Lab #6: Black Body Radiation

 

Reading Assignment:

 

Fundamentals of Physics, Halliday, Resnick and Walker, Chapter 19 (especially 19.11)

Physics for Scientists and Engineers, R. A. Serway:  Chapter 40 (especially 40.1)

 

Introduction:

 

 When a body is heated it emits radiation with a continuous distribution of wavelengths. The intensity of the radiation depends both on the nature of the surface of the body and on the temperature of the body. To simplify the discussion concerning the nature of the surface, we often consider an ideal body, a Black Body, which absorbs and emits all wavelengths of electromagnetic radiation. A good approximation to an ideal black body is a small hole drilled into the side of a closed box. The radiation emitted by such an ideal body is called Black Body Radiation.

Figure 1

 

 

In our discussion of heat transfer, we saw that the intensity I of the radiation, or the power radiated per unit area in W/m2, is given by

 

 

Stefan-Boltzmann Law

 

 

Eq. 1

 

 

where s = 5.6703x10-8 W/m2/K4 is called the Stefan-Boltzmann constant and T is the absolute temperature in K. Note that this equation predicts that every object whose temperature is above absolute zero radiates electromagnetic energy. The power radiated is not uniformly distributed over all wavelengths, but depends upon the wavelength in a way shown in Figure 1, which displays the power radiated per unit area per unit wavelength interval as a function of the emitted wavelength.

 

The figure shows the spectral radiancy or spectral emittance or the intensity of the emitted radiation per wavelength interval Bl. Hence, Bldl is the total intensity radiated with wavelengths between l and l+dl. Integrating over all wavelengths must yield the total intensity radiated given by Eq. 1:

 

 

Stefan-Boltzmann Law

 

 

Eq. 2

 

 

Figure 1 shows that the curves are different for different temperatures, T (in Kelvin, K). We also see that the wavelength lmax at which the intensity is a maximum shifts to lower wavelengths as T increases. The relation is called Wien’s Displacement Law:

 

 

Wien’s Displacement Law

 

 

Eq. 3

 

 

Thus the hotter an object is, the lower the wavelength at which the intensity peaks. An object that glows bluish is hotter and brighter than an object that glows red (the wavelength of blue light is smaller than that of red light).  This is shown in more detail in Figure 2 where the temperature is increasing from front to back.

 

Prior to 1900, two different empirical expressions partly accounted for the curves shown in Figures 1 and 2. This was known as Wien’s Law:

 

 

 

Wien’s Law (for small l)

 

 

 

Eq. 4

 

 

where c1 and c2 are constants.  This expression describes the intensity distributions well at small values of l, but falls well below the curves as l becomes large.

 

 

 

 

 

 

 

 

 

Figure 2

 

In 1900, Planck, based on some revolutionary assumptions, derived an expression that perfectly describes the curves shown in Fig. 1:

 

 

 

Planck’s Law

 

 

 

Eq. 5

 

 

where h = 6.626x10-34 J-s is called Planck’s constant, k = 1.38x10-23 J/K is Boltzmann’s constant and c is the speed of light. The assumptions that Planck made were:

 

·        The atoms in the black body radiator act as electromagnetic oscillators that emit radiation with a frequency f

·        These oscillators can only have energies given by En = nhf, where n = 1, 2, 3, …., an integer.

 

This was the start of Quantum Physics in which the energies of various systems could only have discrete values controlled by an integer, n, called a quantum number.

 

In this lab activity, we will use a simulation of Planck’s Law for you to study Black Body Radiation.

 

 

 

PreLab #6: Black Body Radiation

 

 

Name:_______________________________                                    Section #:________

 

Questions:

 

  1. A cavity radiator has a radiancy RA = 1 W/cm2. Another cavity radiator made of the same material and having the same size has a radiancy RB = 953 W/cm2. What is the ratio of their temperatures, ie what is TA/TB?

 

 

 

 

 

 

 

 

 

 

 

  1. What is the peak wavelength emitted by the human body under normal conditions? In what part of the electromagnetic spectrum is the radiation?

 

 

 

 

 

 

 

 

  1. Estimate the area under the T = 7000 K curve of Figure 1. (Note: to convert the vertical axis to more familiar units, take the vertical scale of 7x106 to be equivalent to 2.2 W/cm2.)  Does your answer agree with that you obtain from Eq. 1?

 

 

 

 

 

 

 

 

Lab #6: Black Body Radiation

 

 

Name:_______________________________                                    Section #:________

Name:_______________________________

Name:_______________________________           

 

 

Goals:

·        To study the properties of Black Body Radiation

·        To study the properties of Planck’s Law

 

Activity 1: Study of the Black Body Intensity:

 

Please access the following Website: http://webphysics.davidson.edu/alumni/MiLee/java/bb_mjl.htm

 

1.      By clicking on the horizontal scale and moving it left or right, you can change the temperature of the black body. When you click on a value, the temperature you have set will appear in the panel on the left of the graph. By clicking at the top of the curve and sliding left or right you can determine the wavelength at the peak (lmax) by superimposing the 2 blue arrows.  Try setting several values of T from 200 nm and up and determining the resulting values of lmax.

 

2.      Open the following table and record T vs lmax for 6 different T values ranging from 3000 K to 10,000 K. Record the color of the wavelength at each of the peak intensities:

 

Data point #

T

lmax

Color at lmax

1

 

 

 

2

 

 

 

3

 

 

 

4

 

 

 

5

 

 

 

6

 

 

 

      

3.      Open the Graph Display and plot your values of lmax as a function of 1/T.

 

4.      Verify that the plot exhibits the behavior you might expect.  Determine the slope of the curve.

 

5.      Copy the graph to the template provided.

 

6.      Compare your value of the slope with that expected from Wien’s Displacement Law.

 

 

 

 

Activity 2: Study of Planck’s Law:

 

1.      Starting with Planck’s Law (Eq. 5), write down the condition that describes the maximum in the black body intensity distribution. (Hint: remember your calculus?)

 

 

2.      Use the condition in step 1 to evaluate the value of lmax at the peak of the intensity distribution. At an appropriate point, you should use Wiens’s approximation, ie assume that l is small, so that:                      

 

 

3.      From this, determine an expression for lmaxT in terms of the constants h, c and k.

 

 

4.      Does your value of lmaxT  agree with the value given in Eq. 3?

 

 

5.      Starting with Planck’s Law (Eq. 5), integrate over all wavelengths to determine the total intensity as a function of T. Does your dependence on T agree with Stefan-Boltzmann’s Law, Eq. 2? (Hint: you may find it easier to try a substitution of variables.)

 

 

PostLab #6: Black Body Radiation

 

 

Name:_______________________________                                    Section #:________

 

Questions:

 

  1. Estimate the surface temperature of the Sun if sunlight has an intensity distribution that peaks around 500 nm.

 

 

 

 

    1. Is your answer consistent with Figure 1?

 

    1. What region of the electromagnetic spectrum is this?

 

    1. Can you think of any obvious connection between your result and some part of the human anatomy?

 

 

2.      Radiation from space has been measured to correspond to black body radiation at T = 2.726 K. (This radiation is usually considered to arise from the ‘Big Bang’ at the start of the universe.)  At this temperature, for what wavelength does the radiation peak and in what region of the electromagnetic spectrum does this radiation lie?

 

 

 

 

 

 

 

 

 

3.      From the result of the calculation you did in step 3 of Activity #2 of this Lab, determine Planck’s constant h using Eq. 3.