A diffraction grating contains 15,000 lines/inch. We pass a laser beam through t

Question

A diffraction grating contains 15,000 lines/inch. We pass a laser beam through the grating. The wavelength of the laser is 633 nm. On a screen 2.63 m away, we observe spots of light.

(a) How far (m) from the central maximum (m = 0) is the first-order maximum (m = 1) observed?

(b) How far (m) from the central maximum (m = 0) is the second-order maximum (m = 2) observed? DO NOT use the “small-angle approximation,” ybright= (L/d)m.

The angles are too large for sin tan to be a good approximation. (unit for answers is in meters) Thank you.

Explanation / Answer

d=1inch/15000lines = 1.693*10^-6 , D=2.63m , = 633nm = 6.33*10^-7m

A) Use equation,

dsin = m

m=1

(1.693*10^-6)sin = 1*(6.33*10^-7)

= sin^-1[(6.33*10^-7)/(1.693*10^-6)] = 21.96 deg

sin = y1/D

y1 is vertical distance of first order maximum from central maximum

y1= D*sin = 6.23*sin21.96 = 2.33m

B) A) Use equation,

dsin = m

m=2

(1.693*10^-6)sin = 2*(6.33*10^-7)

= sin^-1[(2*6.33*10^-7)/(1.693*10^-6)] = 48.4 deg

sin = y2/D

y2 is vertical distance of first order maximum from central maximum

y2= D*sin = 6.23*sin48.4 = 4.66m

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