Interference/Difraction wrong consideration

[jaumzaum]

Every difraction/multiple slit article I've read until now consider the intensity of the middle fringe to be N times the intensity of the single light sources (in difraction I've divided the slit into n small light sources, that means that N.I0 = I = the intensity of the light reaching the whole slit).

This statement in completelly wrong! It consider that all light beams emmited by each light source does not have a phase difference! This would be a good aproximation if we consider the slit size d to be aproximatelly equal to the wave lenght. But it would still be a aproximation. What if the slit size were very bigger than the wave lenght? Let's say a λ=1000nm, d=1mm and L (distance of the screen to the slit) = 1m. This would give
I = 0.88 I0
The result would be very smaller if d = 2, 3 or 10mm
And all those numbers are plausible (the example I calculated λ=1000nm, d=1mm and L= 1m is even in Tipler's book)

http://img22.imageshack.us/img22/5834/sadgdfgdg.png


The path difference is
θ=$(\sqrt{L^{2}+x^{2}}-L)*2\pi/\lambda$

So we have to integrate Sin[θ] from {x, -d/2, d/2}. The problem is that this integral have to be aproximated. A small aprox for the amplitude A is A0 times

http://img803.imageshack.us/img803/3166/sdgfghg.png


So why the books insist to say the intensity does not change?

 

[sophiecentaur]

You are right to be concerned about this but it is not always relevant.
For a narrow aperture or array of sources and a long throw (L), that simple assumption is accurate enough to make the approximation. Look up Fraunhoffer and Fresnel Diffraction to see this discussed in high or low detail - depending on what you want. Here is one link which discusses the problem.
In some cases there is a further consideration and that is that the elements in an array of diffracting sources may not all even 'point' in the same direction; they may not even be considered as point (omnidirectional) sources. Arrays of radio antennae often have non-parallel elements but then, one is more likely to be considering the Fraunhoffer region rather than the near field.