# Rotating Frames: charges in a magnetic field

• ian2012
In summary, the conversation discusses a problem understanding a line of proof in lecture notes regarding a particle with charge +Q and mass m orbiting a fixed particle with charge -Q' in the presence of a magnetic field B. The conversation discusses the equation for \hat{a_{I}} and the substitution of \hat{a_{I}} and \hat{v_{I}} into the equation. The conversation also mentions the importance of visualizing the situation and the confusion regarding the cancellation of terms proportional to \hat{v_{R}}.
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.

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

Born2bwire said:
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.

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.

DaleSpam said:
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).

## 1. What is a rotating frame in the context of charges in a magnetic field?

A rotating frame is a frame of reference that is constantly rotating at a constant rate. In the context of charges in a magnetic field, it is used to describe the motion of charged particles in a magnetic field.

## 2. How does a rotating frame affect the motion of charged particles in a magnetic field?

In a rotating frame, the charged particles experience a force known as the Coriolis force, which causes them to move in a circular motion. This is due to the fact that the magnetic field is also rotating in the same direction as the frame, causing the particles to experience a perpendicular force.

## 3. What is the relationship between the rotation rate of a frame and the strength of the Coriolis force?

The strength of the Coriolis force is directly proportional to the rotation rate of the frame. The faster the frame is rotating, the stronger the Coriolis force will be on the charged particles.

## 4. Can a rotating frame affect the direction of motion of a charged particle in a magnetic field?

Yes, a rotating frame can change the direction of motion of a charged particle in a magnetic field. This is due to the Coriolis force, which can cause the particles to move in a circular or spiral motion rather than a straight line.

## 5. Are there any real-world applications of the concept of rotating frames in relation to charges in a magnetic field?

Yes, rotating frames are used in many real-world applications, such as in the design of particle accelerators and gyroscopes. They are also used in the study of atmospheric and oceanic currents, as well as in the understanding of the Earth's magnetic field and its effects on charged particles in space.

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