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

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 3

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 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.