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Let us consider this:

One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect. — Richard Feynman

From relativity, this is infact true. Then why electric field lines are straight, pointing radially outwards, and magnetic field lines are in the form of a ring or a loop?

i)In other words why are they both different?

I should be able to see them the same way, no matter the frame of reference, right?

Now let me assume that the field lines are the way they are.

So let me get on my trolley and move along with the charged particle. I will see the charge getting repelled by electrostatic force. And I see the electric field lines(radially outwards)

Now I slowly decrease my speed....and finally come to rest.

At rest I will see the charge moving, as well as getting repelled by magnetic force. And now I will see magnetic field lines in the form of concentric circles.

ii)During my journey how will I see the lines getting transformed into loops?

P.S: The entire question is revolving around the phenomenon of a charged particle moving alongside a current carrying wire

Edit:From relativity, is magnetic field just a moving electric field? Or it just relates the forces?

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  • $\begingroup$ “So let me get on my trolley and move along with the charged particle.” Permission granted 👨🏻‍🏭👮🏼‍♂️👨🏼‍⚖️ $\endgroup$
    – Al Brown
    Commented Aug 24, 2021 at 17:54
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    $\begingroup$ @AlBrown this question is over a year ago. I try not to cringe at them lmao. $\endgroup$
    – DatBoi
    Commented Aug 25, 2021 at 2:54

2 Answers 2

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I should be able to see them the same way, no matter the frame of reference, right?

No. When you change to a boosted reference frame which has velocity $\mathbf v$ with respect to the old one, the electric and magnetic fields transform as follows:

$$\mathbf E_\parallel' = \mathbf E_\parallel$$ $$\mathbf B_\parallel' = \mathbf B_\parallel$$ $$\mathbf E_\perp' = \gamma(\mathbf E_\perp + \mathbf v \times \mathbf B)$$ $$\mathbf B_\perp' = \gamma(\mathbf B_\perp - \frac{1}{c^2}\mathbf v \times \mathbf E)$$

where $\parallel$ and $\perp$ indicate the directions parallel to and perpendicular to the boost velocity $\mathbf v$, and $\gamma \equiv \frac{1}{\sqrt{1-v^2/c^2}}$ is the Lorentz factor.

From relativity, is magnetic field just a moving electric field? Or it just relates the forces?

The magnetic field and electric field are two aspects of one physical entity which we call the electromagnetic field. Different observers see different combinations of electric and magnetic fields, but they combine in such a way that all observers will agree on how a charged particle will move, in the sense that the particle's coordinates in one frame are related to its coordinates in another frame by the appropriate Lorentz transformation.

From a comment,

Can you please provide additional details to your answer about electromagnetic field and how it changes it's form- from purely magnetic to purely electrostatic ?

If the EM field is purely magnetic in one reference frame, then you won't be able to find a reference frame in which it's purely electric. The reason is that even though $\mathbf E$ and $\mathbf B$ are not Lorentz-invariant (which means they change when you change reference frames), the quantity $\mathbf B^2 - \mathbf E^2/c^2$ is Lorentz invariant. In particular, if the field is ever purely magnetic then this quantity is positive; in order for it to be purely electric, this quantity would have to change sign, but it remains constant during Lorentz boosts.

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  • $\begingroup$ I must first appreciate the beautiful explanation given by you. In the third part of your answer, where you have mentioned that "If the EM field is purely magnetic in one reference frame, then you won't be able to find a reference frame in which it's purely electric." Well in my case where a charge is moving alongside a current carrying wire- from a frame at rest, the wire has neutral charge hence no electric force and purely magnetic. While the frame moving with the charge can account only electric force, hence purely electric. Am I going wrong somewhere? $\endgroup$
    – DatBoi
    Commented May 6, 2020 at 16:33
  • $\begingroup$ @DatBoi When you boost to the rest frame of your charged particle, the magnetic field doesn't vanish - it simply does not exert a force on the particle because the particle's velocity is equal to zero. $\endgroup$
    – J. Murray
    Commented May 6, 2020 at 16:40
  • $\begingroup$ Moving charges create magnetic field. So in the first case, it is produced by the electrons moving. In the next case,the same magnetic field is developed by protons this time, is that what you're saying? $\endgroup$
    – DatBoi
    Commented May 6, 2020 at 16:43
  • $\begingroup$ That's not what I meant, but yes, in a real current-carrying wire there is no frame in which both the electrons and protons are at rest. What I was saying is that if you observe a charged particle moving near a current-carrying wire, you observe a magnetic force on it; if you boost to that particle's rest frame, the magnetic force goes away (and is replaced by an electric force) but that doesn't mean that the magnetic field goes to zero. $\endgroup$
    – J. Murray
    Commented May 6, 2020 at 17:13
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    $\begingroup$ Minor point. Force (including) the Lorentz force is not a relativistic invariant. So observers don't agree about the forces on charged particles (unless the force is zero). $\endgroup$
    – ProfRob
    Commented May 6, 2020 at 20:02
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I have read on this website that the electromagnetic field is an entity which changes its appearance when viewed from a different moving coordinate system, but both components are real. Keep in mind that magnetic field lines are a mathematical construct designed to represent a complex situation. To find a magnetic force you have to go through two “right hand rules”. One to define the direction of the field, and another to find the direction of a force. The actual force between two parallel current carrying wires is either attraction or repulsion.

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  • $\begingroup$ Can you please provide additional details to your answer about electromagnetic field and how it changes it's form- from purely magnetic to purely electrostatic ? $\endgroup$
    – DatBoi
    Commented May 6, 2020 at 15:49
  • $\begingroup$ There is no reasonable transformation that makes a magnetic field into a radial electrostatic field that's like the electrostatic field around a charged particle. $\endgroup$
    – S. McGrew
    Commented May 6, 2020 at 16:18

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