# Conservation of momentum without newton's third law

1. Aug 7, 2014

### davidbenari

So I've recently read newton's third law violates the principles of relativity. I only know how to prove conservation of momentum if Newton's third law holds. I was hoping someone could explain to me this (proving conservation of momentum when newton's third law is false) without using extremely hefty mathematics, and specifically addressing E&M:

" It turns out that we can rescue'' momentum conservation by abandoning action at a distance theories, and instead adopting so-called field theories in which there is a medium, called a field, which transmits the force from one particle to another. In electromagnetism there are, in fact, two fields--the electric field, and the magnetic field." source: http://farside.ph.utexas.edu/teaching/em/lectures/node28.html#e311

Thanks.

2. Aug 7, 2014

### Staff: Mentor

In a nutshell, the electromagnetic field carries momentum. We define this in terms of a momentum density (momentum per unit volume), as defined at the top of this page:

http://farside.ph.utexas.edu/teaching/em/lectures/node91.html

The total momentum of a system of particles and electromagnetic fields is the sum of the mechanical momenta of the particles plus the integral of the electromagnetic momentum density over all space. Momentum can be transferred between these two forms by way of the electromagnetic (Lorentz) force $\vec F = q(\vec E + \vec v \times \vec B)$, but the total is conserved. Unfortunately, to prove that the total is conserved, you need to use a lot of "hefty mathematics" in the form of vector calculus.