# Relativistic Induction

Q:
There is an interesting article about the relativistic explanation of the origin of magnetic forces: http://physics.weber.edu/schroeder/mrr/MRRnotes.pdf Similar to the textbook by E.Purcell. It allows to "skip" the notion of a magnetic field as such and put it as electric field in another reference frame. Now my question is: what kind of "relativistic" distorted electrical field we are dealing with in the case of charges changing speed? In other words how one loop of wire can induce emf in the other one just by changing its current. The relativistic approach introduces the radiation type of electric field that sort of "lags" the accelerating charge thus pointing backward. This can be an explanation of induction of emf by one wire in the other one. If so (but maybe I'm wrong with this radiation field shape reasoning) how a single wire produces induction on itself? Is it still radiation? When I think about it this way I'm trying to visualise the whole circuit and eventually deduct why a loop shape tends to escalate this phenomenon? As we know the magnetic flux through a loop grows with its area. Maybe the radiation field shape has a lot more impact if it is distributed along 360 deg turn. Why is it that the longer the turn is the more of it (induction) happens? This subject (like electric field distortion) is hard to search in the net. Most explanations go around the issue of changing the flux by moving the flux generator in space (magnet etc). Then it is "natural" to be the same thing when flux changes by changing its source current. In Faraday's law it is the same. But what is the relativistic approach explanation of induction in air core inductor?
- Andrew W (age 34)
A:
You bring back fond memories of my freshman E&M course, taught by Purcell from his book.

Think of the EMF produced on a loop by changing current in another loop of the same size. I think you've already followed Purcell's relativistic argument about why that exists. Move the loops so they are next to each other.The EMF produced by a loop on itself is then just the same as that produced on the other loop, because they almost coincide and thus see the same fields.

Maybe that isn't a full answer. It's hard to think of a better explanation than just to look at Purcell again.

Mike W.

(published on 06/30/2009)

## Follow-Up #1: EM induction

Q:
OK that would explain self induction with "another current" in the same wire and it makes sense. I'm still troubled with the field shape that causes this. Is it radiation? Is it that as some charges accelerate the slower ones (just yet) see their fields pointing more towards them? And in total the current experiences initial "resistance" which can be measured as back emf...
- Andrew W (age 34)
A:
There's no way I can give a better explanation than chapter 7 of Purcell. He's got good pictures, too.

The effect is not really radiation, in the sense that we are usually considering time scales very short compared to the dimensions of the device divided by c.

Rather than think of some charges as being slow and others as being fast (which is not relevant, since we've just seen that the time scales are very short, so that everything happens almost simultaneously). Let's look at what happens in one part of a wire due to a part across the way from it. Initially each sees a magnetic field at right angles to the wire, from the current in the opposite wire. As the current changes, the magnetic fields change. Seeing a changing magnetic field is like moving in a non-uniform magnetic field. Moving in a magnetic field makes a Lorentz force pushing on the electrons. The Lorentz forces on the opposite wires point opposite ways. So if there's a loop, they both point clockwise or both counterclockwise. If you go through the math carefully, you'll find that the direction of the force opposes the direction of the change in current.

Again, Purcell has a more complete  and elegant version of this.

Mike W.

(published on 07/02/2009)

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