Solving the Motion of an Electron in a Magnetic/Electrical Field

[Icheb]

I have a constant magnetic field pointing in the direction of the z axis and I'm supposed to find a formula for the way an electron with arbitrary starting position and velocity would travel in this field.

The formula in this case would be

F(x, x', t) = q(v × B(x, t))

and I'm stuck at the point where I have to split this formula into the three equations for motion. Would I just use

F(x, x', t) = q(x' × B(x, t))

and so on or am I missing something? And how would I proceed afterwards?Then there's also a similar problem, but with an electrical field pointing in the direction of the y axis. Which formula would I use here? The only one I can think of is F = q*E, but don't I need a formula that contains information about the particle involved?

 

[quasar987]

First, there is no x and t dependence on B; the question says that it is constant in the z direction. Then you have to split that vector equation of yours into 3 scalar equations: one for each components. I'll do one for you since this seems to be confusing to you:

$$\vec{F}=q(\vec{v}\times B\hat{z})$$

$$\Leftrightarrow F_x\hat{x}+F_y\hat{y}+F_z\hat{z}=qB(v_x\hat{z}-v_y\hat{x})$$

Therefor, the scalar equation corresponding to the x-component is

$$F_x=-qBv_y$$

But since v_y=dy/dt and by Newton's second law, F_x=md²x/dt², that equation is equivalent to

$$\frac{d^2x}{dt^2}=-\frac{qB}{m}\frac{dy}{dt}$$