Interference for for N slits formula

In summary, the formula for intensity in interference for a lattice with N slits is found by taking the ratio of the sine functions of the angle of incidence and the number of slits. When the angle approaches a multiple of pi, both the numerator and denominator approach zero, resulting in the maximum intensity at that angle. This is because when the numerator and denominator approach zero, the limit of the ratio is equal to the number of slits, N.
  • #1
Nikitin
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Interference for a lattice with N slits

OK, so check out the first formula (the one for intensity) in the following pdf:

http://folk.ntnu.no/magnud/OvLf/bolge/oving11.pdf

It's the formula for Intensity as a function of angle from the interference for a lattice with N very thin slits.

Mathematically, why is ##I_{max}## found only when BOTH the numerator and denominator are approaching zero?

I agree that the intensity is largest when the ratio between the numerator and denominator is biggest, but why does this only happen when both approach zero?
 
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  • #2
Replaces previous post, which was misleading.

Consider when [itex]\theta[/itex] is close to [itex]n\pi[/itex]. Put [itex]\theta = n\pi + \epsilon[/itex].

Then [itex]\frac{sin N\theta}{sin \theta} = \frac{sin (Nn\pi +N\epsilon)}{sin (n\pi + \epsilon)} = \frac{cos (Nn\pi) sin(N\epsilon)}{cos (n\pi) sin\epsilon} = ± \frac{sin N\epsilon}{sin \epsilon}[/itex]

The limit of this as [itex]\epsilon[/itex] approaches zero is simply [itex]±\frac{N\epsilon}{\epsilon} = ±N[/itex].
 
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  • #3
Oh I just figured it out. When the upper term approaches zero, then so must the lower turn because N is just a whole number.

thanks
 

What is the interference for N slits formula?

The interference for N slits formula is a mathematical equation used to calculate the interference pattern produced by light passing through N parallel slits. It takes into account factors such as the distance between the slits, the wavelength of the light, and the angle of observation.

How is the interference for N slits formula derived?

The interference for N slits formula is derived from the principles of wave interference and Huygens-Fresnel diffraction. It involves summing the contributions of waves from all the slits to determine the overall interference pattern.

What is the significance of the number of slits in the interference for N slits formula?

The number of slits in the formula represents the number of parallel paths that the light can take. A larger number of slits results in a more complex interference pattern with more bright and dark fringes.

Can the interference for N slits formula be applied to other waves besides light?

Yes, the interference for N slits formula can be applied to any type of wave, as long as the waves are coherent and follow the principles of wave interference. This includes sound waves, water waves, and electromagnetic waves.

Are there any limitations to the interference for N slits formula?

The interference for N slits formula is based on certain assumptions and ideal conditions, such as a perfect single wavelength of light and parallel slits. In real-life situations, these conditions may not be met, leading to deviations from the predicted interference pattern.

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