Rotating Frames: charges in a magnetic field

I've got a problem understanding a line of proof in my lecture notes.

Given that you have a charge of +Q and mass m orbiting a fixed particle of charge -Q' in the presence of a magnetic field B. The particle is moving slowly enough for relativistic effects to be ignored.

Given that:

$$m\hat{a_{I}}=-\frac{QQ'}{4\pi\epsilon_{0}r^{2}}\hat{r}+Q\hat{v_{I}} \times \hat{B}$$

where $$\hat{v_{I}}$$ is the particle's velocity in the inertial frame.

Substituting

$$\hat{a_{I}}=\hat{a_{R}}+2\hat{\omega} \times \hat{v_{R}}+\hat{\omega} \times (\hat{\omega} \times \hat{r})$$

into the first equation along with $$\hat{v_{I}}=\hat{v_{R}}+\hat{\omega} \times \hat{r}$$

gives:

$$\hat{a_{R}}+2\hat{\omega} \times \hat{v_{R}}+\hat{\omega} \times (\hat{\omega} \times \hat{r})$$ = $$-\frac{QQ'}{4\pi\epsilon_{0}mr^{2}}\hat{r}+(\frac{Q}{m})[\hat{v_{R}}+(\hat{\omega} \times \hat{r})] \times \hat{B}$$

Apparently the terms porportional to $$\hat{v_{R}}$$ cancel if $$\hat{\omega}=-\frac{Q}{2m}\hat{B}$$

Why is this so? Visualizing the situation will probably help a lot.

Last edited:

Born2bwire
Gold Member
What is \omega \times r? Are these scalars or vectors? You are mixing scalars and vectors here.

What is \omega \times r? Are these scalars or vectors? You are mixing scalars and vectors here.

I've corrected it now. The \times is supposed to be the cross product. I didn't know what the latex is for cross product.

Dale
Mentor
2021 Award
This is just a guess. To me that looks like the cyclotron frequency in the presence of a net charge. So the motion is purely radial in a reference frame which is rotating at the cyclotron frequency.

This is just a guess. To me that looks like the cyclotron frequency in the presence of a net charge. So the motion is purely radial in a reference frame which is rotating at the cyclotron frequency.

Well, the omega frequency is known as the larmor frequency. It is half the cyclotron frequency. The charge +Q is precessing around the charge -Q. I don't understand how the velocity in the rotating frame cancels, giving that it is orbiting in the rotating frame (which means there must be a v subscript R).