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RE: Superposition (again)



Hi Mark-
In response to:
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As I understand the principle of superposition for electrostatic
forces, which I believe was the original question, all it says is that
there are no "many-body" terms in the force. I.e. if you have a pair of
charges q1 and q2 then

F12 = K*(q1*q2)/(r12)^2*(unit vector along r12)

and for another pair q1 and q3

F13 = K*(q1*q3)/(r13)^2*(unit vector along r13)

now if q1, q2, and q3 are present simultaneously

F1 = F12 + F13 (vector sum)

not F1 = F12 + F13 +F123(r12, r13, theta) (vector sum)

where F123 is a three body force that depends additionally on the angle
theta between the unit vectors r12 and r13. This is a result that can be
determined only from experiment, not from Maxwell's equations (that would
be a circular argument, because one of Maxwell's equations depends on the
nature of the Coulomb force). Note that it has nothing to do with the
fact that forces can be represented by vectors.
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I think that superposition says a bit more than that, although
I think that the last statement is correct (nothing to do with vectors).
Suppose that (F's stand for vectors)
F1 = [1 + |F13|**2]F12 + [1 + |F12|**2]F13 where |F|**2 means
the squared norm of the force. This is easily generalized to n sources,
violates superposition, and does not depend upon the angle between the
sources. It is also violates Gauss's law for multiple charges.
I don't understand your remark about "circular argument". Gauss's
law demands that the field (and, hence, theforce) be proportional to the
source strength and, therefore, implies the superposition principal for
multiple charges.
Rephrasing the argument in terms of fields, I think that
superposition follows from the fact that Maxwell's equations are linear
in the fields. To the extent that Maxwell's equations codify the results
of all experiments in classical E&M, I agree that superposition was
found "experimentally".
Regards,
Jack