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

• 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?

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## 4 Answers

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.

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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.

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No explanation of WHY or how it can be seen in Maxwell's equations. – FrankH Oct 27 '11 at 17:00

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It is explained in each and every textbook. – Vladimir Kalitvianski Oct 27 '11 at 17:07

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.

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-1 You are repeating Vladimir's answer, and further, you are muddling things--- the field equations are nonlinear in the usual sense in strong field electromagnetism, because of pair creation. – Ron Maimon Oct 27 '11 at 16:46

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@ron, I am not repeating Vladamir's answer, the question was WHY, all Vladimir said was it is true without any explanation as to Why. Furthermore, the question was "Can we read it off of Maxwell's equations?" which Vladimir also did not address. – FrankH Oct 27 '11 at 16:59
Static field strength has nothing to do with the wave function. It is an external field. – Vladimir Kalitvianski Oct 27 '11 at 17:09
The question on why is layish, trying to answer it even more. This principle is plainly a observation of experiments. Maxwells equations came later, and if Maxwell tried some nonlinear ansatz, he would have realized soon that this is wrong. "Truth" is not a useful expression in physics. – Georg Oct 28 '11 at 11:02
You are right. "Truth" is not useful expression in physics. On the other hand, I was not asking why the p. of superposition is true in an absolute sense... just why physicists believe it is valid, in which situation we can use it, whether it is related to some more general principle (are other fundamental forces also linear?). You seem to suggest that there were mainly empirical reasons to accept it and then Maxwell embodied the principle in his theory. This is fine and answers the question. To me, it seems that the my question is perfectly admissible. – quark1245 Oct 30 '11 at 21:28

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.

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