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SamRoss
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I am having difficulty working out one of the steps in Einstein's original SR paper, in the Section entitled "Dynamics of the Slowly Accelerated Electron". Einstein describes an electron in motion in an electromagnetic field. Its equation of motion in a rest frame K (not moving with the electron) is
(1) m(d^2 x/dt^2)=F
where m is the mass of the electron and F is the force on the electron in the x direction. (I am ignoring a constant e that Einstein put in). This is just f=ma.
At a given instant of time, the electron is moving with velocity v. In a frame of reference k, traveling at the same velocity, the equation of motion looks like this...
(2) m(d^2 x'/dt'^2)=F'
Einstein then lists the previously obtained Lorentz transformations...
(3) x'=B(x-vt) and t'=B(t-vx/c^2) where B=(1-v^2/c^2)^-1/2
as well as the transformation for the force on the electron, which turns out to simply be...
(4) F=F' (for this post I am leaving out the y and z directions).
Now, using these transformations, Einstein transforms equation (2) from system k to system K. The result is
(5) mB^3(d^2 x/dt^2)=F
In case this is hard to read the way I wrote it, it turns out to be equation (1) with the left side multiplied by B^3.
My question is, how did Einstein get from (2) to (5)? I'm pretty sure the chain rule is involved but I keep getting the wrong answer. If anyone can take me step by step from (2) to (5) I would greatly appreciate it. Thanks!
(1) m(d^2 x/dt^2)=F
where m is the mass of the electron and F is the force on the electron in the x direction. (I am ignoring a constant e that Einstein put in). This is just f=ma.
At a given instant of time, the electron is moving with velocity v. In a frame of reference k, traveling at the same velocity, the equation of motion looks like this...
(2) m(d^2 x'/dt'^2)=F'
Einstein then lists the previously obtained Lorentz transformations...
(3) x'=B(x-vt) and t'=B(t-vx/c^2) where B=(1-v^2/c^2)^-1/2
as well as the transformation for the force on the electron, which turns out to simply be...
(4) F=F' (for this post I am leaving out the y and z directions).
Now, using these transformations, Einstein transforms equation (2) from system k to system K. The result is
(5) mB^3(d^2 x/dt^2)=F
In case this is hard to read the way I wrote it, it turns out to be equation (1) with the left side multiplied by B^3.
My question is, how did Einstein get from (2) to (5)? I'm pretty sure the chain rule is involved but I keep getting the wrong answer. If anyone can take me step by step from (2) to (5) I would greatly appreciate it. Thanks!