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For the first time I am trying to derive by myself the general relation between the magnetic and electric constants as shown in the following:

$$ F = \frac{k_E q_1 q_2}{r^2} \hspace{2mm} , \hspace{2mm} F = \frac{k_M I_1 I_2}{r} $$

To do this, I am considering a thought experiment where I have two lines of moving charges spaced from one another by a distance $r$ with charge densities $\lambda_1$ and $\lambda_2$ and moving with velocity $v$ in the $x$ direction such that the corresponding currents are $I_1$ and $I_2$. Using $I_k = \lambda_k v$, I calculate the total force between the wires in the rest frame as (positive pulling the wires together):

$$ F = \frac{k_M \lambda_1 \lambda_2 v^2}{r} - \frac{2k_E \lambda_1 \lambda_2}{r} $$

Then I consider a frame in which the observer is moving at the same velocity as the charges in the wires so that there is no current. I then, using $\lambda_k^{'} = \gamma \lambda_k$, I calculate the force in the moving frame as:

$$ F' = -\frac{2k_E \lambda_1 \lambda_2 \gamma^2}{r} $$

And since the force is perpendicular to the velocity, we should have that $F = F'$. My problem is that when I set these equal to one another, I find that I don't get the nice relation that I have seen elsewhere ($c^2 k_M = 2k_E$). I have noticed that if I instead have:

$$F' = -\frac{2k_E \lambda_1 \lambda_2}{\gamma^2 r} $$

Then I get the correct result. However, I don't see why that would be the case, as I believe my charge density transformation to be correct. Can anyone help me find where I have gone wrong? Thanks in advance!

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    $\begingroup$ Why are the lengths of the lines of charges not involved in the two first equations? $\endgroup$ – stuffu Feb 26 at 22:17
  • $\begingroup$ @stuffu I wrote in force per unit length here, but that reminds me that I should scale the right side by a factor of $\gamma$ or $\frac{1}{\gamma}$ in order to rescale it? The only problem I see is that if I do this, then it still won't give me the factors I need. $\endgroup$ – user132849 Feb 27 at 6:34
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    $\begingroup$ Ok, but now the first equation looks like the one that talks about point charges. Are we talking about point charges arranged on a line so that the points are far away from each other, and then another similar line close to the first line? $\endgroup$ – stuffu Feb 27 at 11:46
  • $\begingroup$ I did the integration over two infinite lines of parallel charge and found the force per unit length. But one of the other commenters helped me figure out what my issue was. Thanks for your help though! $\endgroup$ – user132849 Feb 27 at 17:01
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A possible error in your calculations is that you wrote $\lambda_k^{'} = \gamma \lambda_k$, whereas you had better write $\lambda_k^{'} = \alpha \lambda_k$, where $\alpha=1/\gamma$. If $\lambda_k$ is the charge density of the moving charges/wire, the distances between the charges are Lorentz contracted and the density is great. Moreover, from the viewpoint of an observer who is at rest WRT the charges, the length between the charges is the proper length, which is greater than the Lorentz contracted one, and thus the density reduces.

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  • $\begingroup$ But if the charge density is increased in the moving frame, then should it not be what I have written since a factor of $\gamma$ will increase the charge density? $\endgroup$ – user132849 Feb 27 at 6:31
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    $\begingroup$ @user132849 The observer who moves at the same velocity as the charges is not a moving frame WRT the charges. I think you have mistaken a moving frame for the rest one. $\endgroup$ – Mohammad Javanshiry Feb 27 at 8:18
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    $\begingroup$ Ah okay I get it now. I had assumed that the charge density will increase no matter what, but I get it now. Thank you so much for your help! $\endgroup$ – user132849 Feb 27 at 9:26

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