The Science Asylum -- How Special Relativity Fixed Electromagnetism.

[aetherist]

How Special Relativity Fixed Electromagnetism…….. The Science Asylum…… 636K subscribers… 19K… 369K views 3 years ago 2,058 Comments
Electricity and Magnetism…….Electrodynamics (electricity and magnetism) is governed by Maxwell's equations and the Lorentz force law, but that left it a little broken. It would take Albert Einstein inventing special relativity to fix it. If magnets are based on motion and motion is relative, how does that work?

Nick explains how Einstein's STR explains the magnetic field around a wire.
Nick's squirrel (which has a positive charge) moving along parallel to the wire sees that the positive charges in the stationary wire are closer together & hence the wire has a positive charge & hence the squirrel is repelled from the wire.
This repulsion is in effect identical to the repulsion described by the conventional right hand rule, based on the wire being surrounded by a cylindrical magnetic field.
But i see at least 2 problems for the STR theory.

Problem-1A.  If i placed a (stationary) Cu sheet between the squirrel & the wire then the Cu sheet would block/shield the charge repulsion (ie block/shield the faux magnetic force). Whereas we all know that a Cu sheet has almost zero blocking/shielding effect re the magnetic field around a wire.
Problem-1B.  In other words, 2 parallel wires carrying currents experience a force of attraction or repulsion, & i reckon that if u placed a Cu sheet between the wires then that attraction or repulsion would not be much affected.
Problem-1C.  But if the attraction or repulsion is due to Nick's charge effect then a Cu sheet would reduce the attraction or repulsion to almost zero. Hence the STR explanation for magnetic force near a wire is wrong.

Problem-2A.  We know that a charge moving away from a current carrying wire experiences a magnetic force parallel to the wire (as per the right hand rule).
Problem-2B.  But, if Nick's squirrel is moving away from the wire (instead of parallel to the wire) then Nick karnt invoke some kind of STR length contraction of the wire & charges, hence Nick's squirrel would not experience a force in any direction, which would be contrary to what we know happens (as per 2A).

My inclusion of Problem-1ABC might be wrong, praps the Cu sheet would not block charge.
But my Problem-2AB i reckon on its own kills the STR theory.

Has anyone here ever seen any discussion of these 2 problems or similar?

[jasonRF]

My intro physics course (I took the ‘honors’ version) included special relativity, relativistic transformation of the EM field, etc.   We used Electricity and Magnetism, 2nd edition, by Purcell.  Not a great book for a first exposure to electromagnetism, in my opinion.  It ‘derives’ the magnetic force using special relativity and coulomb’s law.  There were only a few EEs in the class, as it was mostly full of physics and engineering physics majors. 

A book that does that same derivation but also includes a lot of the history, including details of the experimental basis for Coulomb’s law and special relativity, is Electromagnetics by Elliott.  He was an EE prof that also wrote a very good book on antennas.    The crazy thing is that he shows the table of data in Coulomb’s paper, and it just has a handful of data points 

I do hate the title of that youtube video, though,  since Maxwell’s equations were already consistent with special relativity.  It feels click-baity. 

[switchabl]

I do hate the title of that youtube video, though,  since Maxwell’s equations were already consistent with special relativity.  It feels click-baity.

It's true that Maxwell's equations turned out to be fine after all but that wasn't so clear at the time. Electromagnetism at the turn of the century was a theory in crisis. The interpretation of the equations was very different from how people think about them today. It was generally assumed that the electrical and magnetic fields were disturbances in some fixed "ether". So in a sense the problem described in the video wasn't really a problem at all. Because the moving reference frame would be moving relative to the ether and so it wasn't supposed to be equivalent.

Unfortunately, experiments started to contradict the ether hypothesis (most famously the Michelson-Morley experiment). Physicists scrambled to make sense of it all. Heaviside FitzGerald suggested that objects moving relative to the ether might experience length contraction. That would have fixed some problems but not others (of course, a similar phenomenon occurs naturally in SR and features prominently in the video). Lorentz and Poincaré tinkered with the definitions of force and even time, trying to come up with something that would not depend on movement relative to the ether. A lot of ideas and mathematics (notably the Lorentz transform) was already in place at that point but Einstein was the first to present a coherent theory, based on a few simple assumptions, instead of a bunch of ad-hoc modifications.

There seems to be this narrative that SR "fixes" Newtonian mechanics while electromagnetics was relativistic all along. But that is mostly a didactic fiction. Historically, special relativity is very much the culmination of an effort to find a consistent theory of electromagnetics. Deriving EM from electrostatics and relativity is elegant and fun but it is also egregious retcon. 

EDIT: It was FitzGerald who first suggested deformation of objects depending on their movement relative to the ether, probably inspired by the work of Heaviside on moving charges. I mixed that up.

[aetherist]

My intro physics course (I took the ‘honors’ version) included special relativity, relativistic transformation of the EM field, etc.   We used Electricity and Magnetism, 2nd edition, by Purcell.  Not a great book for a first exposure to electromagnetism, in my opinion.  It ‘derives’ the magnetic force using special relativity and coulomb’s law.  There were only a few EEs in the class, as it was mostly full of physics and engineering physics majors. 

A book that does that same derivation but also includes a lot of the history, including details of the experimental basis for Coulomb’s law and special relativity, is Electromagnetics by Elliott.  He was an EE prof that also wrote a very good book on antennas.    The crazy thing is that he shows the table of data in Coulomb’s paper, and it just has a handful of data points 

I do hate the title of that youtube video, though,  since Maxwell’s equations were already consistent with special relativity.  It feels click-baity.

How did Purcell & Elliott deal with my Problem-2AB?

[IanB]

How did Purcell & Elliott deal with my Problem-2AB?

They only had to deal with their problems. Your problems are yours alone to deal with.

[aetherist]

I do hate the title of that youtube video, though,  since Maxwell’s equations were already consistent with special relativity.  It feels click-baity.

It's true that Maxwell's equations turned out to be fine after all but that wasn't so clear at the time. Electromagnetism at the turn of the century was a theory in crisis. The interpretation of the equations was very different from how people think about them today. It was generally assumed that the electrical and magnetic fields were disturbances in some fixed "ether". So in a sense the problem described in the video wasn't really a problem at all. Because the moving reference frame would be moving relative to the ether and so it wasn't supposed to be equivalent.

Unfortunately, experiments started to contradict the ether hypothesis (most famously the Michelson-Morley experiment). Physicists scrambled to make sense of it all. Heaviside suggested that objects moving relative to the ether might experience length contraction. That would have fixed some problems but not others (of course, a similar phenomenon occurs naturally in SR and features prominently in the video). Lorentz and Poincaré tinkered with the definitions of force and even time, trying to come up with something that would not depend on movement relative to the ether. A lot of ideas and mathematics (notably the Lorentz transform) was already in place at that point but Einstein was the first to present a coherent theory, based on a few simple assumptions, instead of a bunch of ad-hoc modifications.

There seems to be this narrative that SR "fixes" Newtonian mechanics while electromagnetics was relativistic all along. But that is mostly a didactic fiction. Historically, special relativity is very much the culmination of an effort to find a consistent theory of electromagnetics. Deriving EM from electrostatics and relativity is elegant and fun but it is also egregious retcon. 

Let us assume that length contraction due to velocity is included in all aether era relativity theories, eg in Heaviside FitzGerald Searle Voigt Cohn Larmor Poincare Lorentz & others.
Q1.  Did any of the above aetherists (i will call them aetherists, but that might be wrong, eg Cohn probly did not believe in aether) ever invoke" length contraction of charges in a wire" to explain the magnetic field or magnetic force?
Q2.  If the answer to Q1 is No, then my next question is -- could aetherists have legitimately invoked length contraction?
Q3. If the answer to Q2 is Yes, then my next question is – why did Nick (& Purcell)(& Elliott) say that Einstein's STR solved the problem?
If every aetheric relativity already included length contraction, then when Einstein came along with STR his STR was just another form of relativity, & each & every form of relativity could explain the "length contraction of charges in a wire" idea.
Q4.  Why didn’t Nick & Purcell & Co simply say that say FitzGerald's relativity (FitzGerald had priority i think) explained?

No experiment has ever proven length contraction. What i mean is that velocity might produce length dilation, or it might produce a mix of length contraction & width contraction, etc . In which case Purcell & Co could have if they wished used any mix of length or width contraction or dilation, & Purcell & Co would still have obtained the same result for magnetism near a wire.
Q5.  So, my next question is – why is magnetism near a wire spruiked to be a proof of STR?

[aetherist]

How did Purcell & Elliott deal with my Problem-2AB?

They only had to deal with their problems. Your problems are yours alone to deal with.

Their problem was that they had a (silly)(faulty) STR explanation for their pseudo magnetic field near a wire when a charge was moving parallel to the wire, but, they could not (using STR) produce a pseudo magnetic field near a wire when a charge was moving perpendicularly to the wire.
So, they had a pseudo magnetic field when a charge was moving parallel,
but a real magnetic field when a charge was moving perpendicularly.
Which is what i call Problem-2AB.

Or more simply, they used STR for a charge moving parallel,
but did not need STR for a charge moving perpendicularly.

And, as i mentioned in my other posting, they did not need STR for a charge moving parallel, any (older)(pre-STR) relativity would have done the job.
And, that there job is/was/would be wrong, koz, even if all relativity theories are correct re length contraction, the silly Purcell explanation using length contraction is wrong, length contraction duznt make/cause/create a pseudo magnetic field.
A magnetic field is real.
And length contraction can affect measurements, but length contraction plays no (essential) part in the mechanism that creates a magnetic field near a wire.

[switchabl]

I will take the question in good faith despite your rant because it is interesting and others might like to know. And it is actually discussed in the book by Purcell at the end of the chapter.

So assume the squirrel is moving perpendicularly away from the the wire and change to the squirrel rest frame. In that frame, the wire is now moving away from the squirrel but the current flow is still there. Therefore the overall movement of the electrons is at an angle. The electric field of a moving charge is compressed along the axis of movement (this follows from Maxwell's equations and was known before SR but is often derived using relativity because it is convenient). So looking at an electron on its left and an electron on its right, it will see those at different angles and experience different electric fields. The maths is somewhat cumbersome but if you integrate over the whole wire you get the correct force.

This is hopefully easier to grasp if you look at the drawing (please excuse the crudity of the squirrel).

The title of the video is maybe somewhat misleading in that the "problem" it discusses is more a didactic illustration of how electromagnetism is consistent with special relativity, rather than something that would have been historically relevant to its development. The notion that Maxwell's equations should be equally valid in all inertial frames is one of the key postulates of Einstein's theory. In the old ether theory this was never expected to be true and this was not considered a problem. The actual problem was that some predictions of the fixed ether interpretation contradicted  the new experimental evidence (like Michelson-Morley, Rayleigh-Brace, Trouton-Noble).

[aetherist]

I will take the question in good faith despite your rant because it is interesting and others might like to know. And it is actually discussed in the book by Purcell at the end of the chapter.

So assume the squirrel is moving perpendicularly away from the wire and change to the squirrel rest frame. In that frame, the wire is now moving away from the squirrel but the current flow is still there. Therefore the overall movement of the electrons is at an angle. The electric field of a moving charge is compressed along the axis of movement (this follows from Maxwell's equations and was known before SR but is often derived using relativity because it is convenient). So looking at an electron on its left and an electron on its right, it will see those at different angles and experience different electric fields. The maths is somewhat cumbersome but if you integrate over the whole wire you get the correct force.

This is hopefully easier to grasp if you look at the drawing (please excuse the crudity of the squirrel).

The title of the video is maybe somewhat misleading in that the "problem" it discusses is more a didactic illustration of how electromagnetism is consistent with special relativity, rather than something that would have been historically relevant to its development. The notion that Maxwell's equations should be equally valid in all inertial frames is one of the key postulates of Einstein's theory. In the old ether theory this was never expected to be true and this was not considered a problem. The actual problem was that some predictions of the fixed ether interpretation contradicted  the new experimental evidence (like Michelson-Morley, Rayleigh-Brace, Trouton-Noble).

Well written re compression. I like the idea of the compression of an electric field. I suppose that (in aether theory) we have real compression (due to the charge moving through the aether), & we have pseudo (apparent) compression (due to squirrel moving through the aether) – but STR of course wont allow 2 kinds.

Question-1.   In Purcell's parallel velocity case. Why did Purcell ignore axial compression for the field from his protons?
This would introduce an xx force, ie in addition to Purcell's (phony) yy force.

Question-2.   In a parallel velocity case. When there is no current on the wire, & the squirrel is static, the force is zero.
After the current is switched on, & the squirrel is static, the force is zero.
But axial compression requires that there is a force. Partly axial & partly perpendicular.

U call my arguments/claims a rant. Then allow me to call claims that MMXs were null lies.
No gas mode MMX has ever been null – they all detected an aetherwind.

[TimFox]

The canonical version about aether wind below.  Remember, we're talking about an axiom that the speed of light in vacuum is independent of the reference frame:

Maurizio Consoli of the Italian National Institute of Nuclear Physics in Catania, Sicily, argues in Physics Letters A (vol 333, p 355) that any Michelson-Morley type of experiment carried out in a vacuum will show no difference in the speed of light even if there is an aether. According to him, electroweak theory and quantum field theory suggest that light could appear to move at different speeds in different directions in a medium such as a dense gas in contradiction with special relativity; the speed of light would be sensitive to motion relative to an aether and the refractive index of the medium. Consoli and Evelina Costanzo propose an experiment with laser light passing through cavities filled with a relatively dense gas. With the Earth passing through an aether wind, light would travel faster in one direction than in the perpendicular direction..... Consoli and Constanzo have not run the proposed experiment. The mathematical treatment of their paper does not use the relativistic dragging coefficient to account for the speed of light in a moving medium, and most physicists regard this as an elementary error that leads to their incorrect conclusions. Their paper is very similar to another similarly flawed paper by Reg Cahill ("R.T. Cahill A New Light-Speed Anisotropy Experiment: Absolute Motion and Gravitational Waves Detected, in Progress in Physics, vol 4 , 2006" ), another proponent of an experiment that would detect the elusive "preferential frame". Cahill claims to have detected absolute motion with respect to a preferential frame but his paper suffers from the same mathematical shortcomings as the Consoli-Constanzo paper as well as from lack of experimental error bars in his experimental data processing. Consequently, their research had no impact on the physics community.

[switchabl]

Question-1.   In Purcell's parallel velocity case. Why did Purcell ignore axial compression for the field from his protons?
This would introduce an xx force, ie in addition to Purcell's (phony) yy force.

Question-2.   In a parallel velocity case. When there is no current on the wire, & the squirrel is static, the force is zero.
After the current is switched on, & the squirrel is static, the force is zero.
But axial compression requires that there is a force. Partly axial & partly perpendicular.

The compression of the electric field plays no rule in the parallel case. It is still there at the level of the individual (moving) electrons and ions but if you sum up their contributions, the effect just averages out instead of creating an imbalance. An easy way to see this is to notice that the charge and current distributions are stationary, so this is a magnetostatics problem. And in this case, the Maxwell equations decouple, the magnetic field depends only on the currents and the electric field depends only on the charge distribution. In particular the movement of the ions can have no impact on the electric field and there is no additional force.

[SiliconWizard]

I'd rather hug some photons.

[switchabl]

Well, you're in luck. Einstein's paper on the photoelectric effect was published in the same year as the special relativity one (1905). It was an exciting time to be a physicist.

[iMo]

Why to mess with the STR when the speed of electrons (and of the squirrel in that video) in the copper wire is a couple of millimeters per second?? 

[switchabl]

The relativistic length contraction at these speeds is absolutely miniscule. But in a neutral wire there are a lot of electrons and a lot of ions which are perfectly balanced. Tipping that balance just very, very slightly is enough to give a signficant macroscopic effect and produce the "missing" force on the squirrel.

From the Feynman lectures (part 2, chapter 1)

And all matter is a mixture of positive protons and negative electrons which are attracting and repelling with this great force. So perfect is the balance, however, that when you stand near someone else you don’t feel any force at all. If there were even a little bit of unbalance you would know it. If you were standing at arm’s length from someone and each of you had one percent more electrons than protons, the repelling force would be incredible. How great? Enough to lift the Empire State Building? No! To lift Mount Everest? No! The repulsion would be enough to lift a “weight” equal to that of the entire earth!

[TimFox]

Well, you're in luck. Einstein's paper on the photoelectric effect was published in the same year as the special relativity one (1905). It was an exciting time to be a physicist.

Yet humanity managed to screw it all up with a war that lasted until at least 1945 (with only pauses).
During 2014, I binged on a new crop of history monographs at the centennial of the official start of WW I.
The scariest thing was an account of popular sentiment at the time of the Archduke's assassination in 1914:
There had been several crises before that, such as the two Balkan wars and the now-forgotten conflict between France and Germany in Morocco (two crises in 1905/6 and 1911).
They had all blown over, so surely this one would as well.

[IanB]

They had all blown over, so surely this one would as well.

To quote Blackadder: "The whole thing started because it was just too much effort not to have a war."

[aetherist]

The canonical version about aether wind below.  Remember, we're talking about an axiom that the speed of light in vacuum is independent of the reference frame:

Maurizio Consoli of the Italian National Institute of Nuclear Physics in Catania, Sicily, argues in Physics Letters A (vol 333, p 355) that any Michelson-Morley type of experiment carried out in a vacuum will show no difference in the speed of light even if there is an aether. According to him, electroweak theory and quantum field theory suggest that light could appear to move at different speeds in different directions in a medium such as a dense gas in contradiction with special relativity; the speed of light would be sensitive to motion relative to an aether and the refractive index of the medium. Consoli and Evelina Costanzo propose an experiment with laser light passing through cavities filled with a relatively dense gas. With the Earth passing through an aether wind, light would travel faster in one direction than in the perpendicular direction..... Consoli and Constanzo have not run the proposed experiment. The mathematical treatment of their paper does not use the relativistic dragging coefficient to account for the speed of light in a moving medium, and most physicists regard this as an elementary error that leads to their incorrect conclusions. Their paper is very similar to another similarly flawed paper by Reg Cahill ("R.T. Cahill A New Light-Speed Anisotropy Experiment: Absolute Motion and Gravitational Waves Detected, in Progress in Physics, vol 4 , 2006" ), another proponent of an experiment that would detect the elusive "preferential frame". Cahill claims to have detected absolute motion with respect to a preferential frame but his paper suffers from the same mathematical shortcomings as the Consoli-Constanzo paper as well as from lack of experimental error bars in his experimental data processing. Consequently, their research had no impact on the physics community.

Prof Reg Cahill duz include aether drag & refractive index in his historic explanation of the correct calibration of oldendays gas-mode MMXs in about 2001.
His re-calibration lifted the 1887 "null" aetherwind speed of about 7 km/s to over 200 km/s. Silly Michelson was looking for 30 km/s.

And then i came along & in about 2017 explained that Cahill was wrong. He (& Fizeau & Co) talked of aether drag, when the correct term & mechanism is simple photon drag (a minor error which duznt change the numbers).

Michelson & Morley & Miller got a systematic signal, periodic in a half turn, which was consistent with aetherwind, ie consistent with Earth's spin & orbit through a (supposedly) static (supposed) luminiferous aether.
U say that the MMXs (& Cahill's own experiments) had too much error. No.
U say that Cahill had the same math shortcomings as Consoli & Constanzo. No.

U said that C&C said that vacuum mode MMXs will show no difference in the speed of light. Yes. C&C are correct. A vacuum mode MMX is merely an experiment that is a good test of relativistic length contraction, ie a null result supports the standard equation for (the Lorentz) gamma. But in reality it karnt falsify any other gammas (if they exist) that are based on say width dilation (eg the Voigt relativity gamma), or are based on a mix of length contraction & width dilation etc.
But i must have a look at the C&C paper. What year was it? Cahill was about 2001, did C&C have priority/precedence over Cahill?

In any case, VV Demjanov made a twin-media interferometer in 1968-72, which used a laser & a dense gas (carbon bisulphide), & his X was 1000 times as sensitive as the oldendays klumzy klunky air-mode MMXs. Demjanov found an aetherwind of 140 km/s to 480 km/s on 22 June at Obninsk. His error bars were less than 1 km/s. He published in English in about 2005.

An aetherwind means that we have an aether.
An aether gives us a preferred frame.
A preferred frame kills STR (& thusly kills GTR too).
And kills Purcell's silly relativistic pseudo magnetic field.
Game over, thank u ball boys & ball girls.

[aetherist]

Question-1.   In Purcell's parallel velocity case. Why did Purcell ignore axial compression for the field from his protons?
This would introduce an xx force, ie in addition to Purcell's (phony) yy force.

Question-2.   In a parallel velocity case. When there is no current on the wire, & the squirrel is static, the force is zero.
After the current is switched on, & the squirrel is static, the force is zero.
But axial compression requires that there is a force. Partly axial & partly perpendicular.

The compression of the electric field plays no rule in the parallel case. It is still there at the level of the individual (moving) electrons and ions but if you sum up their contributions, the effect just averages out instead of creating an imbalance. An easy way to see this is to notice that the charge and current distributions are stationary, so this is a magnetostatics problem. And in this case, the Maxwell equations decouple, the magnetic field depends only on the currents and the electric field depends only on the charge distribution. In particular the movement of the ions can have no impact on the electric field and there is no additional force.

No No No No & No.

I don’t agree with u that electron drift duznt introduce charge compression. If the test charge is static then the approaching negative charges compress, & the receding negative charges dilate. If the test charge is drifting then the approaching positive charges compress & the receding positive charges dilate. But here i don’t believe any of that, i am merely parroting what i think is the standard (false)(silly)(Einsteinian Mafia gatekeeper's) protocols foisted on skoolkids. But i will leave it there, koz i want to jump to Purcell's big lie.

It seems to me that u don’t agree with Purcell.
Purcell's big lie says that after the current is switched on (& while Nick's squirrel is static), there is or should be an electric force (due to length contraction of the train of drifting electrons) – so, to eliminate the unwanted force, Purcell invokes that the negative charges (the drifting electrons) re-arrange themselves (ie spread out), to exactly negate the length contraction effect (before his finger leaves the switch).

In addition, Purcell uses that there silly lie to invoke an initial charge dilation, which allows him to double the final relativistic force. So, his big lie kills 2 big problems with one fell swoop.

Then Purcell finally uses gamma for a third time, to vary the raw invariant electric force (i forget whether the raw force was too big or too small).

If needed (apparently it was not needed) Purcell could have i suppose invoked gamma a fourth time to say introduce some kind of (silly) time dilation effect (Einsteinists are stupid but cunning). But now i am being silly, ignore that.

[bsfeechannel]

The title of the video is maybe somewhat misleading

How about "When EM mortally wounded the ether and SR put the last nail in it's coffin"?

[aetherist]

The title of the video is maybe somewhat misleading

How about "When EM mortally wounded the ether and SR put the last nail in it's coffin"?

I am changing my mind re compression of charge field of moving point charge (electron).
I suspect that compression duznt exist.
I suspect that compression has never been proven/tested.
Duzz anyone know of a convincing experiment?

[PlainName]

Silly Michelson...

...the standard (false)(silly)(Einsteinian Mafia gatekeeper's)...

Purcell's big lie

Purcell uses that there silly lie

some kind of (silly) time dilation effect

Einsteinists are stupid but cunning

the oldendays klumzy klunky air-mode MMXs

This is straight out of Trump's playbook - using political assassination to remove competing theories.

[TimFox]

The technical term for this logical fallacy is "ad hominem" argument.

[switchabl]

I am changing my mind re compression of charge field of moving point charge (electron).
I suspect that compression duznt exist.
I suspect that compression has never been proven/tested.
Duzz anyone know of a convincing experiment?

Okay, then you've just thrown Maxwell's equations over board as well. 

The formula for the electric field of a moving charge was first published by Heaviside in 1888 (https://en.wikisource.org/wiki/Page%3AElectromagnetic_effects_of_a_moving_charge.djvu/6). It is stated there without proof but you can plug it into Maxwell's equations to see that it is correct. Or you can derive it directly by calculating the retarded potentials and then differentiating them, without ever using a Lorentz transform. It is not particularly difficult but very tedious, so you rarely find it in textbooks. If you want to see all the gory details, you can try to do it yourself (enjoy!) or borrow the old Mason/Weaver book (§57-59) from archive.org (https://archive.org/details/electromagneticf0035maso/mode/2up).

[HuronKing]

I am changing my mind re compression of charge field of moving point charge (electron).
I suspect that compression duznt exist.
I suspect that compression has never been proven/tested.
Duzz anyone know of a convincing experiment?

Okay, then you've just thrown Maxwell's equations over board as well. 

The formula for the electric field of a moving charge was first published by Heaviside in 1888 (https://en.wikisource.org/wiki/Page%3AElectromagnetic_effects_of_a_moving_charge.djvu/6). It is stated there without proof but you can plug it into Maxwell's equations to see that it is correct. Or you can derive it directly by calculating the retarded potentials and then differentiating them, without ever using a Lorentz transform. It is not particularly difficult but very tedious, so you rarely find it in textbooks. If you want to see all the gory details, you can try to do it yourself (enjoy!) or borrow the old Mason/Weaver book (§57-59) from archive.org (https://archive.org/details/electromagneticf0035maso/mode/2up).

Here is a link to Mason/Weaver that doesn't require a login:
https://archive.org/details/electromagneticf030976mbp/page/n301/mode/2up

Thanks for the reference. Love textbooks like these.