Electron in Constant B-Field (Cylindrical Coordinates)

[Opus_723]

Homework Statement

The position of a proton at time t is given by the distance vector

[itex]\vec{r}(t) = \hat{i}x(t) + \hat{j}y(t) + \hat{k}z(t)[/itex]

A magnetic induction field along the z-axis, [itex]\vec{B} = \hat{k}B_{z}[/itex] exerts a force on the proton

[itex]\vec{F}[/itex] = e[itex]\vec{v}[/itex][itex]\times[/itex][itex]\vec{B}[/itex]

a.) For initial conditions:

x(0) = x[itex]_{0}[/itex]
[itex]\dot{x}[/itex](0) = 0
y(0) = 0
[itex]\dot{y}[/itex](0) = v[itex]_{y0}[/itex]
z(0) = 0
[itex]\dot{z}[/itex](0) = v[itex]_{z0}[/itex]

and using cartesian coordinates calculate the orbit of the proton.

b.) Rephrase this entire problem in circular cylindrical coordinates and solve in circular cylindrical coordinates

The Attempt at a Solution

I think I managed to solve it in Cartesian by solving a coupled system of linear differential equations. My result was:

x(t) = [itex]\frac{v_{y0}m}{eB_{z}}cos(\frac{eB_{z}}{m}t) + (x_{0}-\frac{v_{y0}m}{eB_{z}})[/itex]

y(t) = [itex]\frac{v_{y0}m}{eB_{z}}sin(\frac{eB_{z}}{m}t)[/itex]

z(y) = v[itex]_{z0}[/itex]t

But I don't see how to do this in cylindrical coordinates. When I set up the differential equations using Newton's Law, I get:

[itex]\ddot{\rho}-\rho\dot{\varphi}^{2} = \frac{eB_{z}}{m}\rho\dot{\varphi}[/itex]

[itex]\rho\ddot{\varphi}+2\dot{\rho}\dot{\varphi} = \frac{-eB_{z}}{m}\dot{\rho}[/itex]

[itex]\ddot{z} = 0[/itex]

And of course the initial conditions

[itex]\rho(0) = x_{0}[/itex]
[itex]\dot{\rho}(0) = 0[/itex]
[itex]\varphi(0) = 0[/itex]
[itex]\dot{\varphi}(0) = \frac{v_{y0}}{x_{0}}[/itex]
z(0) = 0
[itex]\dot{z}(0) = v_{z0}[/itex]

But those equations are nonlinear, and I don't see any way to do this problem, although I assumed the problem would be easier in cylindrical coordinates because of the symmetry involved. I can see how the solutions for [itex]\rho(t)[/itex] and [itex]\varphi(t)[/itex] could get really weird in the general case where the helix isn't centered on the z-axis. But I don't see how to simplify this particular problem.

 

[Simon Bridge]

Sketch the orbit - exploit the symmetry.

 

[Opus_723]

I already know the orbit is a helix. I don't see how that helps me though, unless I just assume that the initial speed is exactly what is needed to center the helix on the z-axis?

 

[Simon Bridge]

The question does not specify a choice of axis though ... why not center your z-axis on the center of the helix?

 

[Opus_723]

I thought about that, but it sort of felt like cheating. Sure, I know that the path is going to be circular, and I know that if I move my z-axis so that x[itex]_{0}[/itex] is related to v[itex]_{y0}[/itex] in a particular way then I'll have a constant radius. But moving the z-axis to the right spot requires knowing how the radius of the helix depends on v[itex]_{y0}[/itex]. Which I do know, but heck, if I assume that I have a constant radius and I know the relationship between x[itex]_{0}[/itex] and v[itex]_{y0}[/itex], I've basically solved the problem already. I could describe the orbit from that information alone without any differential equations.

I guess I just wanted to know if there was any way to do this problem without assuming the answer.

 

[Simon Bridge]

Of course you can do it without assuming the answer - it is just easier if you know where it's headed.
Note - the x,y,z version kinda has an origin.
This origin will be displaced from the axis of rotation - you should know the polar equation of a circle whose center is not at the origin. That should give you a clue how things should go.

Is this a physics course or a math course?

However - you do have to do everything from the start and in your attempt you appear to have got ahead of yourself.

You should probably check how you translated the initial conditions:
i.e. you have the radial velocity at t=0 as zero ... but wouldn't that only be true if the z-axis was the axis of rotation? Same with the angular velocity being vy0/x0 ... which appears to assume the turning radius is x0.

How do you convert between cartesian and cylindrical coordinates?