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Scan10003 - Physics 7C Interference Effects in Diffraction...

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Unformatted text preview: Physics 7C Interference Effects in Diffraction Gratings Light waves from the different slits all start out with the same phase, since they are from the same source. At a given position on the screen, the only phase difference between these waves comes from the difference in the distance they traveled to get there. If you put the screen far away from the grating compared to the distance between two slits, the angle from the normal to any screen position is approximately the same for each slit (see Figure 2). By geometry we then see that the difference in distance traveled for the waves from any two adjacent slits is Ax=dsin9 ' . Eq.1 where dis the distance between slits and 9 is the angle from the normal to the position on the screen. Assuming the grating is placed in air (n=l) this difference in distance causes a phase difference A4) Ax dsinB = _ :2 E .2 [W 271(1) fl( 2 j q where A is the wavelength of the light. If the phase difference at a given position is an even multiple of 1c, the interference there is constructive for all adjacent slits, since equation 2 holds for any two adjacent slits. That means that all the slits constructively interfere with each other, and there is a bright spot on the screen at this position. On the other hand, if the phase difference at some position is an odd multiple of 7:, adjacent slits interfere destructively. Thus since each pair of slits produces zero light there, the screen will be dark at this position.I To summarize, bright spots, called interference maxima, are at screen positions 6m given by d sin 0 A = n: x 2 = 2n ”‘ ¢ ( m) ( A J Eq. 3 => d sin 6," = ml where m is a positive or negative integer or zero. Dark spots, called interference minima, are at A¢ = 7: x (2m +1) = affine“) 1 Eq. 3a => dsinem = (m + EM Notice that the central spot on the screen (9:0) is a bright spot, corresponding to m=0. 1 Even if the grating has an odd number of slits, the one unpaired slit will hardly produce any light on the screen, and we can safely ignore it. DIFFRACTION GRATINGS 3 43 ...
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