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Diffraction pattern When the single slit is illuminated, a central bright band appears at location P0 on the screen, as shown in Figure 19-10 The first dark band is at position P1 At this location, the path lengths r1 and r2 of the two Huygens wavelets differ by one-half wavelength, thereby producing a dark band by destructive interference This model is mathematically similar to that of double-slit interference A comparison of a single-slit diffraction pattern with a double-slit interference pattern using slits of the same width reveals that all of the bright interference fringes of the doubleslit interference pattern fit within the width of the central bright band of the single-slit diffraction pattern Double-slit interference results from the interference of the single-slit diffraction patterns of the two slits An equation now can be developed for the diffraction pattern produced by a single slit using the same simplifications that were used for double-slit interference, assuming that the distance to the screen is much larger than w The separation distance between the sources of the two interfering waves is now w/2 To find the distance measured on the screen to the first dark band, x1, note that the path length difference is now /2 because at the dark band there is destructive interference As a result x1/L /w You can see from Figure 19-10 that it might be difficult to measure the distance to the first dark band from the center of the central bright band A better method of determining x1 is to measure the width of the central bright band, 2x1 The following equation gives the width of the central bright band from single-slit diffraction



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2 L w The width of the central bright band is equal to the product of twice the wavelength times the distance to the screen, divided by the width of the slit

Canceling the 2 s out of the above equation gives you the distance from the center of the central bright band to where the first dark band occurs The location of additional dark bands can be found where the path lengths differ by 3 /2, 5 /2, and so on This can be expressed as xm m L/w, where m 1, 2, 3, etc, as limited by the small angle simplication When m 1, this equation provides the position of the first-order dark band The second-order dark band occurs at m 2, and so forth





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3 public class Honcho { 4 static boolean b1 = false; 5 static int z = 7; 6 static Long y; 7 public static void main(String[] args) { 8 for(int i = 0; i < 4; i++) 9 go(i); 10 } 11 static void go(int x) { 12 try { 13 if((x == 0) && (!b1 && z == 7)) Systemoutprint("0 "); 14 if(x < 2 ^ x < 10) Systemoutprint("1 "); 15 if((x == 2) && (y == null | (ylongValue() == 0))) Systemoutprint("2 "); 16 if(z <= (x + 4)) Systemoutprint("3 "); 17 } 18 catch(Exception e) { Systemoutprint("e "); } 19 } }

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16 Monochromatic green light of wavelength 546 nm falls on a single slit with a width of 0095 mm The slit is located 75 cm from a screen How wide will the central bright band be 17 Yellow light with a wavelength of 589 nm passes through a slit of width 0110 mm and makes a pattern on a screen If the width of the central bright band is 260 10 2 m, how far is it from the slits to the screen 18 Light from a He-Ne laser ( 6328 nm) falls on a slit of unknown width A pattern is formed on a screen 115 m away, on which the central bright band is 15 mm wide How wide is the slit 19 Yellow light falls on a single slit 00295 mm wide On a screen that is 600 cm away, the central bright band is 240 mm wide What is the wavelength of the light 20 White light falls on a single slit that is 0050 mm wide A screen is placed 100 m away A student first puts a blue-violet filter ( 441 nm) over the slit, then a red filter ( 622 nm) The student measures the width of the central bright band a Which filter produced the wider band b Calculate the width of the central bright band for each of the two filters

382 Microwave waveguides and antennas Example 19-3 A rectangular waveguide has dimensions of 3 late the TE10 mode cutoff frequency Solution: Fc c 2a (3 (2) 5 cm 3 108 m/s (2)(005 m) 108 m/s 1m 100 cm 3 GHz 5 cm Calcu-

Single-slit diffraction patterns make the wave nature of light noticeable when the slits are 10 to 100 times the wavelength of the light Larger openings, however, cast sharp shadows, as Isaac Newton first observed While the single-slit pattern depends on the wavelength of light, it is only when a large number of slits are put together that diffraction provides a useful tool for measuring wavelength

You have several unknown substances and wish to use a single-slit diffraction apparatus to determine what each one is You decide to place a sample of an unknown substance in the region between the slit and the screen and use the data that you obtain to determine the identity of each substance by calculating its index of refraction 1 Come up with a general formula for the index of refraction

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