Question

2. You have a diffraction grating with 2500 lines/cm. You also have a light source that emits light at 2 different wavelengths, 540 nm and 690 nm, at the same time. The screen for your experiment is 1.2 meters from the diffraction grating.

A. What is the line spacing for the grating?

B. What is the difference in the angle of the 2nd bright fringe for each wavelength for this grating?

C. Which wavelength is closer to the center of the diffraction pattern?

D. How would the color separation be different if you used a 1200 lines/cm grating? Explain.

E. What is the width of the central maximum for each wavelength if the same light source is used to illuminate a single slit with a width of 0.045 mm?

Answer 1

(A) The line spacing for the grating is, 2500 lines/cm = 4x104 cm = 4x106m (B) The diffraction angle for 2nd bright fringe for wavelength 540 nm is, d sin = m2 sin Q = m2 Q = sinh moi =sin-12 (540x10 m) Sin 4x10 ºm = sin (0.27) = 15.66° The diffraction angle for 2nd bright fringe for wavelength 690 nm is, 6, = sin(") (2(690x10 m) = sin 4x10 m = sin(0.345) = 20.180

The difference in the angle of the 2nd bright fringe is, A0=0,- = 20.18° -15.66° = 4.529 (C) From the equation for diffraction angle =sin-1 m ), the diffraction angle in proportional to wavelength. Thus, the diffraction angle is smaller for the smaller wavelength. Thus, the wavelength 540 nm is closer to the center of the diffraction pattern. (D) The line spacing for the grating with 1200 lines/cm is, 1200 lines/cm = 8.33x10 cm =8.33x10-6 m The line spacing for the grating with 1200 lines/cm is greater than the line spacing for the grating with 2500 lines/cm. From the equation for the diffraction angle =sin the diffraction angle is smaller for the grating with 1200 lines/cm since the line spacing is greater for this grating. Thus, for the grating with 1200 lines/cm, the fringes will be closer to the central diffraction pattern. Therefore, the color separation is not clear with the grating with 1200 lines/cm since the diffraction fringes (color fringes) are closer to the central diffraction pattern due to large line spacing.

(E) The width of central maxima for the wavelength 540 nm is, 22L w= 2(540x109 m)(1.2m) 0.045x10-m = 0.0288m The width of the central maxima for the wavelength 690 nm is, W; - 201 а 2(690x109 m)(1.2 m) 0.045x10 m = 0.0368m