Why is the Lorentz Force always perpendicular to velocity?

[JiuBeixin]

As I understand your scenario, the object is moving in frame $S$ not parallel and not perpendicular to the conductor. But the force must be perpenducular to the object's velocity, because in this frame there is only a magnetic field. It may look like this:

View attachment 370316​

Here is the question. Why force is perpenducular to the object's velocity? In frame $S'$ the force is perpenducular to the conductor(in my case)

 

[Sagittarius A-Star]

Here is the question. Why force is perpenducular to the object's velocity? In frame $S'$ the force is perpenducular to the conductor(in my case)

Does my picture in posting #26 describe correctly your case in lab-frame $S$?

 

[A.T.]

In the frame where the test charge is at rest, the EM-force is perpendicular to the (now charged) conductor.

I don't think so, because in frame S′ the conductor is not oriented horizontally.

You might be right here. My statement above was based on symmetry w.r.t. the normal to the conductor that passes through the test charge. I think it would apply to a long charged rod without a current flowing through it.

But here the negative and positive charges are moving in different directions, so their fields are contracted along different axes, which eventually breaks the above symmetry.

It also makes sense that a boost perpendicular to a force doesn't change its direction.

 

[JiuBeixin]

After thinking about this some, I think you have gotten some good advice when you suggested you wanted to learn more about transformation laws.

Maxwell's equations are fully relativistic. If you know how to properly transform charges and currents (usually expressed as charge density and current density), you can use Maxwell's equations to answer your questions. Judging by your questions, you don't currently even know the name of the Lorentz transform - apologies if I got this wrong).

You give the impression of not having any questions about Maxwell's equations (in the most basic form, two Gauss' laws, Faraday's law, and Ampere's law), but rather about the relativistic aspects.

There's a standardized way of describing the source terms. This is in terms of charge density, which would be coulomb/meter^3, and current density, which would be ampere's per meter^2.

The tool you need to know how these quantities transform when switching from a rest frame to a moving frame is called the Lorentz transform, and that's what you need to research.

It'd also be helpful to know the terms "Lorentz boost", which is the way we describe going from a "stationary" frame of reference to a moving frame of reference, and the term "invariant". Invariants are quantites that don't change when you perform a Lorentz boost, I.e. switch from a stationary frame to a moving frame.

The notable invariant of charge and current doesn't have a catchy name, but is given by the quantity

$$c^2 \rho^2 - |J|^2$$

where c is the speed of light, $\rho$ is the charge density in coulomb's/meter^3, and J is the current density in amperes / m^2.

Note that one point of this is that when we boost a neutral current carrying wire in the direction of current flow, it becomes charged. Which you seem to be aware of.

As far as sources go, the results of a "swarm of particles" are well known, but I don't know of many texts that go through the actual work of analyzing a swarm of particles. Usually they just present the results for the continuum limit.

Thanks! You are right. I really dont know the details about Maxwell's equations and Lorentz transformation, so can you give me some advices that where should I begin my study of these laws, as I am only a beginner.

 

[JiuBeixin]

Does my picture in posting #26 describe correctly your case in lab-frame $S$?

except the force, cause I don't know the force's direction in frame $S$ now.

 

[A.T.]

In frame S′ the force is perpenducular to the conductor(in my case)

That was my initial assumption too, because it's true for a charged rod without a current, but not necessarily for one with a current. See my post #33.

 

[Sagittarius A-Star]

except the force, cause I don't know the force's direction in frame $S$ now.

At @PeterDonis wrote in posting #4, the Lorentz force is
$F = q \left( \vec{E} + \vec{v} \times \vec{B} \right)$.
If there is no electic field in frame $S$, then the cross product remains, which is perpendicular to the velocity and to the magnetic field.

Source:
https://en.wikipedia.org/wiki/Cross_product

 

[pervect]

As I dig into this a bit more, I am becoming a bit confused myself. Purcell famously explains magnetism as due to "length contraction" of the distance between charge, which is what the original poster's question is about. But - length contraction is a second order effect in (normalized) velocity $\beta$, and magnetism is first order in $\beta$. Therefore, I can't help but think that Purcell's explanation is fundamentally flawed.

Going back to the charge-current transformation, if we use geometric units we have:

$$\rho' = \gamma \left(\rho - \beta J \right) $$

I _think_ that in SI units, the second term looks more like -v/c^2 J, when you choose non-geometric units where c is not unity. That would be units of (columobs / meter^2 second) / (meters/second) = (coulombs/meter^3) which looks right.

And to first order $\gamma=1$, so the only first order term is $\approx$ $-\beta$ J. Or - v/c^2 J in non-geometric units. So, why does Purcell attribute magnetism to "length contraction" at all?

On a practical level, I think this only emphasises the need to use the full Lorentz transform - attributing it to "length contraction" seems to me to be misguided.

On a side note, I have the impression that the OP isn't familiar with the Lorentz transformation (I could be wrong on that), but if that's the issue, I think the OP needs to be at least curious enough to open another thread on that topic. Though I also suspect that it's something best learned from something other than a PF post.

 

[SiennaTheGr8]

As I dig into this a bit more, I am becoming a bit confused myself. Purcell famously explains magnetism as due to "length contraction" of the distance between charge, which is what the original poster's question is about. But - length contraction is a second order effect in (normalized) velocity β, and magnetism is first order in β. Therefore, I can't help but think that Purcell's explanation is fundamentally flawed.

If I'm correctly understanding what you're saying, I don't think the "first-order"/"second-order" business means there's a flaw. The "order" by itself doesn't tell you whether we can notice the effects of something when only "non-relativistic" speeds are involved.

Consider, for example, that kinetic energy is a "relativistic effect" in precisely the same way that time dilation is: the equation $E_k = (E - E_0) = E_0 (\gamma - 1)$ has the same form as $(t - t_0) = t_0 (\gamma - 1)$. To lowest order, $E_k = .5 E_0 \beta^2$ and $(t - t_0) = .5 t_0 \beta^2$, but the former is perfectly noticeable while the latter is not at all. There's nothing in the math that says, "In the Newtonian limit, you'd better keep that lowest-order $E_k$ term, but go ahead and set $(t - t_0)$ to zero."

So even though length contraction in itself isn't something we'd notice in everyday situations, it can "give rise" to effects that we do notice. This aspect of Purcell's approach is fine, I think (and rather interesting!).

That said, Purcell does seem to give some people the wrong impression that "electrostatics + SR = magnetism," as if the magnetic field were somehow less fundamental than the electric field. The book even calls magnetism a "relativistic effect," which I think is at least misleading (though some may disagree).

 

[A.T.]

But - length contraction is a second order effect in (normalized) velocity β, and magnetism is first order in β.

Are you taking into account that the distances change for both, the positive and negative charges, in an inverse manner. And both these changes contribute cumulatively to the charge density imbalance and thus net charge.

 

[A.T.]

The "order" by itself doesn't tell you whether we can notice the effects of something when only "non-relativistic" speeds are involved.

Yes, you also have to look at the constant factors involved in E and B, not just the exponents.

 

[Sagittarius A-Star]

But - length contraction is a second order effect in (normalized) velocity $\beta$, and magnetism is first order in $\beta$. Therefore, I can't help but think that Purcell's explanation is fundamentally flawed.

I think that cannot be, because Purcell does a calculation of the electric force in the rest frame of the object via lenght contraction (not only a qualitative explanation). The calculation result fits to the force transformation formula.

Here is a simplified calculation:
https://physics.weber.edu/schroeder/mrr/MRRtalk.html

 

[pervect]

As far as having both change, that would just double the effect, not change the order.

It might be related as to linearizing around v=0 or v not equal to zero when expanding gamma in a taylor series to argue changes in gamma, 1/gamma, and hence "length contraction" are second order, but I'm not convinced that's the explanation at this point.

Personally I'm just more comfortable with using the full Lorentz transformation and getting the result directly rather than trying to decompose it into sub-categories such as "length contraction", "time dilation", or "the relativity of simultaneity".

I'm not sure exactly what to say to someone who may not be comfortable with the advice "just use the Lorentz transform", though.

The best argument I found for why we need to use the Lorentz transform to transform charge and currnet is that that the continuity equation $\frac{d\rho}{dt} + \nabla \cdot J$ must transform covariantly, but that's just not a B-level argument so it doesn't really count for this thread. And I'm not sure how to turn it into a B-level argument.

 

[Jizhenzhang]

Great question. This is where special relativity shows the real nature of the Lorentz force.

Your intuition is basically right: a vertical velocity component does not produce any additional charge density in the wire, because length contraction only appears along the direction of relative motion. So in the rest frame of the test charge, the electric force from the wire is still purely vertical.

But force itself transforms relativistically between frames, and a key rule must always hold: the magnetic force must remain perpendicular to the particle’s velocity in every inertial frame. That is not optional—it is required by the geometry of spacetime.

When we switch back to the lab frame, the vertical force component becomes slightly smaller, and a small horizontal force component appears. This horizontal part does not come from extra charge density. It comes from the relativistic transformation of force, which automatically adjusts the components so that the total force stays perpendicular to velocity.

If you calculate the dot product of force and velocity, you will find it is exactly zero. That means the force does no work, which is the defining property of the Lorentz force.

In short: the vertical force comes from length contraction, and the horizontal component comes from relativistic force transformation. Together, they keep the force perpendicular to velocity in all frames. That is why the Lorentz force is fundamentally a relativistic effect.

 

[PeterDonis]

a key rule must always hold: the magnetic force must remain perpendicular to the particle’s velocity in every inertial frame. That is not optional—it is required by the geometry of spacetime.

How would you express this claim in terms of a covariant mathematical formula? Note that the separation between "magnetic force" and "electric force" is frame-dependent.

 

[PeterDonis]

the force does no work, which is the defining property of the Lorentz force.

This depends on how "Lorentz force" is defined--which has already been the subject of some discussion in this thread. Certainly the $q \vec{E}$ component of the overall EM force on a charged object can do work.

 

[A.T.]

It might be related as to linearizing around v=0 or v not equal to zero when expanding gamma in a taylor series to argue changes in gamma, 1/gamma, and hence "length contraction" are second order, but I'm not convinced that's the explanation at this point.

Isn't the explanation simply that even a second order term can be relevant if it has a large enough constant factor?

You should estimate the order of magnitude of the result (E-force in the test charge frame), not argue based on effect order of one input (length contraction).

 

[Sagittarius A-Star]

It might be related as to linearizing around v=0 or v not equal to zero when expanding gamma in a taylor series to argue changes in gamma, 1/gamma, and hence "length contraction" are second order, but I'm not convinced that's the explanation at this point.

The simplified calculation, I linked to in posting #42, does an approximation for small $v$ in equation (1). But of course, the calculation could be changed to become exact.

Purcell does an exact calculation in the book.
His calculation is also more complicated, because he calculates the more general case that the velocity of the object in the lab frame is not assumed to be equal to the velocity of the electrons in the conductor. He calculates the velocity of the electrons in the rest frame of the object by using the "relativistic velocity addition" formula.

His calculation goes over 3 pages. It gets also correctly, that the force in the lab frame must be by a factor $1/\gamma$ smaller than in the rest frame of the object.

This is his conclusion of his calculation:

​

Source (Electricity and Magnetism 3rd Edition by Edward M. Purcell, David J. Morin, page 263):
https://www.amazon.com/Electricity-Magnetism-Edward-M-Purcell/dp/1107014026?tag=pfamazon01-20

 

[Sagittarius A-Star]

As far as having both change, that would just double the effect, not change the order.

I think the point is, that for the charge line-density, the quadratic term in $\beta'$ is the first nonzero term.

The constant terms of the Lorentz contractions cancel out between the electrons and positive atom ions (I checked, that this is also valid for an exact calculation).

​

Source:
https://physics.weber.edu/schroeder/mrr/MRRtalk.html

In the lab frame, the magnetic force is proportional to the product of the object velocity and the electrons velocity (which are assumed to be equal in the simplified calculation).

Conclusion: In this scenario (with assumption, that object velocity and electrons velocity are equal), magnetic force is a second order effect in (normalized) velocity $\beta$.

 

[Sagittarius A-Star]

length contraction is a second order effect in (normalized) velocity $\beta$

The effect of this on the charge line density might be an artifact of the constraint in the OP and in the linked simplified calculation, that object-velocity and electrons-velocity are equal.

Purcell calculates exactly the more general case that the velocity of the object in the lab frame is not assumed to be equal to the velocity of the electrons in the conductor.

In his calculation, in the restframe of the object, the charge line density is not proportional to $\beta^2$, but to the product $\gamma\beta_0\beta$. The electrons move in the lab-frame with $\beta_0$.

 

[pervect]

Isn't the explanation simply that even a second order term can be relevant if it has a large enough constant factor?

You should estimate the order of magnitude of the result (E-force in the test charge frame), not argue based on effect order of one input (length contraction).

I went over the calculation from Purcell from the references that others have posted - thanks to all who responded, by the way - and I can see why his argument works in the special case when the drift velocity, which I'll call v_d, is equal to the observer velocity. When v_0 = v_d = v, the result which is multilinear in both v_0 and v_d is second order in v. This is the same order as length contraction, so all is good in that case.

I see that others have posted another analysis by Purcell, which handles the case where v_0 is not equal to v_d. have yet to study this more general analysis and will have to read and think about it before I make a furthter response.

 

[pervect]

The effect of this on the charge line density might be an artifact of the constraint in the OP and in the linked simplified calculation, that object-velocity and electrons-velocity are equal.

Purcell calculates exactly the more general case that the velocity of the object in the lab frame is not assumed to be equal to the velocity of the electrons in the conductor.

In his calculation, in the restframe of the object, the charge line density is not proportional to $\beta^2$, but to the product $\gamma\beta_0\beta$. The electrons move in the lab-frame with $\beta_0$.

View attachment 370366​

I'm trying to untangle the notation. I am assuming that the lab frame, S, is unprimed, so all unprimed quantites are measured in S. And I am assuming that the primed quanties are measured in frame S' moving with velocity v relative to S. Further, I am assuming $v_0$ is the drift velocity of the electrons in S, and $\lambda_0$ is the linear charge density of the protons in frame S, (i.e. \lambda_0 is a positive quantity), which in frame S is equal in magnitude but opposite in sign to the linear charge density of the electrons because the wire is neutral.

Assuming this is all correct, we continue with the comparison. Because the results seem to agree with what I expect, I am assuming this is all correct.

[add]. To summarize the result from the attachment of Purcell's analysis:

$$\lambda' = \gamma \beta \beta_0 \lambda_0$$

To compare it to my notation, we need to introduce the current in the lab frame, which I will call $I_0$, which I think should be equal to $-c \beta_0 \lambda_0$, with the minus sign occuring because the electrons have a negative charge. This makes the charge density in the lab frame from this analysis reduce to $-\gamma \beta \frac{I_0}{c}$, which is essentially the result I got in my post #38 if we go to the equations I wrote before I assumed $\gamma \approx 1$.

To recap, and to replacing the original volume densities I wrote in my original post #38 $\rho$ and current densities J by multiplying by the cross sectional area of the wire A, converting them into linear charge densites $\rho$ and currents I, while additionally adding in the factors of c which I omitted, we have:

$$\rho' = \gamma \left (\rho - \beta \frac{I_0}{c} \right)$$

Untis wise, $\rho$ in the above equation has units of charge/unit length, while I/c has units of (charge / second) / (length /second) = (charge / length), which checks.

[add]. For consistency, I should have replaced $\rho$ with $\lambda$, but I didn't. They're both linear charge densities, I had just been using a different symbol.

My overall comment is that the equation I wrote above, which is based on the Lorentz transformation using the charge/ current four vector method, is to my mind easier, but I may be biased because it's the way I'm used to doing it - and it's also the way I trust - because it's the way I'm used to doing it. I think that using charge densities and currents is also friendlier and arguably more fundamental than using drift velocities which are based on the Drude model - the actual motion of the electrons in a physical wire is more complicated, but we can approximate them owith the Drude model as a constant drift velocity.

I am still not particularly happy with how the OP handles the case where the wire is not in the direction of the boost and rather supsicious that their conclusions are suspect, but I'm not particularly in the mood to spend more time on it at this point.

 

[Sagittarius A-Star]

Lorentz transformation using the charge/ current four vector method, is to my mind easier

I agree, and your calculation-result is equivalent to that of Purcell.

I think that Purcell's book is addressed to students, who did not (yet) learn about 4-current.

Length contraction is indeed a second-order effect in (normalized) velocity $\beta$. But in the general case, Purcell derives the magnetic force not only from a length contraction.
The term $\beta\beta_0$ comes into his charge line-density formula via the "relativistic velocity addition" for the electrons.