# Superposition principle to find the force due to a system of charges

1. ### siddharth

1,197
To find the force due to a system of charges, we can find the force on a charge due to the other charges and add the vectors.

In this context, my school text book says, "Superposition principle should not be regarded as obvious, or equated with the law of addition of vectors. It says two things: force on one charge due to another is unaffected by the presence of other charges, and there are no additional three-body, four-body,etc., forces which arise only when there are more than two charges".

But since the force acting between two particels is given by Coulumbs law (which ignores the effect of other particles), isn't the superposition principle simply addition of vectors?
And what does the last sentence about three-body and four-body forces mean?
I think I am missing something in the definition of superposition principle.

2. ### dextercioby

12,314
The superposition principle could be better called "the principle of effects superposition" and yes,by virtue of Coulomb's law,those effects are added just like any vector...The important thing is that each charge acts on the test one as if it were alone in the universe,equivalent,the other charges were missing...This is very intuitive,once one considers this electric interaction through forces as an example of the III-rd principle of (Newtonian) dynamics,which refers to 2 bodies (which interact) only...

Daniel.

3. ### siddharth

1,197
So is there any example of a situation where the interaction (force) between two particles depends on whether there are other particles present?
Thanks for the help

4. ### dextercioby

12,314
No.U can always consider a 2-body problem.

Daniel.

5. ### vinter

76
I am not very certain, but most probably nuclear forces do not follow the law of superposition, i.e. if necleon1 alone exerts a force f1 on nucleon0 and nucleon2 alone exerts a force f2 on nucleon0 then when kept together, the force on necleon0 is not the vector sum of f1 and f2.

6. ### dextercioby

12,314
What it's the expression for the nuclear force...?$F_{nuclear}=...?$

Daniel.

7. ### Galileo

1,999
Maybe a lousy example. But the superposition principle for electric fields leads to a superposition principle for the electric potential, electric flux etc. (You just add all the contributions).

This is not true for the energy stored in a field, which is proportional to $E^2$. So if you double the charge everywhere, you quadruple the energy, so the field energy does not obey a superposition principle.

BTW: We use vectors because they are convenient in applying Coulomb's law. While Coulomb's law states the electric force between 2 charged particles (and force has magnitude and direction) it seems obvious to use vectors. But a vector is more than just a quantity with magnitude and direction. It also has an algebraic structure (for example. we can add vectors and do so according to a specific rule). This is independent from the fact that Coulomb's law describes a force. So superposition is not necessarily obvious.

8. ### dextercioby

12,314
It is a lousy example.The energy is a scalar...

Daniel.

9. ### Galileo

1,999
So? A vector is a set of three scalars. The potential is a scalar, yet it's fine to call the fact that you can add the potentials of individual charges a superposition principle.

10. ### dextercioby

12,314
Okay,it's quadratic in fields...That's a good reason...

Daniel.

11. ### siddharth

1,197
I do not understand this. Could you please elaborate?

12. ### dextercioby

12,314
Point of application (for fixed/"tied" vectors,for free ones,there isn't any),magnitude/modulus,direction (a straight line) and a sense (on that straight line) are the "geometric description" of a vector.It is,however,not the only one.An algebraic description assumed that a vector is an element of a space,called vector/linear space,the latter,in its definition assumes some axioms...
That's what Galileo was referring to.
A course on linear algebra will clear everything up.

Daniel.

76
14. ### dextercioby

12,314
Okay,let's say Yukawa potential is somhow good.How do you prove your statement,though...?
Reminder:"(...)nuclear forces do not follow the law of superposition, i.e. if necleon1 alone exerts a force f1 on nucleon0 and nucleon2 alone exerts a force f2 on nucleon0 then when kept together, the force on necleon0 is not the vector sum of f1 and f2."

Daniel.