# The Naked Scientists Forum

### Author Topic: What is the Mass Density of a Magnetic Field?  (Read 5272 times)

#### Pmb

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##### What is the Mass Density of a Magnetic Field?
« on: 25/05/2012 02:51:03 »
JP asked me about when proper mass can't work as good as relativistic mass or even better.

So to satisfy JP's curiosity the best I can I've created a challenge for everyone. I have an SR text buy Hans C. Ohanian. One of the homework problems is to find the mass density of a magnetic field. That's my challenge to you all. Solve this introductory level SR problem. Let's make it as simple as possible and assume that the magnetic field be uniform. Find the mass density of a the magnetic field. Use whatever definition of mass that you see fit.

Good luck.  :)
« Last Edit: 14/06/2012 08:40:24 by chris »

#### MikeS

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #1 on: 25/05/2012 07:53:26 »
Interesting question Pete, here goes.

"In quantum physics, the electromagnetic field is quantized and electromagnetic interactions result from the exchange of photons."
http://en.wikipedia.org/wiki/Magnetic_field

As I don't believe a photon has mass then nor can the magnetic field.

I don't believe you can extract energy from a stationary magnetic field (as in a fixed magnet).  If it does not have energy then by the equivalence principle it cannot have mass.

If the magnetic field is produced by an electric current and current is quantatized (only one electron passes a given point at any instant)then presumably the magnetic field must be quantatized, so it could be argued is it 'uniform'?  It's a small effect so possibly can be ignored.
« Last Edit: 28/05/2012 11:03:29 by MikeS »

#### JP

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #2 on: 25/05/2012 14:48:00 »
Ok, fair enough point that invariant mass is going to be a nightmare to compute in this case, but my original point wasn't about which was easier to compute.  I wanted to know when relativistic mass provides some unique benefit that other concepts don't cover.

In this case (if I'm remembering my special relativity correctly), relativistic mass is proportional (by c^-2) to energy of the field.  Energy of a magnetic field is easy to compute, so relativistic mass is easy to compute.  But that also means that any computation done by using relativistic mass can also be done by using energy, so all we've done is introduce a new name for energy... Is that correct?

I guess my point of view (and I admit I don't know how cosmologists use relativistic mass) is that invariant mass is useful because the length of the energy-momentum four-vector can be useful in doing computations of particle collisions.  When does the relativistic mass perform a function in computations that energy doesn't?  I feel that it should, or else all we're doing is confusing the subject by introducing a new name for a quantity we already have a name for.

I do have to go back and read the references you cited in the other thread, Pete, if you already answered it there.  :)

#### Pmb

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #3 on: 25/05/2012 21:13:58 »
Ok, fair enough point that invariant mass is going to be a nightmare to compute in this case, but my original point wasn't about which was easier to compute.  I wanted to know when relativistic mass provides some unique benefit that other concepts don't cover.
I understand. That's why I came up with this challenge. When I give my view on this I'm hoping you'll have an answer to your question. But what is benefit for one person is of no benefit to another. But in my view there is a benefit. That doesn't imply that everyone's going to agree with me.

#### Pmb

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #4 on: 27/05/2012 21:01:13 »
So many people keep telling me what mass is and yet nobody wishes to calculate the mass density of a magnetic field? If there's no interest in it then I don't see a purpose in posting the solution. I'm sorry to see such lack of interest.  :(

#### CPT ArkAngel

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #5 on: 28/05/2012 02:41:04 »
It should be the total energy stored in the generated magnetic field divided by C^2.

I found this article yesterday when i was searching for something else. Strangely, the author was a teacher from my University whom i have never heard of before (he left before i had started). There is certainly some truth in his claims (and he had many) but you must understand that the magnetic field is a relativistic effect due to the motion of the electric field. Is it possible that an object having a total charge of zero in motion still produces a hidden magnetic field? After all, in GR, even if the gravitational attraction is zero at one point, time dilation still depends on the gravitational potential  like at the center of a hollow sphere having a uniform mass density... This could be the answer for a total unification of all forces for my theory... (nothing to do with plasma cosmology which is something else that i disagree with)

http://www.newtonphysics.on.ca/magnetic/index.html
« Last Edit: 28/05/2012 03:25:24 by CPT ArkAngel »

#### MikeS

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #6 on: 28/05/2012 12:02:24 »
CPT

An interesting article and well worth the read.

Logic would seem to imply that to maintain a current in a wire requires a continuous input of energy.  That begs the question where does the energy go??
If the wire is a superconductor then there is nothing lost due to heat and the original energy that remains after the current has stabilized remains.  If no energy is extracted from the system then no energy need be applied to maintain the system.

Therefore, in a uniform magnetic field, the mass/energy of the magnetic field is equal to the mass/energy needed to create it.

I agree with CPT, the mass density equals the total energy stored in the magnetic field divided by c2.

#### Pmb

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #7 on: 30/05/2012 20:21:02 »
http://www.newtonphysics.on.ca/magnetic/index.html
You're confusing the relativistic mass of a magnetic field with the density of a magnetic field. I took a look at that paper. It says
Quote
Relativity theory gives a relationship predicting the increase of mass of relativistic moving particles, but no physical model has been given to describe the fundamental physical mechanism responsible for the formation of that additional mass.
That is quite untrue. It demonstrates why some teachers want to toss the idea of relativistic mass away. This guy made the mistake of thinking that there was a physical change in the properties of the particle. There isn't. The change in mass with speed has to do wth the properties of spactime. Same holds with the life time ofa particle and its proper lifetime.

It looks like a complicated paper and those are the ones  love. Too bad I don't have a printer so I can relax and read it. Can't do that without a print out. I've saved it and will read a little bit over time. Thanks for the reference.

http://home.comcast.net/~peter.m.brown/sr/mass_mag_field.htm

Read it carefully and you'll easily see the reason I posted this as the question. It's all about the stress in the magnetic field.
« Last Edit: 30/05/2012 20:37:49 by Pmb »

#### yor_on

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #8 on: 31/05/2012 02:33:19 »
As long as we speak of relativistic mass I can get it Pete. But it surprised me reading that Einstein defined the Electromagnetic field as 'matter', if I now got that right? As we have EM fields all over I mean? Also, a magnetic field is frame dependent as I understands it, so where does that 'matter' go? "We make a distinction hereafter between “gravitational field” and “matter” in this way, that we denote everything but the gravitational field as “matter.” Our use of the word therefore includes not only matter in the ordinary sense, but the electromagnetic field itself."

But you are right there, although I think of it in terms of invariant mass as per definition unchanging in all frames of reference, one could easily define it such as invariant mass is a 'static description' of mass in that it do not contain any correction for temperature etc. One might say that the relativistic mass is a dynamical definition of mass perhaps?
=

And so also argue that the concept of relativistic mass is what best fits GR, whereas 'invariant proper mass' is a ideal definition, assuming some ideal state of eh, 'matter' :) As we normally call our invariant proper mass?

It's a very interesting point actually.
« Last Edit: 31/05/2012 02:49:34 by yor_on »

#### yor_on

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #9 on: 31/05/2012 03:21:48 »
I think you are influencing me over to your point of view here :)
I really need to think more about it.
=

Have you seen this one btw?
It seems interesting too.

Information and the nature of mass
« Last Edit: 31/05/2012 03:30:34 by yor_on »

#### CPT ArkAngel

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #10 on: 31/05/2012 04:04:25 »

But is it possible that the main reason why QM and GR seem to be incompatible is because spacetime is not the cause but an effect?

I wish you a good health!

And Yor_on, the big question is what is the field and where is the energy?

#### Pmb

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #11 on: 31/05/2012 14:56:42 »
As long as we speak of relativistic mass I can get it Pete. But it surprised me reading that Einstein defined the Electromagnetic field as 'matter', if I now got that right?
Yes. If you agree with Einstein that is. Seems that alot of people find his ideas too difficult to accept so they've rejected a lot of them, most noteably the criteria for the existance of a gravitational field.

As we have EM fields all over I mean? Also, a magnetic field is frame dependent as I understands it, so where does that 'matter' go?
The matter doesn't go anywhere. If the stress-energy-momentum tensor doesn't vanish in one frame then it doesn't vanish in any frame. Therefore if there is matter in one frame then there is matter in all frames. You were concerned that since the magnetic field was frame dependant then that might give cause for the matter to disappear. Recall that the even though the magnetic field is frame depenant, the EM field isn't. What is a magnetic field in one frame is an electric field in another frame.

"We make a distinction hereafter between “gravitational field” and “matter” in this way, that we denote everything but the gravitational field as “matter.” Our use of the word therefore includes not only matter in the ordinary sense, but the electromagnetic field itself."

But you are right there, although I think of it in terms of invariant mass as per definition unchanging in all frames of reference, one could easily define it such as invariant mass is a 'static description' of mass in that it do not contain any correction for temperature etc. One might say that the relativistic mass is a dynamical definition of mass perhaps?
Inertial mass (aka relativistic mass) changes with temperature since a change in temperature corresponds to a change in thermal energy. Recall that stress has of inertia too. So inertial mass density is a function of pressure.

And so also argue that the concept of relativistic mass is what best fits GR, whereas 'invariant proper mass' is a ideal definition, assuming some ideal state of eh, 'matter' :) As we normally call our invariant proper mass?
No. I'll post a derivation of an example where inertial mass is not proportional to inertial energy. In the case of invariant mass please see the bottom of this page
http://home.comcast.net/~peter.m.brown/sr/invariant_mass.htm

It's a very interesting point actually.
When I first came across it I was amazed, and delighted too for perhaps obvious reasons. :D

#### Pmb

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #12 on: 31/05/2012 15:05:10 »
No worries. :)

But is it possible that the main reason why QM and GR seem to be incompatible is because spacetime is not the cause but an effect?
Sorry. No idea. There's one thing I stay clear of and that's when people talk about relativity and quantum mechanics in the same breath.

I wish you a good health!
Thanks. I got a shock in the mail yesterday. I learned that I'm high risk for diabetes and if I don't change by exercise habits and my eating habits then I'm on that road to conracting it. What a freaky letter that was. So thanks for the well wishing for my heath. I sure can use it at this point.

#### yor_on

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #13 on: 31/05/2012 15:17:01 »
The funny thing Pete, is that I also found relativistic mass to be more appropriate once. Found myself questioning invariant mass in 2008 looking for it. I'm still not sure how to see invariant proper mass, although I would prefer to have some irreducible part being the same in any frame. But what I mean is that in GR and SpaceTime we exist in there is no such thing as a invariant mass. It changes and becomes a dynamic concept although you still may relate it to invariant proper mass. That's why I thought you might argue that relativistic mass was the more proper definition in GR.

#### yor_on

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #14 on: 31/05/2012 15:21:07 »
"What’s tricky here is that the magnitude of four-vectors aren’t quite like that of other vectors – the signature plays a role.  Specifically, the square of the temporal component of the four-momentum (called the relativistic energy) has the opposite sign to the square of the spatial components (called the relativistic momentum).  So, the magnitude of the vector above works out to

Crucially, this quantity is frame-independent.  As such it is often referred to as the “invariant” mass (also “rest” mass).  Okun’s argument (one I happen to agree with) is that by merely saying mass is the same as energy, implies that momentum is the same as energy (given the above equation), and thus the difference in the sign of the signature is “brushed under a rug.”  This difference is very important since it points out the fact that time and space, though similar, are not quite the same exact thing (that’s a discussion for another time).  Note that the four-momentum is conserved which means that it is conserved by component.  We’ll see what that means in a moment."

I found this one really interesting. Also you have Stanford university (somewhere:) questioning if invariant proper mass and energy is the exact same thing. It's very interesting to me.

#### Pmb

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #15 on: 31/05/2012 21:38:50 »
The funny thing Pete, is that I also found relativistic mass to be more appropriate once. Found myself questioning invariant mass in 2008 looking for it. I'm still not sure how to see invariant proper mass, although I would prefer to have some irreducible part being the same in any frame. But what I mean is that in GR and SpaceTime we exist in there is no such thing as a invariant mass. It changes and becomes a dynamic concept although you still may relate it to invariant proper mass. That's why I thought you might argue that relativistic mass was the more proper definition in GR.
Don't get me wrong. I do. But in some fields it can be a pain in the butt. Carryinng around too many subscripts can be a pain. It doesn't find much use in QM since velocity is not a well defined quantity in QM

#### Pmb

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #16 on: 31/05/2012 21:41:18 »
That's true. For photons E = p (c = 1). But I don't see people saying that energy is just another name for momentum for a photon

#### yor_on

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #17 on: 02/06/2012 15:50:32 »
It's the stress energy tensor all over again :)
And that one is one of the weirdest most difficult to understand properties of SpaceTime I know. And as you write it seems to have a lot to do with pressure.

Now this is just me wondering, and not directed to anyone in particular Pete :)

Then we have the proper invariant mass, defined as invariant through the mechanism of assuming something to be at rest relative something else. In a SpaceTime defined by no preferred frames of reference we still use one, although as a 'relation' between two frames. Where one frame can't be preferred we still find that using two frames of reference, at rest relative each other to work. To assume that we need a clear definition of what motion should be seen as. so do we have that? I'm not sure what motion is myself?

Take a uniform motion, define a 'speed' to it, now take another uniform motion and do the same. Are they the same? If not, are they still equivalent in all other forms of measurement, when measuring being at rest relative them?

If they are, what has speeds, as 'motion', to do with it? And with the stress energy tensor? Do a higher uniform motion, as defined by an arbitrarily made frame of reference, present a higher stress? on what, and where?

I keep getting back to this as it has to do with the question if SpaceTime can be seen as a 'whole indivisible continuum' where all invariant proper mass are like fishes in the ocean, and the stress energy tensor becomes a measure of tension in that ocean, well as I think of it :)

#### Pmb

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##### Re: Challenge - Mass Density of Magnetic Field
« Reply #18 on: 02/06/2012 17:16:23 »
Now this is just me wondering, and not directed to anyone in particular Pete :)
I'm getting tired of posting in forums so I won't be respinding to anything other than a direct and serious physics question directed at me only. That's just my way of staying far away from trollers and flamers.

#### The Naked Scientists Forum

##### Re: Challenge - Mass Density of Magnetic Field
« Reply #18 on: 02/06/2012 17:16:23 »