Relativistic Magnetic Field of a moving charge

[GoodShow]

I believe this delves into relativistic physics so I put this here. If I am incorrect I apologize.

I've been learning about magnetic fields and how they are generated by moving charges. If a charge is moving at some arbitrary speed it generates a magnetic field. This is what I've been taught. So my question is this, what happens to the magnetic field when you move at the same speed as the particle and observe it? Because according to your perspective the particle wouldn't be moving and so (according to what I've been taught) it shouldn't be generating a magnetic field. But from the perspective of a person moving at speeds lower than the particle it would appear to generate a magnetic field.

Does the magnetic field disappear to the person moving at the same speed as the charged particle? And if so then where does it go? If not then why not? If to the person moving at the same speed as the charged particle observes a magnetic field, this would imply that there is an absolute rest at which there is no magnetic field generated. This I know has been shown to be incorrect by Relativistic physics because according to it there is no absolute rest.

So in short, what is going on in this situation?


Another quick question having to do with this. If you have a charge that is at rest but you start moving away from it. Then from the relativistic point of view you cannot tell whether it is you who is moving or the charge. So would a magnetic field be generated as you move away from the charge?

 

[WannabeNewton]

Hi GoodShow! Welcome to the forum! Components of the electric and magnetic field can transform into each other under Lorentz boosts. In a very loose sense, the magnetic field "becomes" an electric field when viewed in the other frame and vice versa. What you say is certainly not incorrect at all!

 

[GoodShow]

Thank you. Lorentz boosts? I'm afraid I'm not all that familiar with them. I've heard before that electric fields and magnetic fields are one in the same i.e. electromagnetic fields. A previous professor of mine even described magnetic fields as relativistic electric fields. How is it that one transforms into another?

Please forgive me as well, I'm not all that familiar with the formal mathematics of higher physics than basic Newtonian physics with some calculus application.

 

[WannabeNewton]

By Lorentz boosts I meant Lorentz transformations along some direction; you won't need much higher math for this my friend! See this subsection for a start: https://en.wikipedia.org/wiki/Lorentz_transformation#Transformation_of_the_electromagnetic_field
This link gives some explicit calculations: http://hepweb.ucsd.edu/ph110b/110b_notes/node69.html
See in particular the last two equations.

A classic EM textbook which deals with this in gory detail is "Electricity and Magnetism"-Edward M. Purcell.

 

[WannabeNewton]

I've heard before that electric fields and magnetic fields are one in the same i.e. electromagnetic fields.

In a sense yes. In the UCSD link I gave you, the $F_{\mu\nu}$ quantity is what codifies the physical electromagnetic field. The calculation is showing how the components of $F_{\mu\nu}$ transform if one applies a Lorentz transformation along the $x$ direction. Notice how the components of $F_{\mu\nu}$ are nothing more than the components of the electric and magnetic fields (save for the diagonal which always has zeros on account of this quantity being antisymmetric).

 

[GoodShow]

What are the different subscripts on the B and F in the first equation of that link? Also one of the B in that equation is to the power of T. What is T?

 

[WannabeNewton]

By first equation of that link do you mean the first equation in the UCSD link? If so, for the purposes of that calculation you can think of the the subscripts in $F_{\mu\nu}$ as denoting the entries of the 4x4 matrix it represents (as depicted in the link) and you can think the same for the quantity $B_{\mu\nu}$ (the subscripts are just place holders); the quantity $B_{\mu\nu}$ represents the Lorentz transformation matrix for a boost in the $+x$ direction and the T superscript represents matrix transpose.

If you aren't too comfortable with the way the calculation is done in that link, see this one instead: http://web.hep.uiuc.edu/home/serrede/P436/Lecture_Notes/P436_Lect_18p5.pdf It will explain everything in full detail and everything up till page 17 will be in the spirit of Purcell's book.

 

[GoodShow]

Yeah sorry. F'_μv=B_μρF_ρσB_σv^T . I was wondering what all those subscripts meant and what exactly the equation was saying.

When is it you usually first learn to do Lorentz transforms? I've taken all three levels of Calculus as well as Ordinary Differential Equations and I have yet to see them. Are they more of an advanced physics math?

 

[WannabeNewton]

Not at all. They are usually taught in introductory mechanics classes which delve into some special relativity (SR) (at some schools honors versions will teach them and at other schools all versions will teach them) or in introductory courses dedicated only to SR; if you know up to calc 3 and ODEs you will have no problem with the mathematics of SR. See the UIUC link I just gave, that one will do everything regarding transformation of electric and magnetic fields in an accessible way.