Newton's Third Law and Relativity

[psholtz]

Suppose I have a proton at the origin, and a (stationary) electron on the x-axis at x=r.

Taking q to be the elementary charge, the force acting on each of these particles is:

$$F = \frac{q^2}{r^2}$$

Now suppose that the electron is heading towards the proton (in the negative x-direction) at a relativistic speed. Suppose gamma=2. The field generated by the electron will be "squished" in the x-direction, and correspondingly the force experienced by the proton will be weaker than it would have been had the electron been stationary.

The electric field generated by a moving charge (as seen in the "stationary" frame) is given by:

$$E = \frac{Q}{r^2}\frac{1-\beta^2}{1-\beta^2\sin^2\theta}$$

where Q is the charge on the moving particle and theta is the angle between the origin of the moving particles coordinate frame and the direction of the particle's motion (in this case, the x-axis).

In our case, we have:

$$E = \frac{1}{4}\frac{q}{r^2}$$

and so the force the proton experiences, due to the moving electron, is given by:

$$F = \frac{1}{4}\frac{q^2}{r^2}$$

Meanwhile, the force on the (moving) electron, due to the stationary proton, is given by the Lorentz force law:

$$F = qE + \frac{1}{c}v \times B$$

Since the proton doesn't generate a magnetic field, the force on the (moving) electron due to the stationary proton, at the moment the electron is passing the point x=r, is:

$$F = \frac{q^2}{r^2}$$

In other words, the force the proton imparts to the electron is 4x the force that the electron imparts to the proton.

What happened to Newton's Third Law?

 

[jtbell]

Newton's Third Law is equivalent to conservation of momentum. In electrodynamics, the total momentum of a particles in the system is generally not conserved. However, electric and magnetic fields also carry momentum. The total momentum of the particles and the fields is conserved.

 

[Entropia]

Newton's Third Law states that for every action, there is an equal and opposite reaction. In this case, the force experienced by the proton due to the electron's motion is not equal to the force experienced by the electron due to the proton's presence. This may seem to contradict Newton's Third Law, but it is actually a result of the relativistic effects at play.

In the stationary frame, the electron appears to be moving at a high speed, causing its electric field to be "squished" in the x-direction. This means that the force experienced by the proton, which is directly in the path of the electron's motion, is weaker than it would have been if the electron were stationary. On the other hand, the electron experiences a stronger force from the stationary proton due to the Lorentz force law, which takes into account the electron's velocity.

This difference in forces is a result of the relativity of simultaneity - the concept that two events that are simultaneous in one frame of reference may not be simultaneous in another frame. In this case, the force experienced by the proton is not simultaneous with the force experienced by the electron, leading to a difference in magnitude.

In conclusion, Newton's Third Law still holds true in this scenario, but the relativistic effects cause a difference in the forces experienced by the two particles. This highlights the importance of considering the effects of relativity in understanding the interactions between particles at high speeds.