Function for the movement of a charged particle in a B field

[Eirik]

Homework Statement

I have a physics next week about force fields. We're making the tasks ourselves, but I wanted to challenge myself a bit by including a B field in the paremeterization of the movement of this charged particle. If a charged particle "shot out" through the zx-field, and there's a B-field along the z-axis, how am i supposed to parameterize the movement of the particle?

Relevant Equations

s(t)=[𝑣_0𝑥 𝑡+1/2 𝑎_𝑥 𝑡^2,𝑣_0𝑦 𝑡+1/2 𝑎_𝑦 𝑡^2,𝑣_0𝑧 𝑡+1/2 𝑎_𝑧 𝑡^2]
s(t)=(r*cos(w*t),r*sin(w*t))
F=q*v*B
a=v^2/r

The movement in the z-direction is easy to solve for, as it's only affected by the gravitational force. However, if there's a magnetic field pointing down along the z-axis, the particle is going to be accelerated along the y-axis (F=q*v *B). The force is always going to be perpendicular to the velocity vector, and it's therefore going to move around in a circle. I don't really know where to go from here. I can't really use the position formulas, as the acceleration isn't constant. Should I start by making a function for the acceleration of the particle and then integrate it? And if so, how would I go about doing that? I'm assuming I could use some trigonometry, s(t)=(r*cos(w*t),r*sin(w*t))?

Also, I'm creating this task myself (lol), so if you have any reccommendations as to what could make it a tad bit easier, I'd really appreciate it! We've only worked in 2d and only with the gravitational force, so I'm not really expected to know this stuff. The only requirement for the task, is that it has to have to do with a cannon firing a ball, but I'd love to incorporate B-fields and/or E-fields!

 

[phyzguy]

Since the velocity in the Z direction is parallel to the B field, the B field has no impact on this motion. So the motion in Z is independent of the motion in the XY plane. So, to start, ignore the motion in Z and assume Vz=0. What would be the motion in the XY plane in this case?

 

[Eirik]

To begin with, it would have velocity in the X direction, and Vy=0. Because of the velocity in the x direction, it's then going to be accelerated along the y-axis. This acceleration should be equal to a=q*v*B/m, right? Is it then correct to say that the acceleration is going to be a(t)=[(q*Vx*B/m)*sin(wt), (q*Vx*B/m)*cos(wt), g], where w is the angular velocity? Also, the force from the magnetic field shouldn't do any work on the particle, and therefore the velocity is constant in the XY field, V=V0x?

 

[phyzguy]

I think everything you said is correct. So you have the acceleration. So what is the velocity as a function of time? Also, you should be able to write w in terms of q, v0x, B, and m.