Calculating trajectory electric field

[aaaa202]

In my class we have so far only dealt with electrostatics, i.e. fields of stationary charge distributions. However one question arose while doing so: Given a set of stationary source charges we can calculate the electric field at each point in space. But will that give us enough information to calculate the trajectory of a particle with known charge q and mass m placed at an arbitrary point in space?

Numerically it wouldn't be hard, but is it possible to do analytically for alle kinds of fields? We neglect magnetic forces even though I know they are present. I'm not really sure how you should do it.

 

[tiny-tim]

hi aaaa202!

… will that give us enough information to calculate the trajectory of a particle with known charge q and mass m placed at an arbitrary point in space?

electric field is force per charge … F = qE

force = mass times acceleration … F = m d²r/dt²

if you know E as a function of r, you should be able to solve that ​

 

[aaaa202]

Well it is definitely less obvious what to do than the constant acceleration problems you did in mechanics.

How would you do it?
You could always do it numerically. Like say make a timestep of 0.001sec and then calculate the acceleration and where to the particle has gone after that little timestep. But how would you do it with calculus?

You mention the differential equation:

qE(r) = md²r/dt²

Can you simply solve that for the correct trajectory?

if so, then I love differential equation solutions

 

[tiny-tim]

qE(r) = md²r/dt²

Can you simply solve that for the correct trajectory?

if so, then I love differential equation solutions

yes, simply solve the differential equation

(for example, if E is the electric field of a single charge, then the trajectories will be the same as the orbits of a planet round the sun)