# Problem with Purcell's model of electromagnetism

I've been discussing this elsewhere recently. In my opinion Purcell's explanation in section 5.9 doesn't actually explain why an electron moves in a circular or helical fashion in a magnetic field:

However this length-contraction explanation has been around for so long that's it's rather taken for granted.

## Answers and Replies

Dale
Mentor
2020 Award
Hi John Duffield, welcome to PF!

An electron has a circular or helical path in a uniform magnetic field, but the field around the wire is not uniform. So the path is pretty complicated even using Maxwells equations directly. I wouldn't expect any simple qualitative explanation to cover it, you would really need math.

davidbenari and bcrowell
Nugatory
Mentor
If you consider an infinite uniform sheet of current, the quantitative calculation is easier.

In both that case and case of the wire and its non-uniform field, we can calculate that when the test particle is moving radially (perpendicular to the current direction) there will be an axial force (parallel or anti-parallel to the direction of the current) while if it is moving axially the force will be radial. In general, the velocity of the particle can be decomposed into axial and radial components, and the resulting radial and axial forces add to produce the circular motion we're looking for.

bcrowell
Thanks Dale and Nugatory. The point I wanted t make was that Purcell's description suggests that an electron moves towards the wire, when actually, it doesn't. Yes, the magnetic field around a single wire is not uniform like it is inside a solenoid, but the electron still moves around the magnetic field lines. It doesn't cross directly over them towards the wire. See images of electron motion. I cannot explain the reason for this discrepancy. Perhaps it's the way electromagnetism is taught. For example see section 11.10 of Jackson's Classical Electrodynamics where he says "one should properly speak of the electromagnetic field Fμν rather than E or B separately". There’s no problem with that, but IMHO it ought to come much earlier, because the real reason for the electron’s circular/helical motion is the "spinor" nature of the electron electromagnetic field.

A.T.
Purcell's description suggests that an electron moves towards the wire
Isn't it more about the instantaneous force on an electron with a specific instantaneous velocity, rather it's path over time?

bcrowell
Nugatory
Mentor
The point I wanted to make was that Purcell's description suggests that an electron moves towards the wire, when actually, it doesn't.

It does in the specific case of the electron moving parallel to and in the same direction as the current, and that's one of the early examples. But there's a lot more to Purcell's description after that.

bcrowell
bcrowell
Staff Emeritus
Gold Member
Thanks Dale and Nugatory. The point I wanted t make was that Purcell's description suggests that an electron moves towards the wire, when actually, it doesn't

I have Purcell in front of me, open to section 5.9. I don't see any claim such as the one you impute to him. Could you tell us what edition you're looking at, and give us an accurate quote so that we can understand what you're referring to?

A.T: I would say it's about its path over time. If we replaced the current-in-the-wire with a positively charged rod, the horizontal component of the electron's motion would persist, and the vertical component would increase. It would swoop upwards toward the rod, but it wouldn't exhibit any rotational motion.

Ben: I'm looking at the second edition entitled Electricity and Magnetism Berkeley Physics Course Volume 2. Section 5.9 on page 192 starts with "We know that there can be a velocity-dependent force on a moving charge. That force is associated with a magnetic field, the sources of which are electric currents, that is, other charges in motion". On page 193 he talks about the current in the wire, page 194 is an illustration, page 195 start with Lorentz contraction thence moves on to electron density and the linear density of negative charge being enhanced "when it is measured in the test charge frame". After equation 20 we can read that "The wire is positively charged", even though the electromagnetic field is frame-independent. On page 196 after equation 24 we read this: "We have found that in the lab frame the moving test charge experiences a force in the y direction which is proportional to the current in the wire, and to the velocity of the test charge in the x direction". In the next paragraph he says the speed doesn't matter, which doesn't tie in with the v² term in the Lorentz factor. I don't have a big issue with page 196 which talks about two wires, they are indeed pushed apart. But on page 197 we read that "Moving parallel to a current-carrying conductor, the charged particle experiences a force perpendicular to its direction of motion". This isn't true. The charged particle doesn't move towards the wire. It moves rotationally around the magnetic field lines. On the diagram on page 198 you can see a vertical arrow and a rightward arrow, but no leftward or downward arrow. On page 199 the explanation ends rather abruptly with "The physics needed is all in Eq 12, but the integration is somewhat laborious and will not be undertaken here".

Nugatory: maybe I'm missing something here. But when I read Minkowski's Space and Time, I don't get the same vibe. Minkowski talks about the field of the electron and then electric and magnetic forces, and mentions a "screw" analogy that IMHO fits with the electron spinor. Please note that I think of myself as something of a relativist. I'm forever referring to Einstein and Minkowski.

Last edited:
Nugatory
Mentor
But on page 197 we read that "Moving parallel to a current-carrying conductor, the charged particle experiences a force perpendicular to its direction of motion". This isn't true. The charged particle doesn't move towards the wire. It moves rotationally around the magnetic field lines

"Moving rotationally around the magnetic field lines" is what you'd expect if the particle "experiences a force perpendicular to its direction of motion". If a particle with an initial velocity parallel to the wire is moving towards the wire then it is not experiencing a force perpendicular to its direction of motion, it is experiencing a force that points towards the wire no matter what direction the particle is moving.

Nugatory
Mentor
Nugatory: maybe I'm missing something here. But when I read Minkowski's Space and Time, I don't get the same vibe.

I'm sorry, but I don't understand which statement of mine is generating the vibe that you're talking about here. If you mean the part about Purcell calculating the force for various directions of travel relative to the current flow and doing the linear combination to get circular motion, there's no "vibe" involved, you just have to do the calculations.

George Jones
Staff Emeritus
Gold Member
I'm looking at the second edition entitled Electricity and Magnetism Berkeley Physics Course Volume 2. Section 5.9 on page 192

Page 259 of the third edition.

But on page 197 we read that "Moving parallel to a current-carrying conductor, the charged particle experiences a force perpendicular to its direction of motion". This isn't true.

Page 265 of the third edition.

The force on a charge ##q## moving with velocity ##\bf{v}## in a magnetic field ##\bf{B}## is ##q \bf{v} \times \bf{B}##. If ##\bf{v}## and the wire are parallel, then, since the cross product is perpendicular to ##\bf{v}##, the force is perpendicular to the wire. See diagram (c) on page 194 of the second, page 260 of the third.

The charged particle doesn't move towards the wire. It moves rotationally around the magnetic field lines.

In a constant magnetic field, a charged particle moves in a circle or helix around the magnetic field lines. The magnetic field around a current-carrying wire is not constant.

On the diagram on page 198

Page 266 of the third edition.

I am not sure what this has to do with a charge moving parallel to the wire. In (a) in this diagram ##v## is perpendicular to the wire, the force ##q \bf{v} \times \bf{B}## is along the line of the wire.

Last edited:
bcrowell
Staff Emeritus
Gold Member
But on page 197 we read that "Moving parallel to a current-carrying conductor, the charged particle experiences a force perpendicular to its direction of motion". This isn't true. The charged particle doesn't move towards the wire. It moves rotationally around the magnetic field lines.

As others have already pointed out in this thread, Purcell is only discussing the force at one point in the motion. He isn't integrating the equations of motion. There is no contradiction between his statement and yours.

Dale
Mentor
2020 Award
The point I wanted t make was that Purcell's description suggests that an electron moves towards the wire,
I don't recall that. I think you are misinterpreting something in Purcell.

ut on page 197 we read that "Moving parallel to a current-carrying conductor, the charged particle experiences a force perpendicular to its direction of motion". This isn't true.
That is true. What makes you think it is not. You should work it out.

The charged particle doesn't move towards the wire. It moves rotationally around the magnetic field lines.
A free charged particle, acted on only by the field, would move the way you describe. However that is irrelevant to the example presented. He specifies that the motion is parallel to the wire and then calculates the force on the particle. A problem where the motion is specified and the force is calculated is a different problem than one where you calculate the equations of motion.

bcrowell
I'm sorry, but I don't understand which statement of mine is generating the vibe that you're talking about here.
I was talking about the difference between Purcell and Minkowski, who referred to the field of the electron, and to electric and magnetic force. He also mentioned a rather cryptic screw analogy. When you dig around you can find Maxwell saying something similar: "a motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw". Then when you look at electromagnetic radiation you read things like this: "the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time". This ties in with Minkowski's time axis. Not Purcell's length contraction. For an analogy imagine you're in a canoe at sea and a wave comes at you. The tilt of your canoe denotes E and the rate of change of tilt denotes B. But there's only one wave there. An electromagnetic wave. In similar vein the electron doesn't have an electric field or a magnetic field, it has an electromagnetic field. And this has a spinor or screw nature. But it isn't depicted in Purcell.

All: I have to go I'm afraid. I'll reply individually later. Meanwhile see if you can find any depictions of the electromagnetic field. I can find depictions of a gravitational field, an electric field, a magnetic field, and a gravitomagnetic field. But I can't find any depictions of an electromagnetic field.

Last edited by a moderator:
I've been discussing this elsewhere recently. In my opinion Purcell's explanation in section 5.9 doesn't actually explain why an electron moves in a circular or helical fashion in a magnetic field [..]
However this length-contraction explanation has been around for so long that's it's rather taken for granted.
Thanks for the handy link!

On a side note, but maybe in line with your thinking, I'm also puzzled by that section in Purcell's book; I get the impression that when discussing the force between current carrying wires, he apparently fails to account for the magnetic force between moving charges in the two wires, only accounting for electric fields. Does he pretend that in certain reference frames the forces between those wires are merely due to electric fields?

That is at odds with SR: in any inertial reference frame there are moving charges in both wires and thus also magnetic forces between the wires. The misunderstanding addressed in an earlier thread does seem to be caused by Purcell...

My pleasure. A guy called Emilio gave it to me.

On a side note, but maybe in line with your thinking, I'm also puzzled by that section in Purcell's book; I get the impression that when discussing the force between current carrying wires, he apparently fails to account for the magnetic force between moving charges in the two wires, only accounting for electric fields. Does he pretend that in certain reference frames the forces between those wires are merely due to electric fields?
I think so. But it's hard to be sure.

That is at odds with SR: in any inertial reference frame there are moving charges in both wires and thus also magnetic forces between the wires. The misunderstanding addressed in an earlier thread does seem to be caused by Purcell...
I'm not sure if he caused it. Somebody told me he was the first to come up with the length-contraction explanation for magnetism, but I haven't checked that. I get the sense anyhow that the underlying issue is to do with the general theme of talking about electrostatics then moving onto magnetism. For example it's only at the end of section 5.9 that Purcell introduces the magnetic field. But see Wikipedia where you can read that "the electric and magnetic fields are better thought of as two parts of a greater whole - the electromagnetic field". This unified approach is there in Minkowski's Space and Time where he referred to the field of the electron and to electric and magnetic forces. It's also there in Jackson's Classical Electrodynamics where you can read that E and B have no independent existence. But it's coming quite late in the day, not up front. See Jackson's chapters here.

George Jones said:
I am not sure what this has to do with a charge moving parallel to the wire. In (a) in this diagram v is perpendicular to the wire, the force qv×B is along the line of the wire.
Your comments noted. We see forces like this → and this ↑ but not like this ⥀ .

DaleSpam said:
I don't recall that. I think you are misinterpreting something in Purcell...
I don't think so Dale. I've read it through carefully. And I'm not the only one to refer to this. I contributed to the Electromagnets are a relativistic phenom? thread here because it has come up elsewhere. Somebody then split it off into a new thread.

bcrowell said:
As others have already pointed out in this thread, Purcell is only discussing the force at one point in the motion. He isn't integrating the equations of motion. There is no contradiction between his statement and yours.
The contradiction is between what Purcell describes and what actually happens when you throw an electron past the current-in-the-wire. You might argue that that's because he doesn't give a full description, but I think the problem goes deeper than that.

Nugatory
Mentor
Your comments noted. We see forces like this → and this ↑ but not like this ⥀ .
They are the same thing, as the magnitude and sign of the forces in the horizontal and vertical directions change with the direction of motion of the particle. Their vector sum is always a force that is perpendicular to the direction of motion.... And that's circular motion.

The thing is that when you replace the wire with a charged rod, adding two vectors like this → and this ↑ gives you this ↗ with an increasing upward component as per my OP depiction. Look closely at page 193. See where it says the electrons are moving to the right at a speed of v₀ and the test charge at rest near this wire experiences no force whatever? Now imagine the test charge also moves to the right at a speed of v₀. Motion is relative. We now have a situation wherein the ions are effectively moving to the left at a speed of v₀. So the test charge still experiences no force whatever. But if you throw the test charge along the wire at the drift speed, it doesn't keep going in a straight line. It goes around the magnetic field lines. There are issues here. Particularly when you try applying the wire appears to be charged! to a loop. It contains n electrons whether they're motionless or circulating. The charge density doesn't increase. Page 19 of this paper looks interesting: ‘It is important to note that no contractions are involved from the standpoint of the laboratory frame, but only from the standpoint of a frame moving relative to the laboratory. The only difference between a wire carrying a current and a wire not carrying a current is the existence of a drift velocity for the electrons. The mean distance between the electrons remains unaffected as measured in the laboratory frame.’

Dale
Mentor
2020 Award
I don't think so Dale. I've read it through carefully
And yet you cannot actually produce a quote where he says what you claim. I'm sorry, but that seems like misinterpreting something to me.

Look closely at page 193. See where it says the electrons are moving to the right at a speed of v₀ and the test charge at rest near this wire experiences no force whatever?
This is another correct statement by Purcell. Your examples all point to correct statements.

Now imagine the test charge also moves to the right at a speed of v₀. Motion is relative. We now have a situation wherein the ions are effectively moving to the left at a speed of v₀. So the test charge still experiences no force whatever
This is not correct. Motion is relative, but the scenario is not symmetric between these two frames.

I don't know how familiar you are with relativity. Are you familiar with four-vectors? The charge density and current density together form a four-vector. It makes it easy to see that there is a nonzero charge density in the drift frame.

There are issues here. Particularly when you try applying the wire appears to be charged! to a loop. It contains n electrons whether they're motionless or circulating.
Have you actually worked this out? It is a worthwhile exercise. For simplicity, consider a square loop, with two sides parallel to the direction of motion. Run the numbers and see if there is actually an issue applying it to a loop.

John, I am not a big fan of Purcell's approach as a teaching tool. But it is factually correct. If it is not helping you learn, then I would drop it and read a different book. But you are "tilting at windmills" here.

Last edited:
bcrowell
A.T.
See where it says the electrons are moving to the right at a speed of v₀ and the test charge at rest near this wire experiences no force whatever? Now imagine the test charge also moves to the right at a speed of v₀. Motion is relative. We now have a situation wherein the ions are effectively moving to the left at a speed of v₀.
As DaleSpam said, there is no symmetry between the rest frames of the ions and the electrons. The ions are fixed in a grid, so they keep their spacing in their rest frame when the current is switched on. The electrons aren't fixed so when the current is switched on, they change their spacing in their own rest frame.

There are issues here. Particularly when you try applying the wire appears to be charged! to a loop. It contains n electrons whether they're motionless or circulating.

See this diagram by DrGreg:

From this old thread:
https://www.physicsforums.com/threa...-of-electrostatics.577456/page-3#post-3768045

AT: try your scenario for a loop of current. It just doesn't work.

DaleSpam said:
And yet you cannot actually produce a quote where he says what you claim. I'm sorry, but that seems like misinterpreting something to me.
I can give you more quotes. There's no misinterpretation, he said what he said. I can put up more screenshots if you like.

DaleSpam said:
This is not correct. Motion is relative, but the scenario is not symmetric between these two frames.
It's symmetric enough. Draw it out:

++++++++++++++++++++
--------------------------------->
------------------

++++++++++++++++++++
--------------------------------->
------------------ >

<++++++++++++++++++++
---------------------------------
------------------

<++++++++++++++++++++
---------------------------------
-----------------+

DaleSpam said:
I don't know how familiar you are with relativity. Are you familiar with four-vectors? The charge density and current density together form a four-vector. It makes it easy to see that there is a nonzero charge density in the drift frame.
I consider myself to be a "relativist", and a guy who "roots for relativity", and yes, I'm familiar with four-vectors. And Lorentz invariance.

DaleSpam said:
John, I am not a big fan of Purcell's approach as a teaching tool. But it is factually correct.
What can I say Dale? I'm with Vanhees71 on this. You cannot derive rotational magnetic force from length contraction of a linear wire. You have to start with the electromagnetic field, then everything becomes clear.

A.T.
AT: try your scenario for a loop of current.
The diagram above shows a loop.
It just doesn't work.
What exactly doesn't work?

Dale
Mentor
2020 Award
I can give you more quotes. There's no misinterpretation, he said what he said. I can put up more screenshots if you like.
A single screenshot or quote which actually supports your claim would be best. So far, you have not provided a single one that either shows a mistake by Purcell or that shows that he claims that a free particle moves straight towards the wire.

It's symmetric enough.
This is factually wrong, see below.

I consider myself to be a "relativist", and a guy who "roots for relativity", and yes, I'm familiar with four-vectors.
Then briefly. In the lab frame (and in units where c=1) the proton four-current is ##(\rho_0,0,0,0)## while the electron four-current is ##(-\rho_0,-\rho_0 v_0,0,0)##, where ##\rho_0## is the density of conduction electrons in the lab frame and ##v_0## is their velocity in the lab frame. This gives a total four-current of ##(0,-\rho_0 v_0,0,0)##.

Boosting to a frame moving at ##v## wrt the lab in the direction of the current we get that the proton four-current is ##(\gamma \rho_0, -v \gamma \rho_0,0,0)## and the electron four-current is ##(\gamma (v_0 v-1) \rho_0, \gamma (v-v_0) \rho_0,0,0)##. This gives a total four-current in the moving frame of ##(\gamma v_0 v \rho_0, -\gamma v_0 \rho_0,0,0)##

vanhees71
Gold Member
This is a great example for why Purcell has too much pedagogics in his approach. It's much simpler to introduce special relativity before starting electrodynamics and introduce electrodynamics as the full set of Maxwell equations as the outcome of about a century of careful investigations concerning electricity and magnetism, culminating in the comprehensive understanding of the subject in terms of the field paradigm by Faraday and its mathematical foundation by Maxwell. Then it's very clear, how the quantities behave under proper orthochronous Poincare transformations, and as soon as you formulate the complete electrodynamical and mechanical problem in a relativistic way, there are no apparent paradoxes.

A simple straight wire (or even two parallel wires) with DC is, however, not as trivial a task as it is almost always treated, because if you want to make it fully relativistic, you have to take into account the Hall effect and use a kind of "two-fluid picture". The classical "Jellium model" is of course fully sufficient to get a principle picture.

Of course, the preferred frame of reference in this model is the restframe of the wires. The model consists of a "rigid" homogeneous positively charged background (the ions making up the lattice of the metal) and a fluid consisting of the conduction electrons. For a DC current you have a stationary problem, i.e., everything is time-independent. The full set of macroscopic Maxwell equations under the simplifying assumption that ##\epsilon=\mu=1## read
$$\vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{E}=0,$$
$$\vec{\nabla} \times \vec{B}=\vec{j}, \quad \vec{\nabla} \cdot \vec{E}=\rho.$$
Now we have the constitutive equations
$$j^{\mu}=Ze n_+ u_+^{\mu} -e n_- u_-^{\mu}.$$
$$\vec{j}=\sigma \left (\vec{E} + \frac{\vec{v}} \times \vec{B} \right).$$
This you can solve with the ansatz ##\vec{j}=j \Theta(a-\rho) \vec{e}_z##, ##\vec{u}_+=0##. Note that ##n_+## and ##n_-## are scalar particle densities, i.e., measured in the (local) restframe of each of the fluids. There is no common rest frame of both fluids of course!

Then it's immideately clear that you need the magnetic field in Ohm's Law, i.e., taking into account the Hall effect self-consistently, to make everything relativistically consistent although for the typical household currents it's totally negligible since ##\vec{u}_-## is tiny even on everyday scales of speeds.

However, with this fully relativistic solution there is no problem anymore with relativistic covariance, and you can boost to any frame you like, particularly also to the frame where the conduction electrons are at rest and the positive ions are moving.