A diffraction grating with 12,000 lines per inch is used to analyzethe spectrum

Question

A diffraction grating with 12,000 lines per inch is used to analyzethe spectrum of sodium.

(a.) How much width of the grating must be illuminated to justresolve the yellow sodium D-line doublet in second order? (5889.95Aand 5895.92A)

(b.) By what angle with the centers of the D-lines be separated bythe grating when viewed in second order with a spectrometertelescope similar to the ones we use in our lab?

(c.)If the two D-lines from part (a.) are porjected on a screen2.00m from the grating, how far apart in cm will they appear?

Explanation / Answer

(a.)A typical grating contains several thousand lines percentimeter. the given grating has n = 12,000 lines per inch = (12,000/0.0254) lines/meter a garting ruled with "n" lines/cm has a slit spacing d equalto the reciprocal of that number;hence,d = (1/(12,000/0.0254))m or d = (0.0254/12,000) = 2.1 * 10^-6 m = 2.1 m (b.)the condition for maxima in the interference pattern atthe angle is d * sinbright= m * m =0,1,2,................ or sinbright= (m * /d) or bright= sin-1(m */d) here m = 2 and = 5895.92 A = 5895.92 * 10^-10 m (c.)the location of the bright fringes ybright= (L/d) * m m =0,±1,±2,....................... the distance between the fringes is y2 - y1 = (m * L/d) * (2 -1) m = 2,L = 2.00m = 2.00 * 10^2 cm = 200 cm d = 2.1 * 10^-6 m = 2.1 * 10^-6 * 10^2 cm = 2.1 * 10^-4cm 2 = 5895.92 A = 5895.92 * 10^-8 cm and 1 = 5889.95 A = 5889.95 * 10^-8 cm

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