Apply the equation ym=Rm/a(m=±1,±2,±3,…) to a simple single-slit diffraction pro

Question

Apply the equation ym=Rm/a(m=±1,±2,±3,…) to a simple single-slit diffraction problem. Suppose that you illuminate the back side of a narrow slit with 565 nm laser light and observe the diffraction pattern on a screen R=5.4m away. You find that the distance between the centers of the first minima (dark fringes) on either side of the central bright fringe in the pattern is 2y1=31mm. How wide is the slit? SOLUTION SET UP The angle in this situation is very small, so we can use the approximate relationship given in the introduction. (Figure 1) shows the distances. The distance y1 from the central maximum to the first minimum on either side is half the distance between the two first minima, so y1=(31mm)/2. SOLVE Solving the equation for the slit width a and substituting m=1, we find that a==Ry1=(5.4m)(5.65×107m)(3.1×102m)/22.0×104m=0.20mm REFLECT It can also be shown that the distance between the second minima on the two sides is 2(31mm), and so on. Part A - Practice Problem: What is the distance between the fifth minima on the two sides of the central bright fringe?

Explanation / Answer

Yn = mR(lambda)/a
now Y5 = 5*5.4*565*10^-9 / 0.2*10^-3 = 76.275 mm

The distance between two 5th order dark fringes = 2*Y5 = 152.55 mm

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