Question

Monochromatic light passing through a single slit forms a diffraction pattern on a screen. If the second minimum occurs at an angle of $15^\circ$, find the angle for the fourth minimum.

$31.2^\circ$

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This is College Physics Answers with Shaun Dychko. Light with a single wavelength is passing through a single slit and it forms a diffraction pattern, the second minimum occurs at an angle of 15 degrees and our question is to find the fourth minimum so this is the formula for finding the minima or the destructive positions in the diffraction pattern and that equals the width of the slit—

*D*— times*sin*of that angle to that minimum and that equals the order times the wavelength where this order can be 1, negative 1, 2 and negative 2 and so on. So in the first case, we have*D*times*sin Θ 2*equals 2 times*λ*because*m*is 2 and*Θ 2*we know and in the second case where you have the fourth order minimum that we're looking for— we have the number 4 replaced in for the letter*m*— and it's the same slit so*D*here does not need a subscript and then we have*sin*of the angle to the fourth minimum and we can divide these two equations and the common factors are going to be canceling so we have*Dsin Θ 4*over*Dsin Θ 2*has the*D*'s cancel and then we have 4*λ*divided by 2*λ*and the*λ*'s cancel and 4 divided by 2 is 2 and we can multiply both sides by*sin Θ 2*and then we get*sin Θ 4*then is 2 times*sin Θ 2*. Then we take the inverse*sin*of both sides and we get that the angle to the fourth minimum is inverse*sin*of 2 times*sin*of the angle to the second minimum. So it's the inverse*sin*of 2 times*sin*of 15 and that's 31.2 degrees.