Suppose we know the position, velocity of a charge at time $t=0$ and its rest mass is also known. It is attracted by a (nearly) infinitely massive known opposite charge which is at rest and always be at rest (because no force can move it) in our frame of reference. How can we predict the path of the moving charge? I got a differential equation which was really hard to solve especially due to the complicated force momentum relationship. The equation which I got is : $$ m \dfrac{d}{dt} \dfrac{\vec{u}}{\sqrt{1 - \dfrac{|\vec{u}|^2}{c^2}}} = k \dfrac{\vec{r}}{|\vec{r}|^3} $$ where $\vec{u}$ is the velocity of the particle, $\vec{r}$ is its position with heavy particle at origin, and $k$ is the product of charges and Coulumb law constant.
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$\begingroup$ The reason for choosing very high mass is to ignore retarded potentials. $\endgroup$– PhysicsCommented Mar 24 at 9:39
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$\begingroup$ Can you please include in the question what differential equation you got, and how you got it? And, also, do you want to take into account radiative effects? I imagine you do, but just to make sure! $\endgroup$– Gabriel Ybarra MarcaidaCommented Mar 24 at 9:45
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$\begingroup$ Can you tell me How readiative effects take place. Then I can tell you to consider or not. $\endgroup$– PhysicsCommented Mar 24 at 10:25
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$\begingroup$ When a charged particle is accelerated, it radiates energy (i.e. creates an EM field). But this might be too advanced for you, so you might just ignore it for now! $\endgroup$– Gabriel Ybarra MarcaidaCommented Mar 24 at 10:43
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$\begingroup$ The energy of a particle is only altered when a force acts on it. And I have already considered all forces so I think I have already considered readiative effect on particles energy. Am I correct? $\endgroup$– PhysicsCommented Mar 24 at 10:55
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1 Answer
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Ok, I'm going to give you some hints for now. Everything should be familiar from when you solved the classical two body problem.
First, you might want to rewrite your equation as $$ m \dfrac{d}{dt} \dfrac{\dot{\vec{r}}}{\sqrt{1 - \dfrac{|\dot{\vec{r}}|^2}{c^2}}} = k \dfrac{\vec{r}}{|\vec{r}|^3}, $$ and do a change of variables to polar coordinates i.e. put the infinite mass in the center of your coordinate system.
Then, you might want to take a look at conserved quantities: Energy and Angular momentum (Question: How do these look in SR?)
I hope this is enough for now, and that you might be able to reach the end on your own ;)
answered Mar 24 at 10:58