Feynman field transformations

[harpf]

I'm trying to follow Feynman's explanation on page 26-10 of Volume 2 of The Feynman Lectures on Physics. He describes the electric and magnetic fields in a FoR S' moving between the plates of a condenser. Feynman writes that we see a reduced E and an added transverse B in S'. I've attached a copy of the figure in question.

The transformation formula provided in the text is E' = γE, because B = 0 in S. My thinking is that, according to the formula, E' increases. This effect is also consistent with the S length contraction in S'.

I am confused. Thank you.

 

[harpf]

I have attached additional information in the hope that someone will kindly point out the error in my thinking.

 

[PeterDonis]

My thinking is that, according to the formula, E' increases.

You're right, it does. The "reduced" in the text is in error; possibly a typo or a mis-transcription (since the text was originally transcribed from Feynman's actual lectures).

Another way of seeing that E' has to increase is that $E^2 - B^2$ is an invariant; it's the same in all frames. In frame S, B = 0, so the invariant is just $E^2$, the square of the electric field. In frame S', moving relative to the plates, B' is nonzero, so E' must increase to keep the invariant the same.

 

[harpf]

I appreciate your response. Thank you.

 

[sardar]

I understand your confusion and would like to provide some clarification on Feynman field transformations. These transformations allow us to understand how electric and magnetic fields behave in different frames of reference (FoR). In this specific example, we are considering two FoRs, S and S', where S' is moving with a velocity relative to S.

In S, we have a condenser with two plates and a uniform electric field E between them. In S', the same condenser is moving, so we need to transform the fields to understand how they appear in this FoR. According to the transformation formula E' = γE, we see that the electric field in S' is increased by a factor of γ, which is the Lorentz factor that takes into account the velocity of S' relative to S.

Furthermore, we also see an added transverse magnetic field in S', which is consistent with the length contraction of S in S'. This means that the distance between the plates of the condenser appears shorter in S' due to the relative motion, resulting in an additional magnetic field component.

It is important to note that these transformations are not intuitive and may seem counterintuitive at first. However, they are based on the fundamental principles of relativity and have been experimentally verified. Therefore, it is important to trust in the equations and their predictions.

I hope this helps clarify the concept of Feynman field transformations and their application in understanding electric and magnetic fields in different frames of reference. If you have any further questions, please do not hesitate to ask.