Conservation of Momentum in Electrodynamics
In my previous article about conservation laws in electrodynamics, I discussed a derivation of the Poynting Theorem, the so-called “work-energy theorem” of electromagnetic theory. I also described how it represents the conservation of energy and further, derived its differential form. However, the conservation of momentum still remains a mystery.
Let us first discuss how the momentum of charge carriers alone does not describe the full picture. I will prove that result by showing how electrodynamics seems to violate the third law of motion.
Electrodynamics and the Third Law of Motion
Recall the exercise from the end of my previous article. We start with two point-charges moving towards the origin along x and y axes. For the sake of simplicity, imagine that they have the same sign. Can you draw the directions of electric and magnetic forces on each of the charges due to the other?
Remember, for a moving charge, these fields aren’t given by Coulomb’s law or Biot-Savart law. However, at any instant, the electric field points radially outward, and the magnetic field circles around the loop in tangential direction. The exact derivation involves a two-page proof with Liénard-Wiechert Potential and retarded time. But here is the instantaneous representation of these fields:
Now, use Fₘ = q (v × B) to figure out the direction of magnetic force on the charges. Here’s the diagram you’ll end up with:
Do you see the problem? Evidently, the magnetic forces are equal in magnitude but not even remotely opposite in direction, which is a blatant violation of the third law.
Does it matter?
To the untrained physicist, the third law appears simple; a mere aesthetic symmetry in the vast world of dynamics. If only that were so! In fact, the third law of motion plays a more vital role in dynamics of a system than the second law itself.
The proof of conservation of momentum rests on the cancellation of internal forces, which follows from the third law.
Thus, tampering with the third law would be blasphemy!
How Do We Overcome This Hurdle?
The answer, it turns out, is trivial in words but decidedly tiresome to derive. In a nutshell, you should note that electromagnetic fields themselves are carriers of energy, to which, we can attach a “momentum”, in addition to the mechanical momentum of the charge carriers themselves. This should satisfy our needs for the third law to hold.
At the same time, one can’t simply use the empirical relation of P =√2mK since fields have no concept of “mass”. Instead, we start with the expression for force per unit volume, the specific force, if you will, from Lorentz’s law.
f = ρ (E + v × B) = ρE + J × B
Now, be prepared for a lengthy derivation. We will need to rewrite this equation in terms of electric and magnetic fields alone by eliminating the sources and charges using Maxwell’s equations. Look at the following substitutions.
Substitute these values and you will find yourself with a rather large equation that looks like so:
Now, this looks pretty terrifying. However, we’ll clean this up nicely in just a couple of lines. Recall that I discussed Poynting vector while talking about Energy conservation, which was nothing but S ≡ (1/µ₀) (E × B). Next, recall the identity for the gradient of a dot product. For any vector A,
∇ (A.A) = ∇(A²) = 2 (A.∇) A + 2A × (∇ × A)
Finally, since ∇.B vanishes without fail, we can include it anywhere in a sum or difference without affecting the equation. It even helps us create a pleasing symmetry between the E and B terms. Eliminate the curls using the expression just provided, and we are left with:
We now introduce a tensor quantity in terms of E and B and call it the Maxwell Stress Tensor.
Notice the strange double arrow over the T. This signifies the double indices in the subscript. While a vector is represented by a single index, a tensor carries multiple indices, which can theoretically go beyond three or four in number. In the Cartesian system of co-ordinates, the indices represent the x, y, and z axes.
Thus, we now have an expression for the force per unit volume on a moving charge in terms of the Maxwell Stress Tensor and Poynting Vector, and we are ready to integrate it to find out the total force. Once you do that, your final expression will be this:
The divergence theorem can be used to remove the del operator from the ∇.T term. In the static case, the second term vanishes, and we are left with only the surface integral of T. Thus, the Maxwell Stress Tensor is simply the force per unit area. In terms of components, Tᵢⱼ represents the force per unit area in the iᵗʰ direction, acting on a surface element pointed in the jᵗʰ direction. In simpler words, the diagonal elements of the Tᵢⱼ matrix represent pressures, while the off-diagonal elements are what we call the shears.
The Physical Significance of Maxwell Stress Tensor
An important reason behind deriving Maxwell Stress Tensor is that it simplifies calculations. In cases where only a charge moving along a straight line is under consideration, calculating the electromagnetic force is fairly easy. However, even a slight deviation from this impractically simplistic scenario results in nightmarish equations. For example, if the magnetic field becomes non-homogenous, we immediately find ourselves with the requirement of calculating an arduous curl.
The Stress Tensor eliminates these tedious requirements. And since the final expression involves a surface integral instead of a volume integral, the calculations are simplified further. There are also applications of this quantity in relativistic electrodynamics, where it appears in the expression for electromagnetic stress-energy tensor.
What About Momentum?
The derivation of force totally distracted us from our original task. We had set out to prove the conservation of momentum in the case of non-static electrodynamics. But we have an expression for F now, and we can use the relation F = dp/dt.
Do you notice the perfect symmetry between the expression for force and the Poynting Theorem? The left-hand side here is the time derivative of mechanical momentum. The term with the Poynting Vector thus represents the rate of change of momentum stored in the electromagnetic fields themselves. And the integral of the Stress Tensor should now be the momentum flowing in through a surface in unit time. Representing the mechanical momentum with pₘ we have the integral form of the conservation of momentum:
The Differential Form
Just as we did before, we represent the electromagnetic momentum density with ρₑₘ, and define it as ρₑₘ =μ₀ε₀ S. Similarly, the mechanical momentum density is represented by ρₘ. All that is left now is to bring the momentum terms on one side and remove the integral.
Just as discussed for energy in my previous article, we now have the quantity T, which represents the momentum flux density, just as S represents the energy flux density. Finally, I am going to end this article with a little note.
S and T Play Dual Roles
S represents energy flux density but when multiplied by μ₀ε₀, which is just the inverse of the square of speed of light, it becomes the momentum per unit volume stored in the fields.
Similarly, T is electromagnetic stress, but it can play the role of momentum current density when we take its divergence.
Conclusion
It is now evident that electromagnetic fields carry significant importance, both in the static and non-static case. You already knew that they carried energy and just now, we showed that linear momentum can also be associated with these fields. You can even associate angular momentum to these fields. Thus, any discussion of the conservation of momentum would be incomplete without incorporating the effects of these fields into account.
