On the logical-ness of the principle of superposition

[RubinLicht]

I am aware that there have been a few posts about this, I read most of them so I just want to confirm that what I've picked up is correct.

The principle of superposition only applies to forces that depend linearly on some quantity (charge and mass for example), and since we wouldn't know whether or not the force is linear until we do experiments with three charges, there is no way of knowing until that experiment confirms the linearity of the (in this case) electric force with respect to q.

I haven't really learned much about linearity formally in school, and thanks to how weak American math education is (11th grade now), I had to pick all this up on my own. If the terminology I used or explanation was not satisfactory, please point it out. Thanks friends.

 

[dextercioby]

I think what you call „superposition principle” could be seen as the mathematical rephrasing to the simple fact that forces are genuine vectors, hence members of a mathematical construct called vector space. The sum of two vectors in this space is then defined to be the total force acting on a massive material point. One then has a different, more important principle which states that forces upon a massive material point act independently, more precisely, let A, B, C be 3 bodies of masses mA, mB and mC. Then the total force exerted by B on A is independent of the presence of C, thing which mathematically reads into the fact that the overall potential energy of this 3-particle system is a sum of 3 functions each depending on 2 variables only.

So if the Coulomb force were F(q1,q2) = K q_1 ^2 q_2 ^2 / r^4 r, then this would still be subject to the „superposition principle”, even if the dependence on electric charges would no longer be linear.

 

[mfig]

Superposition is not a feature of particular forces, is it a feature of the differential equations used to describe whatever is under consideration. If the DEs are linear, solutions will exhibit superposition. If the DEs are nonlinear, solutions will not.

 

[H Tomasz Grzybowski]

Think about focusing a coherent EM wave, Let r be the distance from the focus.
According to the Law of Energy Conservation the electric field E should be proportional to 1/r,
but according to the Linear Superposition Principle E should be proportional to 1/r^2.
Does anybody have experimental results (for coherent EM waves)?

 

[Stephen Tashi]

I am aware that there have been a few posts about this, I read most of them so I just want to confirm that what I've picked up is correct.

What you said was unclear. It isn't clear what you mean by "the principle of superposition".

 

[H Tomasz Grzybowski]

The Principle of Linear Superposition for EM fields says that E vectors and H vectors add whenever fields from several point reach a given point.

 

[sweet springs]

The principle of superposition only applies to forces that depend linearly on some quantity (charge and mass for example), and since we wouldn't know whether or not the force is linear until we do experiments with three charges, there is no way of knowing until that experiment confirms the linearity of the (in this case) electric force with respect to q.

Say three particles are charge q at the origin, q1 at r1 and q2 at r2.

In case there is q1 without q2, force acting on q is F=F(q1,r1)
In case there is q2 without q1, force acting on q is F=F(q2,r2)
In case there are both q1 and q2, what force is actiong on q ?
Linearity means that simple addition holds, F=F(q1,r1)+F(q2,r2).
If q3 at r3 is added, F=F(q1,r1)+F(q2,r2)+F(q3,r3) and so on.

 

[RubinLicht]

Wow necroing a two eyar post.

I was just asking why it is not a "logical" next step to assume that the coulomb or gtavitational forces add linearly just from an experiment with two particles.

Ie, if you bring in one charge from infinity, you find some sort of 1/r^2 dependenced but you know nothing ono how the force depends on q, which you can only find after bringing in a second charge. However, if the charge dependence were q^2*q^2, then the force vectors would not add linearly.
Though this seems ridiculous at first, apparently it's not reasonable to rule it out before doing experiments.

Perhaps the was a question I got due to a short paragraph on gravity in kleppners book