Feynman's vs wiki's relativistic resultant

[galvin452]

In Feynman Lectures on Physics vol 2 pg.13.6-13.10 develops the equations for a current carrying wire and a moving charge ( negative test charge) with the same velocity as the electrons in the current. He looks at this situation from two reference frames, 1) the wire still and test charge and electrons moving at velocity v and 2) the test charge and conduction electrons still and the wire with velocity -v.

Using relativity and the Lorentz factor gamma for reference frame 2), the test charge sees an electric field, while from rest frame 1) the test charge sees a magnetic field. Both reference frames result in the same change to the test charge's momentum toward the wire even though the two reference frame's forces (F_1,F_2) are different (i.e. F_2=F_1*gamma) because the relative delta in time (t_1=t_2*gamma) is opposite the delta in forces so momentums p_1=p_2 (p_1=F_1*t_1, p_2=F_2*t_2=F_1*gamma*t_1/gamma=F_1*t_1=p_1).

In http://en.wikipedia.org/wiki/Relativistic_electromagnetism the section "The origin of magnetic forces" has almost the same situation except the current is carried by positive charges and the test charge is positive (rather than negative current and negative charge). However wiki has both a Lorentz contraction for the negative charge and a Lorentz expansion of the positive charge resulting in the same electrostatic force (F_e) as magnetic force (F_m), i.e. F_e=F_m. Or in terms of Feynman's notation, F_2=F_1.

These two do not agree, which one is correct and why? Or if both are why?

 

[Simon Bridge]

Moving lengths are shorter.

The observer at rest wrt the negative charges sees the "proper" spacing between negative charges and a contracted spacing between the positive ones.

The observer at rest wrt the positive charges see the proper spacing for the proper charges, and the spacing of the negative charges is contracted.

I suspect both are correct and you have misread both of them - has Feynman just left the Lorentz factor outside the F while wiki includes it? i.e. who measures each force?

In general - all observers should agree about the net force experienced by the test particle but will disagree about how that force comes about.

 

[galvin452]

In general - all observers should agree about the net force experienced by the test particle but will disagree about how that force comes about.

I gave you the summary but specifically Feynman equation (13.30) reads

F' = F/sqrt(1-v^2/c^2) (I wrote F_2=F_1*gamma)

where F' is reference frame 2 and F is reference frame 1 so this does not agree with your statement "all observers should agree about the net force experience".

So your saying that Feynman Lectures on Physics is wrong?

 

[pervect]

The four-force, the rate of change of momentum with proper time, should be the same for all observers. The four-force is NOT the three force, though!

I'd suggest looking it up in a third source (other than wikki and Feynmann).

 

[WannabeNewton]

In particular, the component of the 3-force that is parallel to the relative motion of the two inertial frames at a given instant of time has the same value in the moving frame as it does in the rest frame of the particle. The 3-force component perpendicular to the relative motion of the two inertial frames at that instant is always smaller by the value $\frac{1}{\gamma}$ in the moving frame as compared to its value in the particle's rest frame. See the entirety of chapter 5 of Purcell "Electricity and Magnetism" as well as appendix G of the same text.

 

[Simon Bridge]

The four-force, the rate of change of momentum with proper time, should be the same for all observers. The four-force is NOT the three force, though!

In particular, the component of the 3-force that is parallel to the relative motion of the two inertial frames at a given instant of time has the same value in the moving frame as it does in the rest frame of the particle. The 3-force component perpendicular to the relative motion of the two inertial frames at that instant is always smaller by the value $\frac{1}{\gamma}$ in the moving frame as compared to its value in the particle's rest frame. See the entirety of chapter 5 of Purcell "Electricity and Magnetism" as well as appendix G of the same text.

I stand corrected - should have been more careful.

 

[TSny]

In http://en.wikipedia.org/wiki/Relativistic_electromagnetism the section "The origin of magnetic forces" has almost the same situation except the current is carried by positive charges and the test charge is positive (rather than negative current and negative charge). However wiki has both a Lorentz contraction for the negative charge and a Lorentz expansion of the positive charge resulting in the same electrostatic force (F_e) as magnetic force (F_m), i.e. F_e=F_m.

Feynman also has a Lorentz contraction for one sign of charge and Lorentz expansion for the other sign of charge when going from the lab frame to the frame co-moving with q. (See Feynman’s 13.24 and 13.26). But, the wiki article makes an approximation for the gamma factors in the charge density (assuming v << c) , whereas Feynman does not make that approximation. You can see that wiki would get the same expressions for the forces as Feynman if wiki did not use the approximation. So, as I see it, there is no discrepancy.

 

[galvin452]

Feynman also has a Lorentz contraction for one sign of charge and Lorentz expansion for the other sign of charge when going from the lab frame to the frame co-moving with q. (See Feynman’s 13.24 and 13.26). But, the wiki article makes an approximation for the gamma factors in the charge density (assuming v << c) , whereas Feynman does not make that approximation. You can see that wiki would get the same expressions for the forces as Feynman if wiki did not use the approximation. So, as I see it, there is no discrepancy.

Thanks, for the feynman equation references. I also missed in the wiki equation for λ has in the middle has ≈ which is where the "(assuming v << c)" is.