Question

A positive particle of charge q and mass m is moving in a region of uniform magnetic and electric fields along the z axis. At time t=0, the particle has a velocity v and angle theta against the x axis. Theres no y component. What will the be the magnitude of v1 of particle velocity at time t=t1?

How would I approach this problem?

Answer 1

wow, hard question. erm if you know the work needed to move a charged particle through the fields you should be able to work out the force that will be exerted on the particle. then you could find the acceleration caused by that. from that you may be able to find v1 at t1.this will all be algabraic mind you. let me get back to you i may have an answer soon just need to brush up on my electromagnatism. good luck. hope that helped.
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Answer 2

First, define the symbols:\[q is the charge, m is the mass,
\[E _{0} is the electric field, B _{0} is the magnetic field, both along the z axis \]\]
If you write the Lorentz force equation and write the components, you will get the following equations to be solved
\[dv _{x}/dt = (qB _{0}/m) v _{y} _________(1)\]
\[dv _{y}/dt = -(qB _{0}/m) v _{x} ________ (2)\]
\[dv _{z}/dt = qE _{0}/m _________(3)\]
I numbered the equations for convenience.
The initial conditions are:
\[t = 0, v _{x0} = v _{0}\cos \theta, v _{y0} = 0, v _{z0} = v _{0}\sin \theta\]
You can solve these equations quite easily. Equation 3 is the easiest. The solution is
\[v _{z}(t) = v _{zo} + (qE _{0}/m)t\]
The x and y components of the velocity are oscillating functions of time:
\[v _{x}(t) = v _{x0}\cos (\omega t)\]
\[v _{y}(t) = - v _{xo} \sin(\omega t)\]
where I have used the symbol:
\[\omega = qB _{0}/m\]
That is the mathematical solution. The picture is that the initial x-component of the velocity, takes the charge in a circle in the xy plane. The initial z component drags it out along a helical path. If there is no electric field, the z-component of the velocity is a constant, and the helix has a constant pitch. With the electric field, the z-component of the velocity is increasing linearly, so the pitch of the helix keeps increasing.
Let me know if you need more details