Conservation Laws in Electrodynamics
One of the most fundamental consequences of nature herself is the plethora of conservation laws that govern the possibilities of a physical process taking place. Widely categorized as either “global” or “local” in nature, these conservation laws are present across all realms of science. Each of them governs a certain physical quantity and most commonly, relates its time differential with the divergence of a closely related vector. The latter of these is what is loosely referred to as the “transport” of that physical quantity.
The most common conservation laws that you are sure to have come across are the conservation of charge, conservation of momentum, and conservation of energy. In the case of electrodynamics, the latter two topics are of immense importance. The conservation of charge is most succinctly described by the following equation of continuity:
As stated before, it relates the time differential of electric charge density with the divergence of current density. In the realm of quantum mechanics, ρ is replaced by P, the probability density and J represents the probability current density.
What About Energy?
Equivalent results can be observed for the transport of energy and momentum when one considers moving charges and fields. I’ll attempt to derive both of those conservation laws in simple terms.
Recall that the force on a charged particle, moving in an electromagnetic field with a velocity v is given by Lorentz’s equation F = q (E + v × B). Thus, the work done while moving a distance dl, will be given by F.dl. Replacing dl with v dt, we can eliminate the second term since (v × B).v = 0. Further, for an infinitesimal volume dV, the charge q can be replaced in terms of the charge density ρ. This leads us to the following expression:
dW = ρ (E.v) dV dt = (E.J) dV dt
Multiplying the charge density with the velocity simply gives us the current density. Differentiate the above equation with respect to time and it will be immediately evident that the dot product on the right-hand side (E.J) is nothing but the expression for power delivered per unit volume.
There is one little trouble, though. While working with electrodynamics, it is in our best interest to work in terms of the fields themselves instead of the sources. That is, the dot product of E.J needs a little readjustment. To do so, recall Maxwell’s equations of Electrodynamics. The curl of magnetic flux intensity is related to J and the time derivative of E. Let us solve it.
What next? Naturally, we would like to solve the first term now. As you might recall, that is simply the scalar triple product of three vectors. It can be expanded like so: E.(∇×B) = -∇.(E×B) — B.(∂B/∂t). Simultaneously, any dot product in the form A.(∂A/∂t) can be rewritten as ½ (∂A²/∂t). Let us take all these values and plug into the expression for power delivered.
This is known as Poynting’s theorem
The work done on the charges by the electromagnetic force is equal to the rate of decrease of electromagnetic energy stored in the fields, minus the energy flow out of the surface. It is the conservation of energy represented in integral form.
One may also consider this as the “work-energy theorem” of electrodynamics. Further, it leads us to the description of an essential physical quantity in the context of electrodynamics, viz. the Poynting Vector.
Energy transported by the electromagnetic field per unit area, per unit time is known as the Poynting Vector.
S ≡ (1/µ₀) (E × B) = (E × H)
Differential Form of Poynting Theorem and the Equation of Continuity
From your earlier lectures, you will remember that the energy stored in an electromagnetic field per unit volume is given by ½ (ε₀ E²+ B²/μ₀). Further, from elementary dynamics, we know that the work done is nothing but the change in mechanical energy of the system. Let us represent the mechanical and electromagnetic energy densities with uₘ and uₑ.
Evidently, the total work done will be the volume integral of uₘ. Thus, we can separate the Poynting Vector on one side, leading us to the differential form of energy conservation. Here’s how it will go.
As you can see, we have arrived at precisely the same form of equation that most conservation laws take. The rate of change of energy is now related to the “transport” of Poynting Vector. This equation represents the conservation of energy just as the conservation of charge was described with the equation of continuity in the beginning of this story.
What About Momentum?
The conservation of momentum is a much lengthier explanation when it comes to electromagnetism. I am going to discuss that in the second part of this article. Until then, here is a little exercise for you:
Imagine two charges of the same sign moving along x and y axes towards the origin. At any instant of time, you can draw the electric and magnetic forces experienced by them. Draw the diagram for this configuration and see if you can figure out what’s wrong with it. (Hint: it has something to do with Newton’s third law of motion).
Conclusion
We have discussed the conservation of energy in the context of electrodynamics and arrived at an expression for the Poynting theorem. Note that in the presence of a linear medium that the field is propagating in, the Poynting vector is represented by E × H, while the energy density is represented by ½ (E.D + B.H). In the second part of this article, I will explain how electromagnetic fields carry momentum and the conservation law for the same.
Update Feb 05, 2022: The second part of this article discussing conservation of momentum in electrodynamics has now been uploaded.
