Can anyone explain the use of inertial frames for problem solving in E&M?

[wotanub]

Can anyone refer me to a discussion of applying the technique of changing reference frames to problem solving? Why it works, and what it means. I'm familiar with using it in some E&M problems, but I guess I don't really "get" it. For example a particle in an E&M field has

$m\vec{a} = q(\vec{E}+\vec{v}\times\vec{B})$

It's common to set $\vec{v} = \frac{\vec{E}\times\vec{B}}{B^{2}} + \vec{u}$ to "cancel" the $\vec{E}$ field and just work out the circular motion for the magnetic field.

What does this say about the physics of the system? Is it right to say the particle really (in the original frame) is moving in a circle "plus" the velocity of the frame? That is, if I want the velocity, I find it in the intertial frame, then just tack on the $\frac{\vec{E}\times\vec{B}}{B^{2}}$ term?

That's right answer, but can someone point me to some formal proof for the general case of any moving particle? I've never seen it stated in general, it's always just the professor going "oh, let's wave our hands like this and make $\vec{E}$ go away..." during an example problem. I use it, but I've never really learned it, so I always have trouble explaining this. I usually say something like "if you add a constant to the velocity, it won't change the $\frac{d\vec{v}}{dt}$"

Or maybe if you've taught a class before, how did you explain it to your students?

 

[Simon Bridge]

The idea is that physics does not care what reference frame you use: stuff still works the same way. Therefore it is a good idea to pick one which makes the math easy to do.

Another common transformation is to make B go away by changing to a reference frame that is stationary wrt to the moving charges. The general idea is to exploit something special about the relationships involved.

A very simple transform is in ballistics when someone throws something from a non-zero initial height and it turns out to be easier just to call the initial position zero or you rotate the coordinates so that "down" becomes "positive" so you don't risk misplacing a minus sign (you've transformed, in the first case, to the frame of someone sitting at that height, and, in the second case, to that for someone upside down).

Every day you do motion and distance calculations taking the Earth as non-rotating - which means you are using a reference frame that is rotating with the Earth (this is actually a non-inertial frame!)

In your example - you'd usually fix your coordinates to the lab frame: the apparatus.
However - you don't have to. Where would the zero of the coordinate system be to make the transformation shown? What is the coordinate system doing? (What effect would the E field normally have?)

 

[A.T.]

I use it, but I've never really learned it, so I always have trouble explaining this. I usually say something like "if you add a constant to the velocity, it won't change the $\frac{d\vec{v}}{dt}$"

http://en.wikipedia.org/wiki/Galilean_invariance

 

[wotanub]

Yes thanks. This is what I was looking for.

 

[PJC01]

Inertial frames are used in problem solving in E&M because they allow us to simplify and analyze complex systems by choosing a frame of reference that is not accelerating. This choice of frame allows us to apply the laws of physics, such as Newton's laws, in a simpler and more straightforward manner.

The technique of changing reference frames is a powerful tool in problem solving because it allows us to transform a problem from one frame of reference to another, where the problem may be easier to solve. This is particularly useful in E&M problems, where the electric and magnetic fields may be changing with respect to time and space.

The equation m\vec{a} = q(\vec{E}+\vec{v}\times\vec{B}) describes the motion of a charged particle in an electric and magnetic field. In order to simplify this equation, it is common to choose a frame of reference where the electric field is zero. This allows us to isolate the effects of the magnetic field on the particle's motion, making it easier to solve for the particle's trajectory.

In terms of the physics of the system, choosing an inertial frame of reference means that the observer is not experiencing any acceleration. This allows us to apply the laws of physics in a simpler and more intuitive way. The addition of the term \frac{\vec{E}\times\vec{B}}{B^{2}} to the velocity in the chosen frame of reference allows us to account for the motion of the frame itself, as well as the motion of the particle in the magnetic field.

To prove this concept in a more general case, we can use the Lorentz transformation equations to transform the equations of motion from one frame of reference to another. This will show that the addition of the term \frac{\vec{E}\times\vec{B}}{B^{2}} is necessary in order to account for the motion of the frame and accurately describe the motion of the particle.

As a scientist, it is important to understand and explain the concepts and techniques we use in our work. In teaching a class, I would explain the use of inertial frames and changing reference frames by first discussing the concept of frames of reference and their importance in problem solving. I would then introduce the technique of changing reference frames and provide examples of how it can be applied in E&M problems. Lastly, I would discuss the general proof for the addition of the term \frac{\vec{E}\times\vec{