Exercise 1: Fundamentals of Electromagnetic Radiation

Objectives:

• Maxwell's wave theory
• Planck's quantum theory
• Planck's radiation law for blackbodies
• Wien's displacement law

If you haven't read virtual handout 1 on the syllabus then read it now.  Then follow the instructions below and answer the questions.

Maxwell's wave theory of electromagnetic radiation (EMR)

where         c = 3 x 10⁸ m s^-1 (speed of light)
                   l= wavelength
                   f= frequency

This relationship allows us to determine the frequency associated with any given wavelength of EMR and vice versa.  An example of the use of Maxwell's equation would be to calculate the frequency at which EMR of wavelength 400 nm vibrates:

so 

       3 x 10⁸ m s^-1
f=  ---------------------
       400 x 10^-9 m cycle^-1

 =   3/4 x 10¹⁵ cycles s^-1

=   7.5 x 10¹⁴ Hz = 7.5 x 10⁸ MHz = 7.5 x 10⁵ GHz

Question 1. Maxwell's equation tells us that frequency and wavelength are _____________ (directly porportional or inversely related) to each other.

Question 2. Does EMR at wavelength 700 nm vibrate faster or slower than EMR at 400 nm?

Question 3. At what frequency does EMR at 700 nm wavelength vibrate?  Show your calculations.

Question 4. At what frequency does middle-infrared EMR with a wavelength of 1.65 mm vibrate?  Show your calculations.

Question 5. KMIK-FM broadcasts at a frequency of 97.5 MHz (megahertz = 10⁶Hz).  What wavelength do these radio waves have? Show your calculations.

Question 6. KEMC-AM broadcasts at a frequency of 870 kHz (kilohertz = 10³Hz).  What wavelength do these radio waves have? Show your calculations.

Planck's quantum theory of EMR:

The energy of a quantum is given as

Q=hf

Question 7. According to Planck's theory, is the energy of a quantum directly or inversely proportional to the frequency at which it vibrates?

Question 8. How many orders of magnitude (i.e. powers of 10) more energetic is the EMR from KEMC-FM (90.5 MHz) compared to the EMR from KEMC-AM (870kHz)?

Question 9.  Since frequency is a term common to both Maxwell's and Planck's equations, the two theories of EMR can be related to one another.  Derive the mathematical relationship between quantum energy (Q) and wavelength (l). Show your reasoning.

Question 10. According to the formula you have derived in question 9, the energy of a quantum of EMR is _____________ (directly or inversely) proportional to wavelenth.

Planck's Radiation Law for Blackbodies

Planck's quantum theory of EMR led him to the mathematical description of radiation EMITTED from a perfect (i.e. ideal) radiator (called a blackbody) at any given temperature.  Planck's Blackbody Law can be expressed as:

Plank's Radiation Law for Blackbodies gives the spectral radiance of an object as a function of its temperature.

If a blackbody is acting as a perfect emitter, the total emitted energy over the whole spectrum is given by the Stefan-Boltzmann law:

Blackbody Model

A blackbody is a perfect absorber and emitter of radiation. That is, in a perfect blackbody, all radiation incident on the object is re-emitted, and emittance is a function only of temperature. In nature, true blackbodies do not exist. However, many objects approximate blackbodies.

 
Question 11. Study the graph above (and read virtual handout 1 if you haven't already).  The Area under each curve equals TOTAL RADIANT EMITTANCE (in watts) from a blackbody at the given temperature.  It should be clear therefore, that the greater the temperature of a blackbody, the _____________ (greater/lesser) the total amount of radiation it emits.

Question 12. Given these observations, which is the more powerful radiator, the sun (6000 K) or the earth (300 K)?

Wien's Displacement Law
The wavelength, in micrometers, at which the maximum spectral radiant exitance occurs for any given blackbody temperature (Kelvin) can be calculated using WIEN's Displacement Law.

If we differentiate Plank's Radiation Law for blackbodies and set it equal to zero, we arrive at a formula which gives the wavelength of maximum radiance for a blackbody of a given temperature. This formula is referred to as Wien's Displacement Law.

Using the formula, complete the following table:
 
 

Question 13. Compare the wavelengths of maximum spectral radiant exitance on the two blackbody curves above with those you calculated using Wien's formula.  Do they agree or disagree?

Question 14. Based on what you have observed so far, the following general statement can be formulated: as the temperature of a blackbody increases, the wavelength of its maximum spectral radiant exitance ________________ (increases/decreases).
 

Question 15. Look at this graphic again.  Our eyes are sensitive to EMR in the wavelenght region of 400 to 700 nm (0.4 to 0.7 mm).  How does the peak spectral radiant exitance of the sun (6000 K) compare to the spectral sensitivity of your eyes?

Question 16. Does the earth EMIT any radiation which we can see with your eyes (the average blackbody temperature of the earth is 300K)?
 
 

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