Rotating Frames: charges in a magnetic field

[ian2012]

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.

 

[Born2bwire]

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

 

[ian2012]

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]

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.

 

[ian2012]

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).