Why is the Principle of Superposition true in EM? Does it hold more generally?

Question

• In the theory of electromagnetism (EM), why is the principle of superposition true? Can we read it off from Maxwell's equations directly?

• Does it have any limit of applicability or is it a fundamental property of nature?

Answer 1

The principle of superposition comes from the fact that equations you solve most of the time are made of Linear operators (just like the derivative). So as long as you are using these operators you can write something like

$$ \mathcal L\cdot \psi = 0$$

where $\mathcal L$ is a linear operator and, let say, $\psi$ is a function that depends on coordinates that $\mathcal L$ is acting on. The superposition principle is the same that saying this

$$\mathcal L \cdot \left(\sum_i \psi_i \right) = \mathcal L\cdot\psi_1 + \mathcal L\cdot\psi_2 + ... = 0$$

holds. An example when it doesn't would be, for example...

$$\mathcal L^2 \cdot\left(\sum_i \psi_i\right) \neq \mathcal L^2\cdot\psi_1 + \mathcal L^2\cdot\psi_2 + ...$$

in general (for the Laplacian in Euclidean space it is equal to). So then, the question is if Maxwell equations are linear. And they are because they are made up with this kind of operators. For instance, Gauss law for two different electric fields can be written as one

$$ \nabla \cdot \vec{E}_1 = \rho_1/\varepsilon \quad; \quad \nabla \cdot \vec{E}_2 = \rho_2/\varepsilon $$

$$ \nabla \cdot \overbrace{\left(\vec{E}_1 + \vec{E}_2\right)}^{\vec{E}} = \frac{\overbrace{\rho_1 + \rho_2}^{\rho_T}}{\varepsilon} \Rightarrow \boxed{\nabla \cdot\vec E = \frac{\rho_T}{\varepsilon}} $$

just because $\nabla$ is a linear operator.

Answer 2

Within the realm of Maxwell's equations, the principle of superposition is exactly true because Maxwell's equations are linear in both the sources and the fields. So if you have two solutions to Maxwell's equations for two different sets of sources then the sum of those two solutions will be a solution to the case where you add together the two sets of sources.

This will only break down when Maxwell's equations break down, for example, when the field strengths are so high that pair production becomes significant. In that case the quantum field theory of Quantum Electrodynamics (QED) would be required. Now, quantum theories are also linear, at least as far as the quantum wave function is concerned, however the probabilities that quantum theories predict depend on the magnitude of the wave function, so in that sense they are nonlinear, and therefore superposition will not apply to the results.

Answer 3

It is true up to very high filed strengths. For too high strengths the field itself is not stable, it can create real pairs. It is like a limit on a field strength in a capacitor. The capacitor dielectric can break.

EDIT: Classical Maxwell equations are linear indeed so the principle of superposition is implemented into them. But break of a dielectric can be introduced too as a resistance depending on the field strength. Thus one can make the Maxwell equation non-linear starting from some threshold strength.

In fact, the dielectric break or capacitor discharge due to cold electron emission (classical non linearity) occurs "well before" creating electron-positron pair in vacuum.

Answer 4

While the first part of the question has been answered satisfactorily, everybody seems to confuse the unconditional linearity of the Maxwell equations with the often observed linearity of the constitutive relations for the material law. The field of nonlinear optics is concerned with the behavior of light in nonlinear media where the constitutive relations are no longer linear.

However, the superposition principle is already violated if even a single electron gets accelerated by the field. So nonlinear media are nothing exotic, even if most media are well described by linear constitutive relations for small field strengths.

Answer 5

Superposition principle is a trouble maker. The problem also comes from the definition of the field. In the beginning the electric field is defined with a test charge. If the test charge exist, if there are two fields we can added them up. We said this is the superposition principle. Up to this point every thing is OK.

But the problem is we later extended the the concept. We claim that even the test charge is not here the field still exist and same as if the the test charge exist. Who knows if the test charge is removed the electric field is there or not? There are two kind theories they debated 100 years for this problem. One is the Faraday and Maxwell, they claim the field is still there even the test charge is removed, field is real substance. Another side is the theory of action-at-a-distance, Which are Weber electromagnetic theory, Schwarzschild-Tetrode-Fokker action principle, absorber theory of Wheeler and Feynman, these theory claim that field is not real, there is only the action between at least two charges!

The later has no the problem of Maxwell equation and superposition. But the theory of action at a distance is to complicate compare the field theory of Maxwell equation. Action-at-a-distance cannot win the war with Maxwell's theory. However they are correct at least for the superposition, only if test charge is there, you can superpose the two fields. Without test charge, the field is not defined how you can superpose two fields?

Feynman notice this problem, but he has not found a way to correct Maxwell theory. Instead, he decided just give up the Maxwell theory. He created QED. In QED the problem is partially solved by normalization, second second quantization. But the problem is still not thoroughly solved by him.

This difficulty now is overcome by the "mutual energy principle" and "self-energy principle". Mutual energy flow is produced by the retarded wave and the advanced wave. the mutual energy principle tell us that the photon energy is transferred only by the mutual energy flow. The self-energy principle tell us the self-energy flow do not transfer any energy.

Interesting is that the new theory belongs to Maxwell's field theory. The mutual energy principle and self-energy principle claim the field is still a real substance. The new theory has absorbed all advantage of the action-at-a-distance and the absorber theory and then updated the Maxwell's field theory.

Any way in the new theory of the mutual energy principle and self-energy principle the supposition principle is avoided. Maxwell equation need superposition principle, without superposition even Maxwell equation is correct for single charge, it still cannot prove it is correct for N charges. The mutual energy principle do not need the superposition principle. Any particles for example photon and electron are consist of 4 waves and 6 energy flows instead of one waves one energy flow.

Please see my publication for the details: http://www.openscienceonline.com/journal/archive2?journalId=726&paperId=4042