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Re: superposition





On Tue, 14 Jan 1997, Leigh Palmer wrote:

Can anyone provide a way that I can justify superposition to
students? Usually we ask students to believe us that adding
electric field vectors from individual charges sum without
accounting for the interaction among the charges. I have usually
stated that this is what we find experimentally. Is there a simple
way to demonstrate its veracity?

The principle of superposition is empirical. It is the result of
any system in which fields (or other quantities) combine additively
like vectors - that's why those quantities are called vector
quantities! Thus the principle of superposition is a matter of
definition.

Leigh


Just so! Does this thread have echoes of the one on the meaning of
momentum? Students asking "why" questions, expecting the teacher to supply
some blinding, insightful, spiritually uplifting revelation which will
make the students say, in chorus, "Oh, that makes it all clear!"

In fact, the only answer one can give for such questions is "That's just
*the way it is*, at least so far as we know at present on the basis of
experiments we've done."

However, students might be reminded that this isn't the first case of
superposition they have seen. The idea goes all the way back at lest to
Galileo. He explained trajectories in this way. Suppose you have a cannon
firing horizontally off a high cliff. If you could turn off gravity, the
ball would move forward in a straight, horizontal line, with constant
speed x = vt and without falling. Now if you turn off the cannon, but turn
on gravity, the ball falls with y = (1/2)at^2 (taking y positive downward).
Now if you fire the cannon with gravity turned on, the position of the
ball at any time is the superposition of the two previous motions, or in
modern vector thinking s = vt + (1/2)at^2 where the displacement s is a
vector, and v and a are vectors also. So the vector displacement s is the
vector sum of the horizontal vector of size vt and the vertical one of
size (1/2)at^2.

Now it's surprising that so many phenomena happen to conform to some kind
of superposition, or our concepts can be formulated so that we can take
advantage of a superposition of them. It's probably the reason vectors
were invented by Gibbs and are now so widely used. But we shouldn't assume
that all physical phenomena obey superposition laws. In particular,
nonlinear phenomena don't usually. The concept of field was invented, and
found to be useful, because it lent itself to superposition. (The force on
C is the vector sum of the force of A on C and B on C.) And if simple
vectors didn't give us a superposition law, we might try something else,
wave functions superposed, for example. We *like* superposition, and will
knock ourselves to find a theoretical framework (with appropriate
concepts) which has it (and which also works).

Exercises for students:

Give as many examples as you know where some kind of superposition works.
State precisely what quantities obey the superposition, and how the
superposition is carried out mathematically.

Give as many examples as you can where it doesn't. And why.

-- Donald

......................................................................
Dr. Donald E. Simanek Office: 717-893-2079
Prof. of Physics Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745 CIS: 73147,2166
Home page: http://www.lhup.edu/~dsimanek FAX: 717-893-2047
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