Experimental work - formulas as the outcome

Every time I run across of (complex to me at least) formulas like those to find inductance for an air-core coil or the noise in a resistor, where so many factors are involved, I wonder what was the process followed by the originator in constructing such an expression.

When there are linear relationship into play I could understand that someone with good insight and experience in the matter could express that, but how when there are roots and powers and fractions with (incredible) values...

And then they came the constants...

a) Thinking of those that came out from experimental work, is it a well outlined process to implement such complex expressions?

b) And an even and more insidious one: how in heaven the proved they were right?

This book answers your questions, and has many references:

https://books.google.com/books/about/Inductance_calculations.html?id=K3KHi9lIltsC

https://www.amazon.com/Inductance-C...TF8&qid=1363061289&sr=1-8&keywords=f+w+grover

This download is a collection of work done by the U.S. Bureau of Standards a hundred years ago:

https://www.electro-tech-online.com/custompdfs/2013/03/Sci169noerr.pdf

atferrari said:

Every time I run across of (complex to me at least) formulas like those to find inductance for an air-core coil or the noise in a resistor, where so many factors are involved, I wonder what was the process followed by the originator in constructing such an expression.

When there are linear relationship into play I could understand that someone with good insight and experience in the matter could express that, but how when there are roots and powers and fractions with (incredible) values...

And then they came the constants...

a) Thinking of those that came out from experimental work, is it a well outlined process to implement such complex expressions?

b) And an even and more insidious one: how in heaven the proved they were right?

Click to expand...

Hello there,

A lot of this stuff comes from experiments and reason. Einstein used experimental evidence combined with pure reason to go further in physics than anyone before him.

Pure reason is a large part of it. If you can reason something out, you can then attempt to prove it. You should have the ground work before you start though so the route you take is at least reasonable in and of itself, so it helps immensely to first study what has been done before this in that area. Knowing that, take the next logical step.

I think a good indicator of just how difficult and counter intuitive it is to extract laws of nature is how long it took humans to figure out how to do it. With the exception of a few very amazing individuals (many from the ancient Greeks, like Archimedes) mathematical models used in the style of theoretical physicists have only existed for about 400 years. Compare that with modern humans evolving over 40,000 years ago and advanced human civilizations existing for 10,000 years.

Perhaps Galileo is the one that really started the ball rolling (or, should we say "the ball dropping"), and Kepler is arguably the first true theoretical physicist of the modern type.

I think the reason it took so long is that many physical processes look complex because many different laws are operating simultaneously and it is hard to see the underlying order. It is human nature to intuitively develop mental models hueristically, but these are usually black-box models that work, but don't reveal the underlying natural laws. It takes the scientific method and isolation of the processes to see the relationships and cast those relationships into simple mathematical equations.

For example, two heavenly bodies will orbit each other in a simple way, and there is some hope to discover the underlying inverse square law of gravity if you observe two bodies in isolation. Newton, did this by noting Kepler's laws of orbits, and realizing that an inverse square law would be consistant with Kepler's laws. Hence, a simple isolated system helped reveal the correct equation.

But, what if we lived in a solar system where many bodies were close together and interacting in complex ways. Or, what if our atmosphere were too opaque to even observe stars and planets at night. Then, discovery of the simple formula for gravity might not ever happen because the special experiment that isolates this particular Law would not be available.

Now that we know how to apply the scientific method and now that we know how important it is to devise experiments that isolalate laws so that we can see them clearly, we progress much faster.

Hello again,


There is also another interesting method for solving some problems which is known as dimensional analysis mainly because we use the dimensions of the problem to solve for some of the variables. This example will illustrate this approach nicely i think.

EXAMPLE:

The speed of sound in a gas might depend on the density, pressure, and volume of gas.
We want to find the exponents in the formula:
v=K*P^x*D^y*V^z

where
v is the velocity,
P is the pressure,
D is the density,
V is the volume,
K is a constant with no dimensions.

How would we go about finding the exponents x, y, and z (and note this is grossly nonlinear)?

Knowing the dimensions of all the quantities v,P,D,and V:
v is length over time: d/t
P can be expressed as mass over length per second squared: m/(d*t^2)
D is mass over length cubed: m/d^3
V is length cubed: d^3

so we substitute the dimensions into the formula above and we get:
d/t=K*d^(3*z)*(m/d^3)^y*(m/(d*t^2))^x

K has no dimension so we drop that and end up with:
d/t=d^(3*z)*(m/d^3)^y*(m/(d*t^2))^x

and we note that we have two m's in the numerators on the right side, and no m's on the left, so
we have to "get rid of" the m's somehow. One way to do this is to make y=-x because that way
the exponents will cancel the two m's, so we have:
d/t=d^(3*z)*(m/d^3)^(-x)*(m/(d*t^2))^x

so we got rid of one variable and now we can cancel the m's:
d/t=d^(3*z)*(1/d^3)^(-x)*(1/(d*t^2))^x

We can pull out the t^2 as t^(2*x) in the denominator:
d/t=(d^(3*z)*(1/d^3)^(-x)*(1/d)^x)/(t^(2*x))

Now we note on the left we have t in the denominator and on the right we have t^2x in the denominator,
so we can equate these two alone:
t=t^(2*x)

and a solution here is:
x=1/2

and since y=-x we also get:
y=-1/2

Now we plug those two (or just x) into the equation and we get:
d/t=d^(3*z+1)/t

multiply both sides by t:
d=d^(3*z+1)

solve this for z we get:
z=0

So the formula works out to:
v=K*sqrt(P/D)

We found this knowing the dimensions of the problem and knowing a formula that might apply.


Another approach to some problems is curve fitting. We plot a bunch of experimental data and then try to find a relationship between all the points. This is sometimes easy to do.

Still yet another approach is what is called evolutionary curve fitting or adaptation. This is where we start with a bunch of experimental data and try to find a formula based on how evolution works. This is like that of animals except here we allow the formula itself to evolve.