# Derieving relativistic electrodynamics equations

I'm trying to derive the Lorentz-invariant field equations, using a point charge (well, a positron actually) centered in the coordinate system. I'm trying to find the electric & magnetic fields generated by it. I've tried using Dirac delta functionfor the charge density.

S' frame of reference is moving relative to S frame of reference with a velocity v along the x axis. At t=0, x'=x. At that moment, an observer in the S system (presumably me) is trying to calculate the electric & magnetic fields in his frame of reference.

Now, is S', positron is still, so
$$\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$
and since it's standing still
$$\vec{B} = 0$$

Then I tried switching the reference system with:
$$\frac{\partial E}{dx \sqrt{1-v^2/c^2}} + \frac{\partial E}{dy} + \frac{\partial E}{dz} = \frac{e \delta(\vec{r})}{\epsilon_0 \sqrt{1-v^2/c^2}}$$

and... I simply don't where to go next? One possibility is I'm down-way wrong.

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robphy
Homework Helper
Gold Member
What is your starting point? (i.e. What do you know?)
Do you know how the electromagnetic field transforms?
Do you know how the charge-current density transforms?
(By the way, you might have left off some component-subscripts in your last equation.)

jtbell
Mentor
gulsen said:
I'm trying to derive the Lorentz-invariant field equations,
Maxwell's equations (which most people mean by the "electromagnetic field equations") are Lorentz-invariant to begin with.

using a point charge (well, a positron actually) centered in the coordinate system. I'm trying to find the electric & magnetic fields generated by it.
Starting from the fields produced by the charge in S, you can calculate the fields in some other frame S', and then show that those fields satisfy Maxwell's equations (in S'). Is that what you're trying to do?

In that case you need the Lorentz transformation for E and B fields.. (The linked page uses the "wedge" symbol for the vector cross product.)

You might also find useful some of the other notes in that collection on relativity and electromagnetism.

Meir Achuz
Homework Helper
Gold Member
You really have to read a graduate textbook. Your start is grossly oversimplied. What you are trying to do is derive the "Lienard-Wiechert" fields of a point charge, which takes several pages in a grad textbook.

robphy said:
What is your starting point? (i.e. What do you know?)
Do you know how the electromagnetic field transforms?
Do you know how the charge-current density transforms?
(By the way, you might have left off some component-subscripts in your last equation.)
-I know basic electrodynamics, some SR, QM -if it helps-, classical fields from classical mechanics, some math (shortly, a 2nd year undergrad).
-yes
-no
(like?)

jtbell said:
http://farside.ph.utexas.edu/teaching/jk1/lectures/node6.html[/quote] [Broken]

Thanks for the URL! I'm going to start studying it tonight.

Meir Achuz said:
You really have to read a graduate textbook. Your start is grossly oversimplied. What you are trying to do is derive the "Lienard-Wiechert" fields of a point charge, which takes several pages in a grad textbook.
And I really don't think so. What is so oversimplified?
Griffiths - Introduction to Electrodynamics, which is an undergradute book, covers the whole subject, but using different thought experiments.

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robphy said:
What is your starting point? (i.e. What do you know?)
Do you know how the electromagnetic field transforms?
Do you know how the charge-current density transforms?
(By the way, you might have left off some component-subscripts in your last equation.)
Here is a place to start

Pete

reilly
Pete -- Meir Ashuz is right. In fact, the Lienard - Wiechart potential is the exact analogue of the potential for a point particle in standard potential theory. It is the solution generated by a point particle driving the full set of Maxwell's equtions. (Just what you wanted to hear -- Jackson does your problem in considerable detail. ) LW is important because it generates retarded fields in a relativisic manner. The solution is generated by the retarded Green's function for the Maxwell associated wave equation. This Greens's function approach permeates relativistic E&M, and virtually all of QFT.

Regards,
Reilly

pervect
Staff Emeritus
I'm not positive I totally understand the orignal question, but if am understanding it correctly, the answer for the electric part of the field of the moving charge is at

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_14.pdf

The Lienerd-wiechert equations would provide an alternate route to the same solution, but I'm assuming that it is' the solution itself that's of interest (I may be misunderstanding the OP here). The route to this solution can be motivated in a more elementary manner than using the LW potentials.

To understand the derivation, one would be better off starting from

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_11.pdf

and working one's way forward, or using the outline index at

http://www.phys.ufl.edu/~rfield/PHY2061/

and starting at the appropriate point.

I personally think that the most important point is that knowledge of the E and B fields at a point in one reference frame allows one to determine the E and B fields at a point in all reference frames - i.e. one does not need the exact source configurations to be able to transform the field. Tensor notation is a convenient way of writing the appropriate transformations down, but the very first thing is to be aware that the task is possible.

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It seems I've asked the question wrong way.
Let me try again.
I know electric field, but I don't know anything about magnetic field, and want it to come out as a consequence or SR on electric field. Then I want to write the electric field equation, that includes this relativistic effect. I want to derive these results using a thought-experiment with a point charge.

George Jones
Staff Emeritus
Gold Member
This issues involved are more subtle than some authors let on. See the discussion and references given in section 12.2, "On the Question of Obtaining the Magnetic Field, Magnetic Force, and the Maxwell Equations from Coulomb's Law and Special Relativity," of the second edition of Jackson's "Classical Electrodynamics." At a glance on amazon.com, I don't see this section listed in the third edition.

Regards,
George

pervect
Staff Emeritus
Gravity is a good example of why you can't derive all of electromagnetism from only the Lorentz transforms and the inverse-square law rule for static fields.

It's unclear to me exactly what, in addition, to the Lorentz transforms is needed to unambiguously derive Maxwell's equations.

A good start might be Gauss's law, which defines a charge as a conserved quantity expressible as a surface intergal - a conserved quantity that's independent of the velocity of the charge.

This would serve to disambiguate E&M from gravity - but I really don't know if Lorentz invariance and Gauss's law are sufficient to unambiguously give Maxwell's equations. I suspect it is not, but I don't have any proof either way.

pervect said:
Gravity is a good example of why you can't derive all of electromagnetism from only the Lorentz transforms and the inverse-square law rule for static fields.
AFIK, the reason was physical, not mathematical. Maxwell is said to have an attempt to apply a similar approach to gravity, but was confused by negative energy (according to what I read in http://arxiv.org/abs/gr-qc/0207065" [Broken])

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gulsen said:
It seems I've asked the question wrong way.
Let me try again.
I know electric field, but I don't know anything about magnetic field, and want it to come out as a consequence or SR on electric field. Then I want to write the electric field equation, that includes this relativistic effect. I want to derive these results using a thought-experiment with a point charge.
arxiv physics/0601028
arxiv physics/0505130

Galileo
Homework Helper
gulsen said:
And I really don't think so. What is so oversimplified?
Griffiths - Introduction to Electrodynamics, which is an undergradute book, covers the whole subject, but using different thought experiments.
Griffiths, in fact, does the whole derivation. In chapter 10.3 by using the Liénard-Wiechert potentials, which involves an entirely straightforward brute-force method, the only drawback is that the derivation is a bit long.
In chapter 12.3 he does it again by transforming from the frame in which the particle is stationary to a moving frame. This is a much more efficient way, but you need to know how the fields transform.

There's also a discussion of this in Duffin, it also assumes the transformation laws of the fields.

Meir Achuz
Homework Helper
Gold Member
George Jones said:
This issues involved are more subtle than some authors let on. See the discussion and references given in section 12.2, "On the Question of Obtaining the Magnetic Field, Magnetic Force, and the Maxwell Equations from Coulomb's Law and Special Relativity," of the second edition of Jackson's "Classical Electrodynamics." At a glance on amazon.com, I don't see this section listed in the third edition.

Regards,
George
That section was wrong and incomprehensible (fortunately) to students.
Good riddance.

reilly said:
Pete -- Meir Ashuz is right. In fact, the Lienard - Wiechart potential is the exact analogue of the potential for a point particle in standard potential theory. It is the solution generated by a point particle driving the full set of Maxwell's equtions. (Just what you wanted to hear -- Jackson does your problem in considerable detail. ) LW is important because it generates retarded fields in a relativisic manner. The solution is generated by the retarded Green's function for the Maxwell associated wave equation. This Greens's function approach permeates relativistic E&M, and virtually all of QFT.

Regards,
Reilly
I was starting with the the classical, i.e. non-tensor, form of the laws of EM. I then transformed them into tensor (i.e. relativistic) form. I assumed that was what the read wanted, i.e. Maxwell's equations in relativistic form. Is that assumption incorrect?

Pete

pervect
Staff Emeritus
It's very easy to show that a force that is a central force in one frame of reference where two particles are stationary is not a central force in another "boosted" frame.

We don't even need to assume at this point that the force is an inverse square law force.

This would be a good exercise if one hasn't already worked it out - consider a particle at (x,y) = (1,1) and another particle at (-1,-1). Draw the 3-force vectors between the particles if the forces are central forces, i.e. directed on a line between the particles.

Now boost the system in the x or y direction, and compute the 3-force vectors in the new frame.

I would suggest converting the 3-vectors into 4-vectors, boosting the 4-vectors like any other 4-vector, then re-converting the 4-vectors into 3 vectors. Remember that a 3-force is dp/dt, the rate of change of momentum with coordinate time, but that a 4-force is dp/dtau, the rate of change of momentum with proper time.

You might take a look at

https://www.physicsforums.com/showpost.php?p=477970&postcount=23

if this is too much work, but it would probably be better to actually work out the problem for oneself.

Or see the attachment that I've copied from that link at the bottom of this post - this will give one a sense of what happens.

In a sense, this shows the existence of a "magnetic" field, i.e. it shows that between a pair of co-moving particles, there must be non-central forces, even though the force between stationary particles is always a central force.

Actually a bit more explanation is needed here, in how we go from forces to fields. We usually describe the "field" as the force on some small test particle. Electric fields can give rise only to central forces, the fact that a central force does not remain a central force under a Lorentz boost shows that there must be some other mechanism that generates force, a mechanism that we can term "the magnetic field", though at this point we have not worked out many of the properties of this field.

However, it is not possible to actually write down Maxwell's equations knowing only the fact that the force between charged particles appears to be a central force when both particles are at rest, and that the theory is Lorentz invariant. One obstacle is that knowing the force between any two stationary particles still does not tell one the force between a stationary particle and a moving one. There are other obstacles, as well - knowing the force between any two particles in arbitrary motion doesn't allow one to calculate the forces in an n-particle system, unless one knows in advance that the force laws are linear.

Gravity is an example where the force-laws are not linear. (Actually I should say GR is an example where the force laws are not linear, though the non-linearities are small under most conditions, i.e. one can get a reasonable model of the solar system by ignoring the nonlinearities that GR predicts).

Gravity also has the problem where the "fields" can't be defined via the "test particle" mechanism I mentioned, because all particles are affected by gravity. Therefore one cannot compare the motion of a test particle and an uncharged particle to determine the gravitational force directly - because all particles are affected by gravity (have gravitational "charge", if you will).

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