PLEASE HELP ME!!! WILL RATE LIFESAVER!!!! A diffraction grating has 2593 lines p

Question

PLEASE HELP ME!!! WILL RATE LIFESAVER!!!!

A diffraction grating has 2593 lines per centimeter, and it produces a principal maximum at = 28.2°. The grating is used with light that contains all wavelengths between 410 and 660 nm. What are the two wavelengths of the incident light that could have produced this maximum? Give the longer wavelength as answer (a)

a)?
b)?

The work function of a metal surface is 4.80 x 10-19 J. The maximum speed of the electrons emitted from the surface is vA = 7.9 x 105 m/s when the wavelength of the light is ?A. However, a maximum speed of vB = 5.5 x 105 m/s is observed when the wavelength is ?B. Find the wavelengths ?A and ?B.

a)? has to be in terms of m
b)? has to be in terms of m

Explanation / Answer

number of lines on grating N = 2593*102 / m for maximum diffraction condition,             d sin = m for maximum m = 1          d sin =    (since d = 1 /N)           sin = N wavelength = sin /N                     = sin28.2 / (2593*102 / m)                     = 1822.4 nm                     = 1822.4*10-9 m      for m = 2 wavelength = sin /2N                     = sin28.2 / 2(2593*102 / m)                     = 911.2 nm                     = 911.2*10-9 m for m = 3 wavelength = sin /3N                     = sin28.2 / 3(2593*102 / m)                     = 607.4 nm                     = 607.4*10-9 m                     = 607.4*10-9 m for m = 4 wavelength = sin /4N                     = sin28.2 / 4(2593*102 / m)                    = 455.6 nm                     = 455.6*10-9 m                     = 455.6*10-9 m answers : the possible wavelengths (in between 410 nm and 660 nm) : A)     = 607.4 nm        = 607.4*10-9 m        = 607.4*10-9 m B)     = 455.6 nm        = 455.6*10-9 m .................................................................... .................................................................... mass of the electron m = 9.11*10-31 kg work function w0 = 4.80 x 10-19 J velocity of the electron v = 7.9 x 105 m/s plank's constant h = 6.63*10-34 J-s speed of light c = 3*108 m/s from photoelectric effect , photon energy E = work function + (K.E)electron                         E = w0 + (1/2)mv2                           = 4.80 x 10-19 J + (1/2)(9.11*10-31 kg)(7.9 x 105 m/s)2                            = 7.64*10-19 J but , photon energy E = hc/ hence , wavelength of the light is                = hc/E                   = (6.63*10-34 J-s)(3*108 m/s) / (7.64*10-19 J)                   = 2.6024*10-7 m                   = 260.24*10-9 m .................................................................................. velocity vA = 7.9 x 105 m/s wavelength A = 260.24*10-9 m ............................................................. if velocity vB = 5.5 x 105 m/s photon energy E = work function + (K.E)electron                         E = w0 + (1/2)mv2                           = 4.80 x 10-19 J + (1/2)(9.11*10-31 kg)(5.5 x 105 m/s)2                            = 6.17*10-19 J but , photon energy E = hc/ hence , wavelength of the light is                B = hc/E                   = (6.63*10-34 J-s)(3*108 m/s) / (6.17*10-19 J)                   = 3.2195*10-7 m                   = 321.95*10-9 m answers : wavelength A = 260.24*10-9 m wavelength B = 321.95*10-9 m .................................................................... mass of the electron m = 9.11*10-31 kg work function w0 = 4.80 x 10-19 J velocity of the electron v = 7.9 x 105 m/s plank's constant h = 6.63*10-34 J-s speed of light c = 3*108 m/s from photoelectric effect , photon energy E = work function + (K.E)electron                         E = w0 + (1/2)mv2                           = 4.80 x 10-19 J + (1/2)(9.11*10-31 kg)(7.9 x 105 m/s)2                            = 7.64*10-19 J but , photon energy E = hc/ hence , wavelength of the light is                = hc/E                   = (6.63*10-34 J-s)(3*108 m/s) / (7.64*10-19 J)                   = 2.6024*10-7 m                   = 260.24*10-9 m .................................................................................. velocity vA = 7.9 x 105 m/s wavelength A = 260.24*10-9 m ............................................................. if velocity vB = 5.5 x 105 m/s photon energy E = work function + (K.E)electron                         E = w0 + (1/2)mv2                           = 4.80 x 10-19 J + (1/2)(9.11*10-31 kg)(5.5 x 105 m/s)2                            = 6.17*10-19 J but , photon energy E = hc/ hence , wavelength of the light is                B = hc/E                   = (6.63*10-34 J-s)(3*108 m/s) / (6.17*10-19 J)                   = 3.2195*10-7 m                   = 321.95*10-9 m answers : wavelength A = 260.24*10-9 m wavelength B = 321.95*10-9 m velocity vB = 5.5 x 105 m/s photon energy E = work function + (K.E)electron                         E = w0 + (1/2)mv2                           = 4.80 x 10-19 J + (1/2)(9.11*10-31 kg)(5.5 x 105 m/s)2                            = 6.17*10-19 J but , photon energy E = hc/ hence , wavelength of the light is                B = hc/E                   = (6.63*10-34 J-s)(3*108 m/s) / (6.17*10-19 J)                   = 3.2195*10-7 m                   = 321.95*10-9 m answers : wavelength A = 260.24*10-9 m wavelength B = 321.95*10-9 m answers : wavelength A = 260.24*10-9 m wavelength B = 321.95*10-9 m wavelength B = 321.95*10-9 m

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