mad_Eye! ah, but velocity not conserved, so there's no reason to wonder where it went.

Energy is conserved …

if I switch on an electromagnet, I can move or heat something at a distance …

so energy leaves the electromagnet and somehow arrives at the distant object.

If energy is conserved (and physicists prefer to say that it is), then where did it go in between?

Energy is "real", in the sense that it may change form, but it doesn't disappear: if it disappears in the form that we're looking at, then we're entitled to look for it, and to expect to find it in some other form.

So physicists say "it's in the field". To paraphrase the famous Eccles:

"everything's … gotta be … somewhere! "​

Dale
Mentor
Obviously the fields are not "nothing at all" as I already noted that the forces the fields produce are very real. But these fields, whatever they are, are known to exist in a perfect vacuum.
This doesn't make sense to me. If a field is there then energy can be stored in it and it is not "perfect vacuum" or "empty space" by definition so there is no difficulty. I just don't know what you mean by this comment.

In the old days it was alleged that a perfect vacuum was filled with a substance termed the "aether" which was assumed to be the media in which light waves propagated and the like. But Einstein and the Michelson-Morley experiment led to the abandonment of the old aether theory. So if a perfect vacuum is not empty, then the question you have to answer is what exactly fills that vacuum space that allows the fields to transmit forces there? I sure don't know for sure what it might be. I think there is some speculation of "strings" or "dark energy" or "zero point energy" or "neutrino sea" or the like but I've heard of no proof that any of this speculation has been established as true. So to me that's a conundrum. What do you say is the material that fills a perfect vacuum that gives rise to the forces of fields?
The fields themselves give rise to the forces. See the Lorentz force equation F=q(E+vxB). The fields are the thing that gives rise to the force by definition, no other material is required.

Which I might mention raises the question that if "something" fills a "perfect vacuum" then that vacuum isn't exactly "perfect", is it?
Yes, that is my point exactly. Your error is only in thinking that something other than the field is required.

Last edited:
This doesn't make sense to me. If a field is there then energy can be stored in it and it is not "perfect vacuum" or "empty space" by definition so there is no difficulty. I just don't know what you mean by this comment.

The fields themselves give rise to the forces. See the Lorentz force equation F=q(E+vxB). The fields are the thing that gives rise to the force by definition, no other material is required.

Yes, that is my point exactly. Your error is only in thinking that something other than the field is required.
First lets get straight who is claiming what and therefore who is thinking in error.

"The fields are certainly not "nothing at all" and a portion of space containing a field is hardly empty. Your mentioned conundrum simply doesn't exist, neither in classical physics nor in quantum physics."

This quote comes from DaleSpam who clearly asserts that in essence what is regarded by science as a "perfect vacuum" is "hardly empty"! You are now saying that the thing that is in the vacuum is a "field". Well that is a bit of a hand wave, since a "field" is an abstract mathematical principle describing a distribution of forces. On the other hand we agree that FORCES are present in the vacuum.

So apparently you think that forces are created by abstract mathematical principles! That hardly makes sense, even though MadEye has already suggested that "fields" are simply some "other kind of existence" whatever that means? What are we talking about here? forces from the astral plane? Creatures from the Id? Or that there may be some "material" objects in a perfect vacuum? Quantum foam perhaps?

My original point that nobody has addressed is that that there are real forces in empty space MAPPED by electric and magnetic fields. But so far as I know there is NO model that suggests how these forces arise. To say they arise from the fields is wrong because the field is only an abstraction. That's like saying my wallet is full of money because I dreamed I was rich! (It's just "virtual" money that can exist in a perfect vacuum and even though imaginary is perfectly real!)

Your errors are that you think an abstraction in more real than reality! So if one asserts that a perfect vacuum is indeed "perfect" in the material sense so no material objects exist there, then one must ask how indeed do the all too real forces arise there? To say that forces come from the "fields" is like saying your car is powered by its "velocity"! I know of no accepted model since the fall of aether theory that provides an explanation of how electric and magnetic forces are transmitted through "empty space". To say "fields do it" is equivalent to saying "it's done by magic". At least magic is not quite as abstract a concept as "fields".

Please note that I did not say I had some answers to these questions. I simply asserted it was a "conundrum"!

Dale
Mentor
You are now saying that the thing that is in the vacuum is a "field".
Exactly. The point is that I agree with you that it is not logically sound to say that "nothing" does "something" like store energy. So since there logically has to be something that is storing the energy what should we call it? In science we call it a "field". No problem, we use weirder names like "quark" and "gamma ray" as well as more common names like "force", so the name isn't important. What is important is its experimental and theoretical properties, which are defined such that a "field" can exert forces on matter and store energy and carry momentum etc.

There is nothing at all that is illogical in any way with that. Nobody is claiming that "nothing" is the cause of "something". And the "something" that is the cause has well defined properties and is given a nice short name which is easy to pronounce and remember.

Well that is a bit of a hand wave, since a "field" is an abstract mathematical principle describing a distribution of forces. On the other hand we agree that FORCES are present in the vacuum.
Technically the force is not present in a vacuum. If you look at the Lorentz force equation you see that there must be a charge in order for the force to exist, and charge, in turn, is always attached to matter. So there must be some matter present for the force to exist. However, this is just a minor problem, hardly worth pursuing. The bigger problem is in thinking that fields somehow have a different scientific standing than forces.

My original point that nobody has addressed is that that there are real forces in empty space MAPPED by electric and magnetic fields. But so far as I know there is NO model that suggests how these forces arise. To say they arise from the fields is wrong because the field is only an abstraction. ...

Your errors are that you think an abstraction in more real than reality! So if one asserts that a perfect vacuum is indeed "perfect" in the material sense so no material objects exist there, then one must ask how indeed do the all too real forces arise there?
You apparently have the mistaken belief that forces are somehow "more real" than fields. That is simply not the case. They have equal footing both theoretically and experimentally. In every sense that a force is real, a field is also real. In every sense that a field is abstract, a force is also abstract.

All physical concepts have two aspects to them, theoretical and experimental. The theoretical aspect is how a concept is defined in an abstract mathematical framework, and its relationship to other similar theoretical abstractions is expressed in equations. The experimental aspect is how a quantity can be physically measured, or in other words, how you can translate the abstract equations of a theory into predictions for the measured outcome of an experiment. So, a field is abstract in the sense that it has a place in the mathematical framework of physical theories. A force is abstract in the same sense. Similarly, both forces and fields have well-defined processes for measuring them experimentally. They enjoy, in every way, the same scientific standing.

I don't think you two actually disagree, just associating different meaning to the same words. Words are very ambiguous, just look at the dictionary, anything can mean anything if you put it in the right *context*.

What is "real"? Force is what we can measure, I'd say force is real, but is actually abstract concept and what is really real is *movement* of these particles we can see. Concept for the "field" comes from fluid dynamics, so we have "flux" and even "rotation" concept applied to these em fields we can not directly see, but even without such direct reference to very "material" properties I'd say there is something very real about electromagnetic fields, at least in the CONTEXT that there is indeed some AREA of INFLUENCE that defines the force, which defines displacement which is reality, and those VOLUMES of SPACE are as real as the "space" and "distance" is real, but the mechanics behind the cause for it is far from obvious, even mechanics behind the fluid dynamics fields like vorticity and whirlpools is very hard to qualify as it in its micro structure follows the principles of chaotic systems, though in macro scale it may give rise to perfectly stable entities like solitons.

The point is, even the fields in fluid dynamics are not some individual objects by themselves, rather they are large-scale *manifestation* of the kinetic state of the medium at smaller-scale, it is an ABSTRACT CONCEPT. It is "real" like the hole in the wall is real, it is "nothing" and yet it is there, thanks to the existence of the wall and its ability do 'deform'. General relativity call this "wall" a 'fabric of space-time', before that they used to call it 'Aether'.

You apparently have the mistaken belief that forces are somehow "more real" than fields. That is simply not the case. They have equal footing both theoretically and experimentally. In every sense that a force is real, a field is also real. In every sense that a field is abstract, a force is also abstract.

All physical concepts have two aspects to them, theoretical and experimental. The theoretical aspect is how a concept is defined in an abstract mathematical framework, and its relationship to other similar theoretical abstractions is expressed in equations. The experimental aspect is how a quantity can be physically measured, or in other words, how you can translate the abstract equations of a theory into predictions for the measured outcome of an experiment. So, a field is abstract in the sense that it has a place in the mathematical framework of physical theories. A force is abstract in the same sense. Similarly, both forces and fields have well-defined processes for measuring them experimentally. They enjoy, in every way, the same scientific standing.
Whoa! Yes I DID have the belief that forces were somehow more real than fields! And I didn't stop to consider just what a "force" might actually be! So now the question has to arise not only "what do we mean by field?" but also "what do we mean by force?" OK. Are they the same?

Well we do know that if we have a vacuum and say there is a capacitor in there and maybe some magnets, we DO know that if say we fire in an electron it will be deflected this way and that. And that there is a mathematical model that predicts and describes that trajectory using the mathematics of vector fields and the like as expressed in say Maxwell's equations. So now we have to ask "what is real"? Well, we know for sure, the vacuum and the material objects [magnets, capacitors, electrons] are real. We also know that the resultant motion (trajectory) of a test electron is real. But what seems to be missing is a logical credible model that links our apparatus (applied voltages, magnetic strengths etc.) to the deflection of the motion. It is absolutely clear that there are mathematics that describe the linking. And it is tempting to think that there must be some underlying real "objects" that correspond to the mathematics. But the mathematics themselves are totally abstract! Lorentz says a "force", does it, but there clearly is a problem here as one can't really say WHERE a force is acting. I've seen calculations that show that you can have a force acting in one spot whereas if you calculate another you find the force located in a different place (say forces on objects vs forces in fields). So you indeed have a point that force may be as abstract as fields!

So the question still remains then, doesn't it? How indeed are trajectories of test charges changed though a vacuum? Exactly WHAT is the model for that? Still seems a puzzle to me. I'm just thinking out loud here.

SpectraCat
Whoa! Yes I DID have the belief that forces were somehow more real than fields! And I didn't stop to consider just what a "force" might actually be! So now the question has to arise not only "what do we mean by field?" but also "what do we mean by force?" OK. Are they the same?

Well we do know that if we have a vacuum and say there is a capacitor in there and maybe some magnets, we DO know that if say we fire in an electron it will be deflected this way and that. And that there is a mathematical model that predicts and describes that trajectory using the mathematics of vector fields and the like as expressed in say Maxwell's equations. So now we have to ask "what is real"? Well, we know for sure, the vacuum and the material objects [magnets, capacitors, electrons] are real. We also know that the resultant motion (trajectory) of a test electron is real. But what seems to be missing is a logical credible model that links our apparatus (applied voltages, magnetic strengths etc.) to the deflection of the motion. It is absolutely clear that there are mathematics that describe the linking. And it is tempting to think that there must be some underlying real "objects" that correspond to the mathematics. But the mathematics themselves are totally abstract! Lorentz says a "force", does it, but there clearly is a problem here as one can't really say WHERE a force is acting. I've seen calculations that show that you can have a force acting in one spot whereas if you calculate another you find the force located in a different place (say forces on objects vs forces in fields). So you indeed have a point that force may be as abstract as fields!

So the question still remains then, doesn't it? How indeed are trajectories of test charges changed though a vacuum? Exactly WHAT is the model for that? Still seems a puzzle to me. I'm just thinking out loud here.
A field is the gradient of a potential, and for your example, the electrostatic potential is "real" .. it is just as real to the electron as an ice half-pipe is to Shaun White ... change the shape of the half-pipe, and you will change Shaun's trajectory .. change the shape of the electrostatic potential, and you will change the trajectory of the electron. If you make a change to the potential, the consequent changes to the particle trajectory are completely predictable .. it is all well-defined and understood, so I guess I don't see the issue. Is it the fact that you cannot *see* the potential that makes you uncomfortable? You could certainly *feel* it if you were to hold a test charge in your hand and move it through the potential gradient ... so what is the problem with the *reality* of the field/potential/force/whatever?

Note also that in this explanation, the existence of the potential implies that, at each point along its trajectory, the electron is experiencing a local interaction ... only the gradient of the potential at its instantaneous position provides the force that it feels. There is no need for "action at a distance" in this case, everything can be described by local interaction of the particle with the potential.

Now, if you want to know *what* the field is, and you are not happy with the results of classical electrostatics and electrodynamics, then you can try the quantum version. I am far from an expert on the subject, but as I understand it, QED provides a description of how electrostatic fields are created as excitations of the vacuum background using photons and creation operators. This again provides a completely consistent description of what is observed in the lab.

Dale
Mentor
Whoa! Yes I DID have the belief that forces were somehow more real than fields! And I didn't stop to consider just what a "force" might actually be! So now the question has to arise not only "what do we mean by field?" but also "what do we mean by force?" OK. Are they the same?
No, they are not the same, but they do have the same status scientifically.

Well we do know that if we have a vacuum and say there is a capacitor in there and maybe some magnets, we DO know that if say we fire in an electron it will be deflected this way and that. And that there is a mathematical model that predicts and describes that trajectory using the mathematics of vector fields and the like as expressed in say Maxwell's equations. So now we have to ask "what is real"?
This question comes up very often on these forums, and I never engage in a debate about the reality of anything without asking for a suitable scientific definition of "real". I have yet to find a satisfactory definition of "real", and until that term is well defined I think the question itself is meaningless and any debate on the topic is pointless.

But what seems to be missing is a logical credible model that links our apparatus (applied voltages, magnetic strengths etc.) to the deflection of the motion.
On the contrary, we have a very logical and very credible model. It is logical because it has a rigorous mathematical framework (theory), and it is credible because it has more than a century and a half of validation (experiment). That is precisely the whole point of the scientific method and the importance of both theory and experiment.

It is absolutely clear that there are mathematics that describe the linking. And it is tempting to think that there must be some underlying real "objects" that correspond to the mathematics. But the mathematics themselves are totally abstract! ... So you indeed have a point that force may be as abstract as fields!
You are missing a critical point: experiment. If these mathematical models and abstract constructs had no bearing on experiment then we would not give them a second thought, they would indeed be nothing more that "totally abstract" curiosities as you have said. But that is not the case; these quantities are all measurable. You can construct physical devices with wires and batteries and pieces of iron and glass or plastic, and and they behave as predicted. So these scientific concepts are not mere mathematical abstractions, they are also physically measurable quantities.

DaleSpam said:
You are missing a critical point: experiment. If these mathematical models and abstract constructs had no bearing on experiment then we would not give them a second thought, they would indeed be nothing more that "totally abstract" curiosities as you have said. But that is not the case; these quantities are all measurable. You can construct physical devices with wires and batteries and pieces of iron and glass or plastic, and and they behave as predicted. So these scientific concepts are not mere mathematical abstractions, they are also physically measurable quantities.
I think you missed the point. He is not complaining about anything like that.

By the way, everyone now agrees both "field" and "force" are abstract (mathematical) concepts but everyone also agree these concepts do refer to some real and measurable phenomena and its properties.

There are two issues:

1.) the CAUSE for the force, everyone agrees it is 'field potential'

2.) ..but what is the CAUSE for the 'field potential'?

bjacoby's complaint is that physics does not even try to answer the second question.

Dale
Mentor
I think you missed the point. He is not complaining about anything like that.

By the way, everyone now agrees both "field" and "force" are abstract (mathematical) concepts but everyone also agree these concepts do refer to some real and measurable phenomena and its properties.

There are two issues:

1.) the CAUSE for the force, everyone agrees it is 'field potential'

2.) ..but what is the CAUSE for the 'field potential'?

bjacoby's complaint is that physics does not even try to answer the second question.
I will let bjacoby speak for himself, but in any case, that is a silly complaint. If X were an answer for the second question then you could simply add a third question "what is the CAUSE for X?" and then "what is the CAUSE for Y" and so forth. You get into a "turtles all the way down" situation.