# Maxwell's equations from continuum limit

In appendix A.6 of Schroeder's Thermal Physics, he mentions (in regards to classical fields):

The usual approach is to first pretend that the continuous object is really a bunch of point particles connected together by little springs, and eventually take the limit where the number of particles goes to infinity and the space between them goes to zero. The result is generally some kind of partial differential equation (for instance, the linear wave equation or Maxwell's equations) that governs the motion as a function of place and time.

I take the part in parentheses to mean that there's some way to derive Maxwell's equations from the continuum limit of some sort of system of springs. I'm familiar with deriving the wave equation from a system of springs, but have never heard of obtaining Maxwell's equations and can't seem to find anything that mention it other than this somewhat vague reference. I'm not even sure what the "springs" would be for this case unless we're talking about oscillations of the electromagnetic field. in which case the derivation would seem kind of circular to me. Is this some remnant of aether theory or something? An explanation of the procedure would be appreciated.

• I am not completely sure but here it goes: While not exactly springs, I think the author might be alluding to the derivation of macroscopic Maxwell equations from microscopic ones, by, for instance, considering a spacial averaging over bounded and free charges/currents, which will, at first approximation, include vibrational effects of atoms/molecules around their center of mass. Take a look of section 6.6 in Jackson's book, Classical Electrodynamics, 3ed, and the references suggested in there. – secavara Mar 9 '18 at 1:37
• Apparently, this is how Maxwell derived his equations. – flippiefanus Mar 9 '18 at 4:22
• It is interesting to note that quatum mechanical averaging is quite different (the opposite) from the "classical" one. – Vladimir Kalitvianski Sep 30 '19 at 15:15

I think you're interpreting him a little too literally. His meaning can be taken more literally if you use his first example, a wave equation for sound waves in a solid. I don't think he literally means that you can get Maxwell's equations by coupling together massive particles with springs -- when you do that, you get sound waves, not light waves. There are fundamental reasons why you can't hope to get a description of the electromagnetic field by literally using this method:

(1) The speed at which wave disturbances propagate will be less than $$c$$. (This holds if the matter that the springs and masses are made of are any reasonable form of matter -- specifically, one that obeys the dominant energy condition.)

(2) You're going to have a hard time getting rid of the longitudinal vibrations and getting the transverse ones. The prohibition on longitudinal vibrations really only makes sense relativistically, for a wave propagating at $$c$$.

(3) You will essentially be constructing an aether theory, which will not be consistent with Maxwell's equations (or at least, not with how we interpret them nowadays).

The discrete degrees of freedom that he has in mind for the EM field are probably more like the normal modes of a cavity, or something of a similar nature in QED.

The derivation of wave equation that you alluded in your question can be adapted to derive Maxwell equations. Perhaps, the best exposition of this technique is given by J.D. Jackson in section 12.7 of his 'Classical Electrodynamics' (3rd edition). It is so lucid that I reproduce it as is below.

The Lagrangian approach to continuous fields closely parallels the techniques used for discrete point particles. The finite number of coordinates $$q_i(t)$$ and $$\dot{q}_i(t)$$, $$i = 1, 2, \ldots n$$, are replaced by an infinite number of degrees of freedom. Each point in space time $$x^\alpha$$ corresponds to a finite number of values of the discrete index $$i$$. The generalized coordinate $$q_i$$ is replaced by a continuous field $$\phi_k(x)$$, with a discrete index ($$k = 1, 2, \ldots, n$$) and a continuous index $$x^\alpha$$. The generalized velocity $$\dot{q}_i$$ is replaced by the $$4$$-vector gradient $$\partial^\beta\phi_k$$. The Euler-Lagrange equations follow from the stationary property of the action integral with respect to variations $$\delta\phi_k$$ and $$\delta(\partial^\beta\phi_k)$$ around the physical values. We thus have the following correspondences: $$i \rightarrow x^\alpha, k$$, $$q_i \rightarrow \phi_k(x)$$, $$\dot{q}_i \rightarrow \partial^\alpha\phi_k(x)$$,
$$$$L = \sum_i L_i(q_i, \dot{q}_i) \rightarrow \int\mathcal{L}(\phi_k, \partial^\alpha\phi_k)d^x$$$$ and $$$$\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_i}\right) = \frac{\partial L}{\partial q_i} \rightarrow \partial^\beta\frac{\partial\mathcal{L}}{\partial(\partial^\beta\phi_k)} = \frac{\partial\mathcal{L}}{\partial\phi_k},$$$$ where $$\mathcal{L}$$ is a Lagrangian density, corresponding to a definite point in space-time and equivalent to the individual terms in a discrete particle Lagrangian. For the electromagnetic field, the "coordinates" and "velocities" are $$A^\alpha$$ and $$\partial^\beta A^\alpha$$.

I think this paragraph beautifully summarizes how one can go from masses-with-springs to field-equations and how to use the latter to get a Lagrangian description of the electromagnetic fields.

Jackson further writes

In analogy with the situation with discrete particles, we expect the free-field Lagrangian at least to be quadratic in the velocities, that is, $$\partial^\beta A^\alpha$$ or $$F^{\alpha\beta}$$, the electromagnetic field tensor.

Lastly, he also argues for a term $$J_\alpha A^\alpha$$ to account for the interaction of charged particles with electromagnetic fields. You may want to go over the rest of the section for his derivation of the inhomogeneous Maxwell equations. (The homogeneous ones follow from the definition of $$F^{\alpha\beta}$$.)

Before closing let me add that I could not find a mass-with-springs analogy in volume 1 of Maxwell's treatise on electricity and magnetism. In fact, his derivation (article 76) of Gauss law is similar to what we find in modern textbooks, except that he calls $$R$$ the magnitude of the electric field.

It's actually not springs but rather writing a Lagrangian and then deriving it. If I understand properly it just refers to the ability to write a Lagrangian density $$\mathcal{L} = \mathcal{L}_{\mathrm{field}} + \mathcal{L}_{\mathrm{int}} = - \frac{1}{4 \mu_0} F^{\alpha \beta} F_{\alpha \beta} - A_{\alpha} J^{\alpha}$$ for classical electrodynamics. Euler-Lagrange equations for the above equations will give the Maxwell's equations. The fact that one can consider the components of the electromagnetic potential $$A^{\lambda}$$ as field quantities which are a function of space and time. Just like in the case of a system of springs as one approaches the "limit where the number of particles goes to infinity and the space between them goes to zero". Under such an approximation one can replace the displacement of the individual particles $$\delta _i$$ by a function of space$$\delta (x)$$.