View source for Introduction to Electrodynamics/Chapter 7/5

[[../4|7.4 Potential Formulations of Electrodynamics]]

[[../../8|8 Electromagnetic Waves]]

= 7.5 Energy and Momentum in Electrodynamics =

Our new equations lead to curious results that seem to contradict some fundamental laws in classical mechanics. Namely, Newton's Third Law, which leads both to the conservation of momentum and the conservation of energy.

== 7.5.1 Newton's Third Law in Electrodynamics ==

We have our axes aligned like this.

Imagine a point charge q traveling on a train track along the x axis at constant speed -- to the left.

We can calculate, roughly, what the Electric and Magnetic fields generated by this charge are. Even though the Biot-Savart Law and Coulomb's Law do not apply, the modifications we make to these laws are pretty close to them. Namely, the E field points out from the positive charge (although it is somewhat weaker ahead and behind) and the B field curls around the velocity (again, weaker ahead and behind.) We'll discuss how to calculate these fields precisely in Chapter 9, but hand-waving should be enough to convince you it is so.

It's probably a good exercise at this point to see how the time derivatives of E and B affect each other. You won't be able to get all the intricate details, but you should see that there is some opposition to any change in the fields.

Let's introduce another charge now, moving down along the Y axis at constant speed v. It is attached to a train track as well. It's E and B fields are similar, except they are rotated 90 degrees.

Question: What forces do these charges exert each other as they close in on the origin? The electric fields oppose each other, so we see Newton's Third holds.

But look at the B fields. The force of the guy on the right on the guy above is in the positive x direction, but the force of the guy above on the guy to the right is up. Newton's Third Law does NOT hold for electrodynamic forces.

This is important. Newton's Third Law does NOT hold in electrodynamics, even though it did in electrostatics and magnetostatics.

This is important! Now one of our most loved laws, the conservation of momentum, is completely useless in electrodynamics, or at least it seems so.  

However, all is not lost. Remember we assigned an energy to the fields themselves? Why not say that the particles do not directly interact with each other, but transfer momentum to the fields, which the fields then carry and transfer to other particles? Perhaps if we look at things this way we can save the conservation of momentum.

7.5.2 introduces the Poynting Theorem which gives us the Poynting Vector which tells us how energy is moving around in the fields.

7.5.3 introduces Maxwell's Stress Tensor, a sort of double-vector, allows us to calculate the force on a volume of space or along a surface.

7.5.4 combines these two concepts with the forces seen to give us the momentum that is carried by the fields themselves.

[[/2|7.5.2 Poynting's Theorem]]

[[/3|Maxwell's Stress Tensor]]