(c) In this problem, the angle is small (less than a few degrees). When the angl

Question

(c) In this problem, the angle is small (less than a few degrees). When the angle is small,tan is approximately equal tosin, or tan ˜sin. With this approximation in mind, what isthe approximate algebraic expression for the distance yfrom the central bright fringe to themth-order bright fringe? Expressyour answer in terms of the distance L between the gratingand the screen, the order m of the bright fringe, thewavelength of the light, and the separationd between the slits of the grating.

Problem Light of wavelength 440 nm(in vacuum) is incident on a diffraction grating that has a slitseparation of 4.2 × 10-5 m. Thedistance between the grating and the viewing screen is 0.12m.

(b1) If the entire apparatus is submerged inwater (nwater = 1.33), what is thedistance y from the central bright fringe to thesecond-order bright fringe?

y ˜

Explanation / Answer

1)The waves from all slits are in phase as they leave theslits.For an arbitrary direction measured from thehorizontal,however,the waves must travel different path lengthsbefore reaching the screen.The path difference between therays from any two adjacent slits is equal to d * sin.If thispath difference equals one wavelength or some integral multiple ofa wavelength,waves from all slits are in phase at the screenand a brigyt fringe is observed.Therefore,the condition formaxima in the interference pattern at the anglebright is d * sinbright= m * m = 0,±1,±2,±3,..................... We can use this expression to calculate the wavelength if weknow the grating spacing d and the anglebright. When the angles to the fringes are small,the positions of thefringes are linear near the center of the pattern.This can beverified by noting that for small angles,tan ˜sin,the positions of the bright fringes as ybright= L * sinbright. ybright= L * (m * /d) (smallangles) This result shows that ybright is linear in theorder number m,so the fringes are equally spaced. 2)the distance y from the central bright fringe tothe second-order bright fringe y = (L/d) = 440 nm = 440 * 10^-9 m L = 0.12 m d = 4.2 × 10-5 m

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