Planck's radiation law tells us that the intensity of radiation per unit area and per unit...

  

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Planck's radiation law tells us that the intensity of radiation per unit area and per unit wave- length A from a black body at temperature T is I(A) - 2hc21-5 ehc/AkBT -1 where h is Planck's constant, c is the speed of light, and kg is Boltzmann's constant. a) Show by differentiating that the wavelength A at which the emitted radiation is strongest is the solution of the equation 5e-hc/AkBT hc + - 5 = 0. AKBT Make the substitution x = hc/AkBT and hence show that the wavelength of maximum radiation obeys the Wien displacement law: b = T' where the so-called Wien displacement constant is b = hc/kpx, and x is the solution to the nonlinear equation 5e+x-5= 0. b) Write a program to solve this equation to an accuracy of = 10-6 using the binary search method, and hence find a value for the displacement constant. c) The displacement law is the basis for the method of optical pyrometry, a method for mea- suring the temperatures of objects by observing the color of the thermal radiation they emit. The method is commonly used to estimate the surface temperatures of astronom- ical bodies, such as the Sun. The wavelength peak in the Sun's emitted radiation falls at A = 502 nm. From the equations above and your value of the displacement constant, estimate the surface temperature of the Sun. Planck's radiation law tells us that the intensity of radiation per unit area and per unit wave- length A from a black body at temperature T is I(A) - 2hc21-5 ehc/AkBT -1 where h is Planck's constant, c is the speed of light, and kg is Boltzmann's constant. a) Show by differentiating that the wavelength A at which the emitted radiation is strongest is the solution of the equation 5e-hc/AkBT hc + - 5 = 0. AKBT Make the substitution x = hc/AkBT and hence show that the wavelength of maximum radiation obeys the Wien displacement law: b = T' where the so-called Wien displacement constant is b = hc/kpx, and x is the solution to the nonlinear equation 5e+x-5= 0. b) Write a program to solve this equation to an accuracy of = 10-6 using the binary search method, and hence find a value for the displacement constant. c) The displacement law is the basis for the method of optical pyrometry, a method for mea- suring the temperatures of objects by observing the color of the thermal radiation they emit. The method is commonly used to estimate the surface temperatures of astronom- ical bodies, such as the Sun. The wavelength peak in the Sun's emitted radiation falls at A = 502 nm. From the equations above and your value of the displacement constant, estimate the surface temperature of the Sun.