A system like that shown in Figure consists of N slits,

Chapter 28, Problem 94GP

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IP A system like that shown in Figure 8-26 consists of \(N\) slits, each transmitting light of intensity \(I_{0}\). The light from each slit has the same phase and the same wavelength. The net intensity \(I\) observed at an angle \(\theta\) due to all \(N\)slits is

                                                    \(I=I_{0}\left[\frac{\sin (N \phi / 2}{\sin (\phi / 2}\right]^{2}\)

                             

In this expression, \(\phi=(2 \pi d / A) \sin \theta\), where \(\lambda\) is the wavelength of the light. (a) Show that the intensity in the limit \(\theta \rightarrow 0 \text { is } I=N^{2} I_{0}\). This is the maximum intensity of the interference pattern. (b) Show that the first points of zero intensity on either side of \(\theta=0\) occur at \(\phi=2 \pi / N\) and \(\phi=-2 \pi / N\) (c). Does the central maximum \(\theta=0\) of this pattern become narrower or broader as the number of slits is increased? Explain.

Equation Transcription:

𝜙

𝜙

𝜙

Text Transcription:

N

I_0

I

theta

N

I=I_0[sin(Nphi/2/sin(phi/2]^2  

phi=(2pid/A) sin theta

lambda

theta rightarrow 0

I=N^2 I_0

theta=0

phi=2pi/N

phi=-2pi/N

theta=0