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Two opposite charges are in a spaceship and are attracted by the electric field $E_s$ But for an observer on earth the Electric force is $$E_e=\gamma E_s$$ Normally the forces are scaled down by $\gamma$ in the earth frame and here also the total force is scaled down. But why the Electric component of force is scaled up? Is it because the Electric fields are now closer together because of length contraction?

But i think this is not the answer because if that was the case,then gravitational fields and hence Forces would have scaled up.But this is not the case in nature. Can you provide a derivation or something which explains how the Electric field is scaled up?

Qmechanic
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Sophile
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  • Comments are not for extended discussion; this conversation has been moved to chat. Please do not use comments to discuss things unrelated to improving the post being commented on. – ACuriousMind May 03 '21 at 15:16

3 Answers3

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enter image description here

In above Figure-01 an inertial system $\:\mathrm S'\:$ is translated with respect to the inertial system $\:\mathrm S\:$ with constant velocity
\begin{align} \boldsymbol{\upsilon} & \boldsymbol{=}\left(\upsilon_{1},\upsilon_{2},\upsilon_{3}\right) \tag{02a}\label{02a}\\ \upsilon & \boldsymbol{=}\Vert \boldsymbol{\upsilon} \Vert \boldsymbol{=} \sqrt{ \upsilon^2_{1}\boldsymbol{+}\upsilon^2_{2}\boldsymbol{+}\upsilon^2_{3}}\:\in \left(0,c\right) \tag{02b}\label{02b} \end{align}

The Lorentz transformation is \begin{align} \mathbf{x}^{\boldsymbol{\prime}} & \boldsymbol{=} \mathbf{x}\boldsymbol{+} \dfrac{\gamma^2}{c^2 \left(\gamma\boldsymbol{+}1\right)}\left(\boldsymbol{\upsilon}\boldsymbol{\cdot} \mathbf{x}\right)\boldsymbol{\upsilon}\boldsymbol{-}\dfrac{\gamma\boldsymbol{\upsilon}}{c}c\,t \tag{03a}\label{03a}\\ c\,t^{\boldsymbol{\prime}} & \boldsymbol{=} \gamma\left(c\,t\boldsymbol{-} \dfrac{\boldsymbol{\upsilon}\boldsymbol{\cdot} \mathbf{x}}{c}\right) \tag{03b}\label{03b}\\ \gamma & \boldsymbol{=} \left(1\boldsymbol{-}\dfrac{\upsilon^2}{c^2}\right)^{\boldsymbol{-}\frac12} \tag{03c}\label{03c} \end{align}

For the Lorentz transformation \eqref{03a}-\eqref{03b}, the vectors $\:\mathbf{E}\:$ and $\:\mathbf{B}\:$ of the electromagnetic field are transformed as follows \begin{align} \mathbf{E}' & \boldsymbol{=}\gamma \mathbf{E}\boldsymbol{-}\dfrac{\gamma^2}{c^2 \left(\gamma\boldsymbol{+}1\right)}\left(\mathbf{E}\boldsymbol{\cdot} \boldsymbol{\upsilon}\right)\boldsymbol{\upsilon}\,\boldsymbol{+}\,\gamma\left(\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{B}\right) \tag{04a}\label{04a}\\ \mathbf{B}' & \boldsymbol{=} \gamma \mathbf{B}\boldsymbol{-}\dfrac{\gamma^2}{c^2 \left(\gamma\boldsymbol{+}1\right)}\left(\mathbf{B}\boldsymbol{\cdot} \boldsymbol{\upsilon}\right)\boldsymbol{\upsilon}\boldsymbol{-}\!\dfrac{\gamma}{c^2}\left(\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{E}\right) \tag{04b}\label{04b} \end{align}

Nothing more, nothing less.

How the Lorentz force 3-vector or the Lorentz force 4-vector are transformed see my answer here Are magnetic fields just modified relativistic electric fields?.

Expressions of the kind $''$...scaled down by $\gamma$...$''$ or $''$...the Electric fields are now closer together because of length contraction...$''$ are misplaced.

Frobenius
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  • Peraphs MathJax does not support \boldsymbol. PS: missing the image. – Sebastiano May 12 '21 at 22:13
  • @Sebastiano : It seems that the problem with `\boldsymbol' and the image is only yours (your browser ??? ). Nobody has reported anything like that. – Frobenius May 13 '21 at 05:36
  • Hi. now I see correct your answer: the image and the formulas. I use always the same browser Chrome. Yesterday I have seen only your two answers with the same problem. – Sebastiano May 13 '21 at 11:24
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    @Sebastiano : Ok, browsing problems solved. – Frobenius May 13 '21 at 11:27
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Normally the forces are scaled down by γ in the earth frame and here also the total force is scaled down. But why the Electric component of force is scaled up?

The actual derivation is based on Lorentz transformation equations but one intuitive way to visualise this is to visulalise the electric field lines. In a charge at rest, the field lines are pointing outward uniformly in all directions.

But, in a charge that is moving , the field lines get "scrunched up" in the transverse direction to the field of motion, so that the electric field strength in that direction increases

silverrahul
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  • Thats what i initially imagined.But,if that was the cause,then gravitational force should also get stronger.But this doesn't happen.why is it so ? – Sophile May 01 '21 at 06:22
  • Gravitation has nothing to do with electric field lines. Why would gravitational force behave like them ? – silverrahul May 01 '21 at 06:30
  • I mean we can represent electric field with electric field lines.Likewise gravitational force can be represented with gravitational field lines.so,if electric field get "scrunched up",then shouldn't gravitational field lines also behave like that? – Sophile May 01 '21 at 07:19
  • I know Gravity is not a force.but,who knows if electricity is a force?Still we can represent both with field lines.. – Sophile May 01 '21 at 07:19
  • Yes, but such field lines are simply our representations that help us predict what is going on. Without going into philosophy of science, these representations of field lines are not "real" and more importantly how the representations of electric field would behave are not necessarily identical with how the representations of gravitational field would behave, because these are 2 distinct and different fundamental forces with different "mechanism" – silverrahul May 01 '21 at 07:28
  • Electric field and (Lorentz) force are not interchangeable terms. – ProfRob May 01 '21 at 07:44
  • @silverrahul I agree that electric field lines are not "real". But,then why did you say >>the field lines get "scrunched up">> //or are you saying that electric field lines are "real" than gravitational field lines? – Sophile May 01 '21 at 08:44
  • Yeah, i meant electric field lines get scrunched up. They are no more or less real than gravitational field lines, but there scrunching up is a way to visualise how electric fields behave,. Gravitational fields do not necessarily behave in a similar way – silverrahul May 01 '21 at 09:27
  • If you want a proper derivation, you can find it in Purcell Morin . But like i said, they use Lorentz transformation which is taken from a separate section. They make use of conclusions from several other places in the book to derive this. – silverrahul May 01 '21 at 09:33
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Electric field E transforms this way:

$$E'=\gamma E$$

Gravity field G transforms this way:

$$G'=\gamma G$$

Force F, be it electric or gravitational, transforms this way:

$$ F'= F / \gamma $$

stuffu
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  • If G transforms as Ge = Gs*gamma , then how in the next line are you saying that gravitational force transforms Ge = Gs/gamma ? Those 2 statements are inconsistent with each other – silverrahul May 01 '21 at 08:32
  • @silverrahul What do you think my G means? I did say what it means, as I thought there might be some confusion otherwise. Read carefully : ) If you think that force can not go down when field goes up, then nature disagrees with you. : ) – stuffu May 01 '21 at 08:47
  • Are you saying that Gravity field goes up , but force of gravity goes down ? How do you figure that ? If the force of gravity goes down, then the gravitational field at that point goes down, by definition. – silverrahul May 01 '21 at 09:29
  • @silverrahul Bob floats in space. A planet flies by with Joe standing on the planet. Bob says: "Planet's force on Joe is decreased, because of motion, planet's force me is increased, because of motion". – stuffu May 01 '21 at 10:39
  • Transformation equations generally refer to the first case, where Joe and planet have same distance between them , only both are moving at higher velocity . Planet's force on Bob increasing, is not because of transformation. It is simply because distance between planet and Bob increases. – silverrahul May 01 '21 at 10:46
  • @silverrahul Transformation equations of fields refer to the case where the field moves. In my example a field strength transforms up, a force goes down. – stuffu May 03 '21 at 06:20
  • That is not how it works. If a field strength transforms up, then the force corresponding to that field will also transform up. If electric field transforms up, electric force will increase. If gravitational field transforms up, gravitational force will increase – silverrahul May 03 '21 at 06:31