# Force and B field

1. Apr 4, 2012

### Sefrez

I am posting this after seeing this thread:

and after having thought of the matter for a while, it makes no sense. Have a test charge placed near a wire, and have that wire have electrons flowing in one direction. The protons are stationary in the reference frame of the test charge. I have seen it stated that the test charge experiences no net force.

Yet if it is moving, which implies a relative velocity with respect to the protons in the given frame, even at the same velocity of the electrons such that in that reference frame of the test charge they are at rest, that there is a force.

But this has got to be total nonsense. What we assign the identity "proton" and "electron" is totally arbitrary. That being said, we could assume that at the start of my post I instead said that protons were moving in the reference frame of the test charge and the electrons were at rest. If that were the case, I certainly see no distinctness such that I could say that the test charge has a force as with the after case of what I mentioned before.

That being said, I conclude that if a wire has current running through it, and we define the current to be such that only one charge carrier is moving relative to the test charge, there is a net force. There is no net force only when you define current as that of both contrasting charge carriers moving in opposite directions in equal magnitude. This would be equivalent to the first case, but with the test charge moving at half the speed of the current.

This seems to be the only way that the net result of the effect of contraction is zero. Because, as I see it, if a wire is neutral with no current, then assuming when current is pushed through it that charge is still conserved, it is still neutral. However, if charge carriers of opposite type are moving relative to the test charge such that there contraction net result is not zero, there is a seen force by the test charge.

Am I right here or totally losing it?

2. Apr 4, 2012

### tiny-tim

Hi Sefrez!
there is no net electric force (for an infinite straight current-carrying wire that is overall neutral), for a frame in which either the positive or negative charges are stationary …

the electric fields of the positive and negative charges (with different velocities) cancel out​

for any other frame, there is an electric force, which on some particles is balanced by the magnetic force

see DaleSpam's link, to D V Schroeder's article, http://physics.weber.edu/schroeder/mrr/mrr.html

3. Apr 4, 2012

### Sefrez

Thanks for the reply. I think I am beginning to see where my confusion is. I believe it is simply my initial condition that is flawed. This is the part that I don't understand:
In the lab frame, why is it that the positive charges are said to be separated by l being equal to the separation of the negative charges - even though the positive charges are in motion? More precisely, why is it that its contracted separation is equal to l - the negatives rest separation? I am assuming that in the lab frame when no current flows, the spacing is equal. But I see this cannot be the case because if the positive charges were then set into motion, they would contract to < l - which disagrees with the diagram.

Thanks!

4. Apr 5, 2012

### tiny-tim

because the total number of electrons is fixed

since the length of the wire (which is stationary) is also fixed, that means that the density must be the same no matter how fast the electrons are moving

the fitzgerald-lorentz length contraction, of course, only applies to rigid bodies

the electrons aren't rigidly connected to each other, so contraction is irrelevant

(compare this with a train made of adjacent but unconnected carriages on a circular track, with the same length as the track when stationary: if one carriage has power, and pushes the others to nearly the speed of light, then a gap will open up between it and the carriage behind it)

5. Apr 5, 2012

### Sefrez

I understand this from a global perspective, but I thought it was irrelevant locally to the test charge. But nonetheless, how come the electrons do change in separation when the test charge is moving relative to the protons? I still don't understand what make the protons dominant. That is, for some reason still not understood by me, electrons can be moving at any rate, but yet as long as the test charge is fixed relative to the protons, they don't contract. In the diagram, you see both the negative and positive charge densities change, but yet the frame with the protons is in charge equilibrium rather than the frame with the electrons. I am having trouble differentiating the rules between electrons and protons. But it is obvious here, the protons must be different other than simply contrasting electrons in charge.

I did not know this. Though, the electrons do contract/detract when the test charge is in motion relative to the protons? How is that?!

6. Apr 5, 2012

### tiny-tim

i'm sorry, i can only repeat that contraction is irrelevant

a rigid body by definition has the same configuration whatever its speed, so we can compare its length at different speeds (and find a contraction)

but the electrons (as a body) are not rigid … in their stationary state and in their moving state, the electrons have different configurations, which can't be compared …

we can't find a contraction because we don't have the same thing to compare with itself
both the proton frame and the electron frame are in charge equilibrium

(other frames aren't, and have a small electric field from the imbalance)
you mean, in the frame of the test charge?

because then we're comparing the same thing in two different frames (the electrons in the stationary frame and in the test charge frame) …

result, lorentz contraction​

before, you were trying to compare two different things (electrons with no current and electrons with current) in the same frame …

nothing to compare​