# Classic Electromagnetism

Hi,

I would like to clarify some points on electromagnetism. Most (if not all) of my teachers say that moving charges create a magnetic field. I would just like to know... relative to what? If two electrons move parallel to each other with the same velocity, would they show any magnetic interaction?

Furthermore, my prof.s also say "Current creates a magnetic field. When a current flows in a conductor, electrons are flowing in them, this creates the field" Again.. electrons are 'flowing'... does that mean the magnetic field is created due to the acceleration (electrons do accelerate across a potiential difference right?) or is it due to the drift velocity as it is. If it is the drift velocity, then shouldnt wires of differnet mterial produce different magnetic fields? Also, we know the Lorentz formula
F = q(E + vxB) - what is the v relative to? for. eg. If I take a stream of electrons that are accelerating with constant acceleration 'a' from x=-[oo] to x=[oo], then all inertial frames that move parallel (or antiparallel) to this stream describe the stream (i.e, parallel to x axis) in the same manner. So how do I decide what the 'v' in Lorentz equation is?

Also, is there any way of absolutely distinguishing between an electric and magnetic field? Is it possible that an electric field in one frame might appear like a magnetic field in another? I was thinking of motional emf and its field when I wrote the previous line... it can be thought of on the basis of magnetic field, but it is an electric field as well ( I think... ).Is it possible to describe all magnetic effects on the basis of induced (mebbe imaginary) electric fields alone?

In replies, please use mathematics or formal theoretical framework wherever possible but.... Im in high school ok Thanks for any help

Kartik

Hi

Yes, Magnetic Fields are created by accelerated charges and no An Alectric field in one frame is not seen as a Magnetic Field in the other.

I still dont get how to apply v in the formula. Relative to what? How do I sense the movement relative to a field? Also, according to This site , it says u can interchange the fields... can someone gimme some feedback on that?

Kartik

Originally posted by kartiksg
Most (if not all) of my teachers say that moving charges create a magnetic field. I would just like to know... relative to what?

Relative to the observer who sees the charges moving and measures the field.

If two electrons move parallel to each other with the same velocity, would they show any magnetic interaction?

In a frame in which both electrons are moving, yes. In the electrons' rest frame, no: there is no magnetic field in that frame.

(Well, actually yes because they still have an intrinsic magnetic moment, so there is a magnetic field in that frame. But there is no velocity-induced magnetic interaction due to their charges.)

Furthermore, my prof.s also say "Current creates a magnetic field. When a current flows in a conductor, electrons are flowing in them, this creates the field" Again.. electrons are 'flowing'... does that mean the magnetic field is created due to the acceleration (electrons do accelerate across a potiential difference right?)
or is it due to the drift velocity as it is.

It's due to the drift velocity.

If it is the drift velocity, then shouldnt wires of differnet mterial produce different magnetic fields?

Well, if you apply a fixed potential difference, then different wires will have different resistances, thus different currents, thus different magnetic fields.

Remember, the current is related to the drift velocity by,

I = nevdA

where n is the electron density, e is the electron charge, vd is the drift velocity, and A is the cross-sectional area of the wire.

If two materials have different drift velocities, then each individual electron will produce a different magnetic field. But the total magnetic field produced also depends on how many electrons are moving. What matters to the total field, is the the rate at which the total charge is moving: that quantity is called the current.

So, for instance, if you fix the current, then you will have the same magnetic field? How is that compatible with differing drift velocities? Well, the electron densities are different too, and if you physically arrange it so that I is constant, then vd and n always have to vary inversely to make their product a constant.

Also, we know the Lorentz formula
F = q(E + vxB) - what is the v relative to?

It's relative to whatever frame is measuring the force, E field, and B field.

for. eg. If I take a stream of electrons that are accelerating with constant acceleration 'a' from x=-[oo] to x=[oo], then all inertial frames that move parallel (or antiparallel) to this stream describe the stream (i.e, parallel to x axis) in the same manner.

No, different frames will see the electrons moving at different speeds, and thus will measure different charge densities for the streams (Lorentz contraction), different speeds, and different magnetic fields.

Also, is there any way of absolutely distinguishing between an electric and magnetic field?

In a given frame, yes. The electric field in that frame induces a force independent of velocity in that frame.

Is it possible that an electric field in one frame might appear like a magnetic field in another?

Yes! Exactly! Maxwell's theory of electromagnetism is a relativistic theory, even though nobody knew that when it was invented. The electric field and magnetic field are unified into the electromagnetic field: the first "unified field theory". What is an electric field in one frame is generally partly electric and partly magnetic in another, and same for a magnetic field in one frame. Different observers see different E and B fields, which are really just different perspectives of the same unified F (electromagnetic) field.

(Einstein invented special relativity by noticing that Newton's laws of mechanics were inconsistent with Maxwell's theory. Most people tried to keep Newton's mechanics by assuming that Maxwell's theory was only true in one special frame -- the "aether frame". Einstein took the opposite approach, and looked for a new mechanics that would allow Maxwell's equations to be true always -- and invented relativity.)

Is it possible to describe all magnetic effects on the basis of induced (mebbe imaginary) electric fields alone?

More or less. If you start with just Coulomb's law, and require that the laws of electromagnetism obey Einstein's relativity postulates, then you can derive not only magnetic effects but all of electromagnetism.

(Well, I'm lying a little. There are some technical assumptions you have to make, like assuming that the fields can depend on position and velocity, but not acceleration, that they can be derived from a relativistically invariant potential, etc. In section 26-1 of volume II of the Feynman Lectures, Feynman kind of derides people who claim they can derive Maxwell's equations this way because there are a bunch of non-obvious extra assumptions you have to make ... but still, the basic idea is valid.)

To see a "derivation" of Maxwell's equations from Coulomb's law and relativity, see Principles of Electrodynamics by Schwartz. To see how electromagnetic fields transform in different reference frames, see most intermediate or advanced texts in electromagnetism, such as the Feynman Lectures.

Thanks... I understood most of it... but I still didnt get the parallel electron part. You said that the magnetic field will exist in the frames where the electrons are in motion. So if in a frame, I see the electron moving with velocity v, I see 2 forces on the electrons
1. Coulombic force (repulsive):- Fe = k e^2 / r^2 where r is the dist of separation and k is the coulomb's constant.
2. Magnetic force (attractive):- Fm... Im not sure what its magnitude will be as I dont know how to calculate the magnetic field produced by an electron... but I know it will be attractive (Using the Maxwell's Grip rule and the Fleming's Left hand rule).

In yet another frame, the observer sees no motion of the electrons. So he sees only the coulomb force.. call this Fe'

Using the first postulate of relativity:

Fe' = Fe + Fm

???

How can this be... I tried asking my teachers (3 of them) this and didt get a good answer...

Is it that the coulomb force is decreasing in the first frame... or increasing in the second.. or is there some other force Im not taking into consideration? ( I suppose u can ignore the effects caused due to the electron spin as they will be common in both... or can u? )

Thanks for the help

Kartik

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Also, can u elaborate on how different observers will measure different charge densities for the stream of electrons?

Can I also have an equation on how to describe the magnetic field of a moving electron? I have equations for fields of a current - Biot Savart Law.. but none for a single charged particle.. or a system of particles. Also, is there a single equation that describes all the electric and magnetic forces together - simultaneously.. rather than four separate equations - as in maxwell's 4 equations. I read stuff on other threads about the EM field tensor... can someone explain what it is and how I ca apply it?

Thanks

Kartik

Originally posted by kartiksg

Using the first postulate of relativity:

Fe' = Fe + Fm

This is a mistake. The first postulate does not say that the total force in one frame is the same as the total force in another. (Just like it doesn't say that the field in one frame is the same as the field in another, or the time in one frame is the same as the time in another, ... etc.)

If I find time, I might sit down and work out the transformations and post them, but don't expect that immediately.

Originally posted by kartiksg
Also, can u elaborate on how different observers will measure different charge densities for the stream of electrons?

Imagine a linear stream of charge. If you boost in the direction of its motion, the boosted observer will see the stream length-contract, so the charge density will increase (the same charge is contained in a smaller volume).

Can I also have an equation on how to describe the magnetic field of a moving electron? I have equations for fields of a current - Biot Savart Law.. but none for a single charged particle..

It's really just another form of the Biot-Savart law:

That's for a constant velocity. Things start getting very complicated once you introduce acceleration.

Also, is there a single equation that describes all the electric and magnetic forces together - simultaneously.. rather than four separate equations - as in maxwell's 4 equations.

Yes. In unified form, Maxwell's four equations become two equations, and if you assume that the fields are derived from a potential, one of those equations is a tautology, so the meat of it is in one equation.

In tensor (matrix) form,

&part;Fab/&part;xa = 4&pi; jb

&part;Fbc/&part;xa + &part;Fca/&part;xb + &part;Fab/&part;xc = 0

where F is the field tensor and j is the current 4-vector. (There is an implicit summation over the 'a' index in the first equation. There are also supposed to be some &epsilon;0's and &mu;0's and maybe c's in there somewhere, but I always choose units in which they're equal to 1 so I don't have to worry about them.)

If you assume that the field tensor F is derived from a potential 4-vector A,

Fab = &part;Ab/&part;xa - &part;Aa/&part;xb

then the second of the equations is automatically true and thus redundant.

I read stuff on other threads about the EM field tensor... can someone explain what it is and how I ca apply it?

It's a 4x4 antisymmetric matrix whose six independent components are the three electric field and three magnetic field components. I don't really want to give a lecture on it, so you should just find a book that discusses electromagnetism in covariant form. Usually you don't bother with it unless you care about transformations between highly relativistic frames or something.

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Thanks a lot.. the site really helped.... as for those weird looking equations... I guess Ill wait till I get to college and do partial derivatives before I try to get what it meant Again, Thanks a ton for the help

Do write about why the forces need not be equivalent in different frames and how they are balanced when u get time.

Kartik

turin
Homework Helper
Originally posted by kartiksg
... I still didnt get the parallel electron part.
..
... I see 2 forces on the electrons
1. Coulombic force ... Fe ...
2. Magnetic force ... Fm ...
...
In yet another frame, the observer sees no motion of the electrons. So he sees only the coulomb force.. call this Fe'

Using the first postulate of relativity:

Fe' = Fe + Fm

How can this be...
...
Is it that the coulomb force is ... increasing in the second.
Yes. (If I understand you correctly). In Newtonian physics, the magnetic force will balance the electric force when the electrons are moving at c parallel to each other (an interesting result, I think). Applying relativistic effects, the equation you suggest will be true. You can look at it two ways: 1) The charge on the electrons will increase (a relativistic phenomenon, kind of like an increase in mass, but different, so don't go around telling people that I said it was the same) the faster they go, thus increasing the electric repulsion, and thus canceling the magnetic attraction 2) You can consider the tensor equation, but that's more sophisticated.

If you know what a vector and a matrix are, then just think of the Newtonian version of the electric and magnetic fields as 3-D vectors and the charge as a scalar. The relativistic version of the electromagnetic field as a 4x4 matrix the charge and current get lumped into one 4-D vector. Then, you operate the 4x4 matrix on the 4-D vector to get the 4-D force.

As for why the forces aren't the same in different frames... well, it should at least be obvious that the accelerations aren't the same: if two electrons are released from rest and repel each other in one frame, then in a frame moving with respect to that one, there will be time dilation, so that observer will see them repelling more slowly. If you want to talk about forces, we have to get into the relativistic definition of force and how it transforms, but hopefully this acceleration argument will be convincing.

It is possible to construct a 4-dimensional spacetime vector called the "4-force" such that your equation Fe' = Fe + Fm will hold, if those symbols refer to the 4-forces. But the ordinary 3-forces aren't the same in both frames.

I was going to type in a derivation of these effects, but I thought it would be quicker if I could just find a web page that derives it, similarly to how Schwartz or Feynman do it (I think Feynman is, as usual, clearer) ... they consider an electron being repelled from a current-carrying wire, though, instead of two electrons:

http://www.astro.warwick.ac.uk/warwick/chapter3/node3.html [Broken]

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Originally posted by kartiksg
Hi,

I would like to clarify some points on electromagnetism. Most (if not all) of my teachers say that moving charges create a magnetic field. I would just like to know... relative to what?
A charged particle radiates an electric field. When the particle is moved there is a lag in the time it takes for the field to catch up. There is a difference in the shape of the electric field from the initial position and the succeding positions. The waves created in the electric field by moving the particle generating the field are electromagnetic waves.
So the answer is that charges moving relative to their own electric field create magnetic fields.

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Originally posted by kartiksg Also, is there any way of absolutely distinguishing between an electric and magnetic field?
A magnetic field is an electric field with transverse waves in it. The electric field is always potentially radiant. The waves that cause it to become "magnetic" are at right angles to the radiant vectors of the electric field's potential.
Is it possible that an electric field in one frame might appear like a magnetic field in another?
This is a very important question. If an electric field is stationary relative to the charge from which it emanates, could there be motion in some other frame such that the electric field seems to be possessed of transverse waves propagating at C?
Is it possible to describe all magnetic effects on the basis of induced (mebbe imaginary) electric fields alone?
Electric fields naturally arise from charged particles. There is no way to induce an electric field in a particle. Electrons have one kind of charge, Protons have the other kind of charge, and Neutrons don't have any charge. There doesn't appear to be any way to induce an electric field in a particle that it doesn't already have one, or to change the one it has.

It is true that all magnetic effects arise from the electric field.

Hi,

Thanks to all u ppl... Im finally beginning to get the hang of this... E field and B field by themselves arent Lorentz invariant but taken together, they are invariant. Force by itself is not invariant, but the four vector of force is. Thanks.... Thanks a ton.... I am finally beginning to see how 'relative' relativity can actually get .

I have seen the formulae for length contraction, time dilation etc. but havnt seen one for charge increase... can I know how the rest charge differs from moving charge? Also, it was stated in www.modernrelativity.com [Broken] that the mass by itself remains the same in the transformation between frames and we should associate the [gamma] with the velocity vector.. or sumthing like that. Is is the same thing here with the charge as well? ( Im not sure I phrased that well... ).

Also... if magnetic field can be thought of as the lag of electric field... how do we think of the lag of a magnetic field? Is there something like that? I think that should happen when u accelerate the charge, resulting in a gradual lag of the lag of the electric field ( all tongue twisted.. ).

Thanks

Kartik

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Originally posted by kartiksg Also... if magnetic field can be thought of as the lag of electric field...
This isn't quite right. The electric field itself doesn't lag. There is a lag in the adjustment of its position when you move the source of the field. The lag in the adjustment of it position is extremely small but measurable: it is the speed of light.
how do we think of the lag of a magnetic field? Is there something like that?
I think that should happen when u accelerate the charge, resulting in a gradual lag of the lag of the electric field ( all tongue twisted.. ).
You can't ever accelerate a charge to the speed of light, therefore there is no risk of accelerating it faster than the magnetic field that results from its change of position can propagate.

Originally posted by zoobyshoe
A charged particle radiates an electric field. When the particle is moved there is a lag in the time it takes for the field to catch up. There is a difference in the shape of the electric field from the initial position and the succeding positions. The waves created in the electric field by moving the particle generating the field are electromagnetic waves.
So the answer is that charges moving relative to their own electric field create magnetic fields.

This is misleading. If a charge accelerates (not just moves), it will emit electromagnetic radiation, which is a self-supporting oscillation of electric and magnetic fields. (And it will create radiation even in the particle's instantaneous rest frame.) But that doesn't account for all magnetism, it just accounts for electromagnetic waves. There are plenty of other magnetic phenomena. The point is not that a changing electric field will produce a magnetic field in some frame; it's that an electric field (changing or not) in one frame is (at least partly) a magnetic field in another frame.

A magnetic field is an electric field with transverse waves in it.

This definitely is not true. You can have electric and magnetic fields without any waves in them at all. I think you're again looking specifically at the case of electromagnetic radiation -- and even then, I wouldn't say that a magnetic field is an "electric field with waves in it".

Originally posted by kartiksg
E field and B field by themselves arent Lorentz invariant but taken together, they are invariant. Force by itself is not invariant, but the four vector of force is.

Yes, that's the idea.

I have seen the formulae for length contraction, time dilation etc. but havnt seen one for charge increase... can I know how the rest charge differs from moving charge?

Charge is invariant, just like mass is. Charge density increases by a factor &gamma; = 1/&radic;(1-(v/c)2), because there is length contraction by a factor &gamma; along the direction of motion, shrinking the volume and increasing the density.

Also, it was stated in www.modernrelativity.com [Broken] that the mass by itself remains the same in the transformation between frames and we should associate the [gamma] with the velocity vector.. or sumthing like that. Is is the same thing here with the charge as well?

It sounds like they're talking about the modern definition of mass, which is to say that it is invariant. There is also relativistic mass, which some people call mass, which has an extra &gamma; implicitly in its definition; if you use invariant mass instead, then that &gamma; factor is explicitly lying around, but it comes from 4-velocity, so maybe that's what they're talking about.

Also... if magnetic field can be thought of as the lag of electric field...

It can't. Electromagnetic waves are described this way, as a propagating "kink" in the electromagnetic fields. Look at these diagrams:

http://www.chem.yale.edu/~cas/jenkins.html [Broken]

Electromagnetic waves (both changing electric and magnetic fields) are generated when a charge is accelerated.

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turin
Homework Helper
Originally posted by kartiksg
(1)
E field and B field by themselves arent Lorentz invariant but taken together, they are invariant. Force by itself is not invariant, but the four vector of force is.
...
(2)
I have seen the formulae for length contraction, time dilation etc. but havnt seen one for charge increase... can I know how the rest charge differs from moving charge?
...
(3)
... the mass by itself remains the same in the transformation between frames and we should associate the [gamma] with the velocity vector.. or sumthing like that. Is is the same thing here with the charge as well?
Just don't forget that you have to put them together in an object alot like a matrix, called a second rank 4-D tensor. A simple vector just won't do anymore. Then, you have something alot like a matrix operating on a vector that gives you another vector as the force.

(2) The relativistic charge density is the rest charge density multiplied by the &gamma; factor (if I remember correctly).

(3) I agree with this, but it is apparently still quite popular to speak of the mass as a component of a tensor (the time-like component of the four-momentum). It is almost the same thing with charge, but charge density really. The charge density is the time-like component of the electric four-current. By charge density, I mean, that part of the electric four-current that effects the electric field components of the Farady tensor, as opposed to the magnetic field components.

Originally posted by Ambitwistor This definitely is not true. You can have electric and magnetic fields without any waves in them at all.
This is the point of contention. The illustration presented at the site you linked to shows the stationary particle with radiant lines that are supposed to represent both electric and magnetic lines. That is: the charged particle is radiating two completely different kinds of lines of force when at rest. Thus, according to that site, when the particle is accelerated, both kinds of energies are "kinked", but the kink in the electric field doesn't have anything to do with the magnetic properties of the field created .
Is this how you understand it?

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When you accelerate a charge, the changing electric field generates a changing magnetic field, which in turn generates a changing electric field, etc., to produce an electromagnetic wave.

However, it is a mistake to claim that all of magnetic fields are produced from changing electric fields, or are related to waves, or something like that. Consider electrostatics and magnetostatics.

And no, the "radiant lines" in the linked site's figures do not represent both electric and magnetic fields. It was a diagram of just the electric field. The magnetic field of a moving point charge isn't radial at all. For that matter, a charge at rest doesn't radiate a magnetic field at all (unless you're including an intrinsic magnetic moment due to its spin, which I'm not; I'm just considering a classical point charge).

Originally posted by Ambitwistor
When you accelerate a charge, the changing electric field generates a changing magnetic field, which in turn generates a changing electric field, etc., to produce an electromagnetic wave.
You say the changing electric field generates a changing magnetic field. We start with electric, how is it transduced to magnetic?
However, it is a mistake to claim that all of magnetic fields are produced from changing electric fields...
Give me some examples of magnetic fields which are not produced from changing electric fields.
or are related to waves
,
The word "kinks" used at the site is acceptable.
Consider electrostatics and magnetostatics.
I'm not sure how electrostatics
supports your point, and I've never heard of magnetostatics so you'll have to fill me in to the extent it's necessary to understand how it supports your point.
And no, the "radiant lines" in the linked site's figures do not represent both electric and magnetic fields. It was a diagram of just the electric field.
Thank God.
The magnetic field of a moving point charge isn't radial at all. For that matter, a charge at rest doesn't radiate a magnetic field at all (unless you're including an intrinsic magnetic moment due to its spin, which I'm not; I'm just considering a classical point charge).
This last concurs with what I already thought to be true.

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Originally posted by zoobyshoe
You say the changing electric field generates a changing magnetic field. We start with electric, how is it transduced to magnetic?

Ampere-Maxwell's law: a changing electric field induces a magnetic field.

Give me some examples of magnetic fields which are not produced from changing electric fields.

The magnetic field of a charge moving at constant velocity.

I'm not sure how electrostatics
supports your point, and I've never heard of magnetostatics so you'll have to fill me in to the extent it's necessary to understand how it supports your point.

Electrostatics is the special case of electromagnetism when the electric field is static. Magnetostatics is the special case when the magnetic field is static. There are no wavelike, propagating disturbances in such fields: that's the definition of "static" (unchanging with time).

Originally posted by Ambitwistor
Ampere-Maxwell's law: a changing electric field induces a magnetic field.
This is simply a description of what happens, not an explanation of how it happenes.
The magnetic field of a charge moving at constant velocity.
This is in conflict with what you said earlier, that a magnetic field only arises from acceleration. You said it was misleading of me to say it arose from simply "moving" the particle.
Electrostatics is the special case of electromagnetism when the electric field is static.
I've never heard it described as a special case of electromagnetism, and I still don't see where it supports your point.
Magnetostatics is the special case when the magnetic field is static. There are no wavelike, propagating disturbances in such fields: that's the definition of "static" (unchanging with time).
I understand the concept,then, but you are ignoring the fact that non-varying magnetic fields are being generated by constantly moving charged particles. The initial question was about this basic kind of field, not oscillating fields. The static field around a wire carrying current results from the moving charges in that wire. Likewise the static field around a permanent magnet results from the electrons in the material constantly moving around tiny circuits in the magnetic domains.
The magnetic fields, in these situations, are in fact "kinks" in the electric field.

Originally posted by zoobyshoe
This is simply a description of what happens, not an explanation of how it happenes.

Fundamental physical theories always explain what happens, not "how it happens". Gravity doesn't say "how mass produces a gravitational field", it just says that when mass is present in some amount, there is a gravitational field. Likewise, Maxwell's theory doesn't say "how a changing electric field produces a magnetic field", it just says that when a changing electric field is present, there is a magnetic field.

This is in conflict with what you said earlier, that a magnetic field only arises from acceleration.

I never said that. In fact, I specifically said the opposite: that electromagnetic radiation from accelerating charges doesn't account for all magnetism.

You said it was misleading of me to say it arose from simply "moving" the particle.

It seemed were implying that magnetic fields always arise from changes in the shape of the electric field, as when you accelerate a charge. That's what I called misleading; it is not true for the magnetic fields arising from a charge in inertial motion. Rather, the electric field in one frame transforms to be a partly electric and partly magnetic field in another frame. The fields do not change shape in this situation.

I've never heard it described as a special case of electromagnetism,

What, electrostatics?? Magnetostatics?? Electrostatics and magnetostatics are what you get when you set the time derivatives in Maxwell's equations to zero: that's the very definition of "electrostatics" and "magnetostatics".

What did you think the definition of "electrostatics" was?

and I still don't see where it supports your point.

The point is that in an electrostatic or a magnetostatic situation, there are no waves, and no changing fields. So your statement, "A magnetic field is an electric field with transverse waves in it" cannot be true.

I understand the concept,then, but you are ignoring the fact that non-varying magnetic fields are being generated by constantly moving charged particles. The initial question was about this basic kind of field, not oscillating fields.

That's my point!! You were the one who was claiming that a magnetic field is "an electric field with waves in it". But there are certainly cases of magnetic fields where no waves are involved.