Particle motion in a magnetic field

[etotheipi]

Homework Statement

A particle of mass $m$, charge $q$ and position $\mathbf{r}(t)$ moves in a magnetic field $\mathbf{B}$ pointing horizontally, and a uniform gravitational field $\mathbf{g}$ pointing vertically downward.

Derive that the particle undergoes helical motion with constant drift $\bot$ to $\mathbf{B}$.

Relevant Equations

N/A

The equation of motion can be integrated w.r.t. $t$ since $\frac{d}{dt} (\mathbf{r} \times \mathbf{B}) = \dot{\mathbf{r}} \times \mathbf{B} + \mathbf{0}$ $$\int (q\dot{\mathbf{r}} \times \mathbf{B} + m\mathbf{g}) dt = \int m\ddot{\mathbf{r}}(t) dt$$ $$\frac{q}{m} \mathbf{r} \times \mathbf{B} + t\mathbf{g} + \mathbf{c} = \dot{\mathbf{r}}$$ I thought it might help to put a basis to it, so I let $\mathbf{B} = B\mathbf{\hat{x}}$ and $\mathbf{g} = -g\mathbf{\hat{y}}$, so that $$\dot{\mathbf{r}} = \frac{qB}{m} \mathbf{r} \times \mathbf{\hat{x}} -gt \mathbf{\hat{y}} + \mathbf{c}$$The $\frac{qB}{m} \mathbf{r} \times \mathbf{\hat{x}}$ is a component of the velocity perpendicular to the $\mathbf{x}$ axis which I assume contributes to the 'circular' part of the motion. It's not obvious to me what the $\mathbf{c} - gt\mathbf{\hat{y}}$ component of velocity will cause. I don't think I can integrate this equation again either.

I wondered whether anyone could point me in the right direction (vector pun not intended)? Thanks!

 

[etotheipi]

Thanks for the reply! It's easier to visualise without the constant term, so yes let's say we boost into another frame with $\dot{\mathbf{r}}' = \dot{\mathbf{r}} - \mathbf{c}$ so that we are left with $\dot{\mathbf{r}'} = \frac{qB}{m} \mathbf{r}' \times \mathbf{\hat{x}} -gt \mathbf{\hat{y}}$. In the absence of the weight, $\dot{\mathbf{r}}' \cdot \ddot{\mathbf{r}}' = 0$ so we would end up with helical motion.

With the weight added back in, conceptually I can see why it will be helical motion with constant vertical translation, though I wonder if there is any way of deriving the equation of the helix to show this?

 

[Replusz]

sorry about previous post, i completely misread your question

 

[etotheipi]

sorry about previous post, i completely misread your question

Ah no worries, I thought it was useful . Didn't even notice!