# Superposition Principle for Electric Fields

If there is a collection of charges $$q_1,q_2,q_3....q_n$$, and we want to calculate the total Electric Field due to all these charges at a point $$P$$ ,then the we sum them all up by the principle of superposition.

$$E_{tot}=E_{1P}+E_{2P}....E_{nP}$$ Where $$E_{nP}$$ is the field due to the $$n^{th}$$ particle at point $$P$$.

I know that this superposition principle has been proved experimentally, but is there a more deeper reason as to why this happens? Why is the field at a given point independent of the fields due to other charges. I mean it could have also been something like: $$E_{tot}=E_{1P}$$ x $$E_{2P}$$ x .... x $$E_{nP}$$. Is there a particular reason for nature to act this way? I am sorry if this is a stupid-silly question.

• It could not have been the product of the individual fields. You need to state a valid reason for that. – ggcg Dec 27 '18 at 17:09
• Also, experimental evidence is all you need. Anything else is fluff. – ggcg Dec 27 '18 at 17:10
• So is it basically some kind of postulate? Cannot it not be derived from something else? – Akaash Srikanth Dec 27 '18 at 17:13
• It is a consequence of linearity of the fundamental equations. Linear equations --> linear superposition of results, e.g. that the effect of multiple causes is the sum of the effects of each. HOWEVER, I refrain from using this as a "proof" since the equations we have are based on observation and some day we may see them violated. So, at least to the degree that Maxwell's equations are valid, yes linearity implies the principle of superposition. – ggcg Dec 27 '18 at 17:16
• One of most famous proposed alternatives is Born-Infeld Electrodynamics. Asymptotically linear, and relativistic invariant. Electron itself might be a soliton solution, and born infeld action can be 'derived' as a brane action in string theory. Biggest flaw is that nobody knows how to quantize such a theory (see diracs small book on quantization) – lalala Dec 27 '18 at 18:02

The effect of each charge appears to be completely independent of the effects of other charges.

So those effects get summed.

The effects of different charges sort of cross right over each other without affecting each other. Like ripples on a pond cross each other and continue, each unaffected by the others.

It could have been that they affected each other, and that would be more complicated -- if you had to take account of the way they affect each other that would require more work than when they don't affect each other at all.

But it didn't turn out that way.

It sounds like you are asking for a complete theory about electric fields that would have this result fall out of it. You could imagine "OK, THIS is what electric fields are like! Now I know.".

You could get a theory like that, but it would have to be compatible with relativity theory, since the numbers come out right from that. People have a hard time envisioning relativity, and if you had an alternative that got the same results, it might likely be hard to envision that too.

It would only be a way to think about it. The math gets the right experimental results within experimental error. Concepts about what's happening with the math are useful for your intuition, and they are likely to suggest interesting applications of the math.

But probably there will be multiple concepts that fit, with no way to show which of them is right. They're just ways to think about it that are compatible with the math.

The particular thing we're seeing here is that the forces that each charge puts on all other charges after a delay, do not change each other. They all act independently. That's what the math describes.

Superposition principle has not been proven experimentally, it’s the other way around: experimental evidence suggested that the interaction between multiple electric charges could be modeled assuming the effect were incremental, so electrostatics has been developed with this feature in mind.

Superposition principle is, indeed, a principle (i.e. a postulate), which is justified only in a specific range of conditions, that is, whenever we can neglect nonlinear effects in the interaction between charges: macroscopically this is practically always verified, and it’s the reason of the success of the theory.

• So does that imply that superposition "might" turn out out be something different from just a simple summation of electric fields, even though currently it is a marvelous approximation for all practical purposes, modelling the real world very well? – Akaash Srikanth Dec 27 '18 at 17:48
• superposition is not supposed to turn out to be anything exoteric; it is a feature of the solutions of all linear differential equations: sum of solutions will still be a solution; when experimentalists revealed that interactions were well explained (up to the available experimental precision) by superposition, theorists started developing a model that incorporated such superposition feature, i.e. Maxwell equations are linear; – Francesco Bernardini Dec 27 '18 at 17:53
• I made a slight error in the comment. Rather than calling it "Superposition" I should have called it "The way electric fields behave". – Akaash Srikanth Dec 27 '18 at 18:11
• yes, it might; I'm not sure if it already has been found, but as always happens in good science, a principle is valid until it's found a condition in which it's not valid anymore; and after that, it still holds valid but gets confined in a validity region; – Francesco Bernardini Dec 27 '18 at 18:50

According to classical electromagnetism, the principle of superposition applies to electric fields because the Maxwell's equations are linear. Those equations hold the status of postulates in the classical treatment. And as far as normal day to day life is concerned, we know the classical equations work very well. Therefore, their implication that the fields can be superimposed is a direct consequence (not an assumption).

There is indeed a deeper meaning to this though. Superposition principle for electric fields implies that the presence of other charges does not affect the field of a charge. And although Quantum Electrodynamics (the more accurate EM theory) does not involve such things as forces and fields it does explain why that may be so. Photon, the mediator of EM force in QED, does not couple with itself.

What does that mean? Let's say you have two charges. In QED, they interact with each other by exchanging photons. Classically, we think of each of them setting up a field and then each charge interacts with the field. Now, you introduce a third charge. According to QED, it will exchange photons with the first two charges but these photons will not affect the photons involved in the interaction between the first two charges. All the interactions add up (linearly). Classically, you say the fields setup by the first two charges have not been affected by the third charge.

There is a slight catch to this. While photons do not couple with each other directly they can interact with each other indirectly via vaccum polarization. An example of such an interaction is Delbrück Scattering. Fortunately for us such effects are considerably weaker when compared to the primary charge-photon coupling and therefore do not change our calculations too much.

Edit: I must clarify that QED does not explain why photons do not couple with each other. This is too an assumption. You may call this kicking the can down further and you won't be wrong. We just don't know why this is so.

• This is an interesting chicken-and-egg issue: are M’s equations linear because we observe that the superposition of fields holds, or does the superposition holds because M’s eqs are linear? You argue the latter but I would say the experimental observation that there is a superposition principle is more fundamental and is the seed of linearity. – ZeroTheHero Dec 27 '18 at 17:57
• @ZeroTheHero That is fair comment. It could very well be that M (in reality, others before him) modelled their equations to explain that fact. But I wanted to point out that the way we now think of classical EM, we start with M's equations and then derive properties of fields. – Apoorv Khurasia Dec 27 '18 at 18:00
• So if the Maxwell's equations have been modeled to make sure that the experimentally observed "Superposition" hold good,then if we attempt to explain the cause Superposition of electric fields, we might need something more fundamental right? – Akaash Srikanth Dec 27 '18 at 18:16
• @AkaashSrikanth Yes. But we don't have it yet. You may call this whole photons do not couple with other photons thing a more fundamental postulate (because it explains more phenomena) but it is yet another postulate that we need to explain through some other theory. People are working on it but for now we don't know why this assumption holds. – Apoorv Khurasia Dec 27 '18 at 18:20
• In a sufficiently small range of variable values, practically any function is effectively linear. Because we usually want to describe things in the simplest possible way, we start with linear equations. It's only when experimental observations over a wider range of values disagree with the linear equations, that we look for higher-order equations to describe things. As it turns out, vacuum electrodynamics has proven to be linear over an enormous range of values. – S. McGrew Dec 27 '18 at 18:24