Transformations of Electromagnetic Fields: Griffiths' Claims

[sophiatev]

TL;DR

Looking for an intuitive explanation as to why we can generalize how a certain field transforms when moving from one inertial reference frame to another to how other fields would transform

In Griffith's Introduction to Electrodynamics, chapter 12, he discusses how electromagnetic fields transform when we move from one inertial reference frame to another. On page 553, he claims

We shall assume also that the transformation rules are the same no matter how the fields were produced - electric fields associated with changing magnetic fields transform the same way as those set up by stationary charges. Were this not the case we'd have to abandon the field formulation altogether, for it is the essence of a field theory that the fields at a given point tell you all there is to know, electromagnetically, about that point; you do not have to append extra information regarding their source.

He then considers how the electric field inside a conductor made up of two parallel rectangular plates would change when we move from a frame where the conductor is stationary to one where it is moving at speed v. He considers two cases - one in which the conductor plates are parallel to the direction of motion, meaning the field between them is perpendicular to the motion, and one where the plates are perpendicular to the direction of motion, meaning the field between them is parallel to the motion. He derives equations for how the perpendicular component of the field transforms using the first scenario, and how the parallel component transforms using the second. He then generalizes these results to any situation where a system of charges is stationary in one frame and moving in another. Specifically, he uses them to derive how the electric field of a stationary point charge transforms in a frame where that charge is moving. I suppose I can see a vague justification for this generalization - the dimension of the charge configuration parallel to the direction of motion will be Lorentz contracted, leading to the same change in the perpendicular/parallel components of the fields that we saw when this dimension was Lorentz contracted for the conductor. However, I don't see generally why his initial claim about the transformation rules being the same no matter how the fields were produced holds. I understand that the fields are able to tell you the force a charged particle will experience at any point in space, meaning they give you a complete picture of how that charge will move. I suppose in this sense there is no need to append "extra information regarding their source" - once you know the mathematical expression for the fields, you're good. But to get that mathematical expression, you generally need to know about the charge configuration that produced the fields. And it's how this charge configuration changes when you move from one inertial reference frame to another that determines how the fields transform. So I don't really see how you can divorce the two, and more importantly I don't see why you can just generalize the transformation of fields produced by a specific charge configuration to other fields produced by other charge configurations. I don't really know anything about field theory, so I apologize in advance if the answer to this question follows directly from the mathematical tenants of a field theory (which I guess is what Griffiths is implying).

 

[kent davidge]

I have studied through that book. Griffiths' point is that the fields transform in the same way no matter what created them. Of course, you need to know what is generating them to find out what they are in the first place! But once you have found them in one frame, it doesn't matter what has created them. You only need to know how they transform from the original to the new frame to get them in the other frame.

 

[PeroK]

Summary:: Looking for an intuitive explanation as to why we can generalize how a certain field transforms when moving from one inertial reference frame to another to how other fields would transform

In Griffith's Introduction to Electrodynamics, chapter 12, he discusses how electromagnetic fields transform when we move from one inertial reference frame to another. On page 553, he claims He then considers how the electric field inside a conductor made up of two parallel rectangular plates would change when we move from a frame where the conductor is stationary to one where it is moving at speed v. He considers two cases - one in which the conductor plates are parallel to the direction of motion, meaning the field between them is perpendicular to the motion, and one where the plates are perpendicular to the direction of motion, meaning the field between them is parallel to the motion. He derives equations for how the perpendicular component of the field transforms using the first scenario, and how the parallel component transforms using the second. He then generalizes these results to any situation where a system of charges is stationary in one frame and moving in another. Specifically, he uses them to derive how the electric field of a stationary point charge transforms in a frame where that charge is moving. I suppose I can see a vague justification for this generalization - the dimension of the charge configuration parallel to the direction of motion will be Lorentz contracted, leading to the same change in the perpendicular/parallel components of the fields that we saw when this dimension was Lorentz contracted for the conductor. However, I don't see generally why his initial claim about the transformation rules being the same no matter how the fields were produced holds. I understand that the fields are able to tell you the force a charged particle will experience at any point in space, meaning they give you a complete picture of how that charge will move. I suppose in this sense there is no need to append "extra information regarding their source" - once you know the mathematical expression for the fields, you're good. But to get that mathematical expression, you generally need to know about the charge configuration that produced the fields. And it's how this charge configuration changes when you move from one inertial reference frame to another that determines how the fields transform. So I don't really see how you can divorce the two, and more importantly I don't see why you can just generalize the transformation of fields produced by a specific charge configuration to other fields produced by other charge configurations. I don't really know anything about field theory, so I apologize in advance if the answer to this question follows directly from the mathematical tenants of a field theory (which I guess is what Griffiths is implying).

If this is not valid then:

a) You could have two scenarios where the fields are the same, yet they transform differently. What happens if you subtract one of those fields from the other? In the original frame, the field is zero everywhere, yet it transforms to something that isn't zero. No forces transform to a scenario where there are forces and vice versa.

b) In terms of relative motion, there is the direction parallel to the relative motion and the directions perpendicular to the relative motion. By the principle of superposition, the total field is the sum of these two. If you analyse how a parallel field transforms and a perpendicualr field transforms, then by the principle of superposition again you have analysed the entire field.

 

[sophiatev]

a) You could have two scenarios where the fields are the same, yet they transform differently. What happens if you subtract one of those fields from the other? In the original frame, the field is zero everywhere, yet it transforms to something that isn't zero. No forces transform to a scenario where there are forces and vice versa.

Ah, okay, I see. This counterexample was quite helpful, thank you!