Thermal Radiation 1. the Electromagnetic Spectrum

Module 1, Lesson 3

Objective: By the end of this lesson you will be able to use Stefan-Boltzmann Law to calculate the radiation emitted and absorbed by various objects. This will require you to be able to identify the different parts of the electromagnetic spectrum, as well as describe which parts of this spectrum are transmitted through the earth’s atmosphere.

Image courtesy STCI/JHU/NASA.

1. The Electromagnetic Spectrum

As you read your eyes are being bombarded by electromagnetic radiation. You know it as visible light, but it’s really just a small part of the much larger spectrum of radiation. Visible light is radiation that has wavelengths of 300-700 nm. We see the shorter wavelengths as red light and the longer wavelengths as blue light, with many of the other colours in between. Though you might not have realized it, the other parts of the electromagnetic spectrum have names that you’re familiar with: X-rays , radio waves, heat, are some examples. They’re all just different wavelengths of electromagnetic radiation. Figure 1. This chart shows many of the typical characteristics of the electromagnetic spectrum. Notice that only certain wavelengths penetrate the atmosphere. There are two windows, one near visible light and one near the radio waves. Our eye evolved to see the light that passes easily though the atmosphere and we use radio waves to transmit signals. The reason we didn’t evolve to see radio waves is a matter of resolution, and out of the scope of this course, though your instructor may tell you if coaxed.

2. Blackbody Spectrum

Some people are astonished to ﬁnd out that all bodies warmer than absolute zero (-273oC) not only reﬂect light, but emit radiation. For example an average human emits about 300 W of thermal radiation. So why don’t we notice it? The wavelength and power emitted by a body depends on the temperature. We need temperature above about 800oC for the body to emit visible light. Bodies at temperatures below that, including room temperature, emit mainly in the infrared wavelengths. Very cold bodies like liquid helium (-269oC) emit mainly microwaves.

A blackbody is an object that absorbs all incoming electromagnetic radiation and balances this incoming radiation by emitting thermal radiation. The exact wavelengths of radiation that the body emits depends on the body’s temperature and is described by Planck’s law. Formally, Planck’s law gives the intensity of radiation as a function of frequency � and temperature T,

where h, c, and k are Plank’s constant, the speed of light, and Boltzmann’s constant. This looks like a complicated formula, but it’s really not.

PhET has a wonderful simulation of the blackbody radiation spectrum. Let’s explore this simulation more. The simulation shows a plot of Plank’s Law for a given temperature, which describes the intensity of radiation emitted for a given wavelength. Notice that the distribution has a peak at some wavelength, which is where most of the radiation is emitted. The integral of this curve is the total radiation emitted by the body.

The key element of the simulation is a temperature slider to the right of the graph. By changing the slider you can plot Planck’s law for different temperatures. One can see that as the temperature of the body decreases, the wavelength of the peak radiation increases. Very hot objects radiate near the visible spectrum. The temperature for an incandescent light bulb is marked on the slider. The point of a light bulb is to light up dark areas, but it actually radiates very little light in the visible spectrum. That which it does radiate is mostly in the red wavelengths, which gives that warm, comforting glow. At higher temperatures the peak moves into the centre of the visible spectrum which makes white light, which is why objects look “white hot”. At even higher temperatures the radiation is dominated by blue light, then eventually moves out of the visible spectrum.

Question: The human body is about 37 oC. At what wavelength does the peak of the radiation lie? Hint: You’ll have to use the + and - buttons to play with the axis of the plot. You may also ﬁnd it helpful to click the show ruler box in the bottom right hand corner of the simulation.

Answer: 10 micrometers.

You’ll notice this is nowhere near the visible spectrum, which is why we don’t glow.

The relationship between the temperature and the wavelength for which maximum power is emitted is known as Wien’s displacement law:

Notice that higher the temperature the shorter the wavelength of the emitted thermal radiation. The temperature of the sun is 5778 K [2]. From the formula above we can calculate that the wavelength at maximum power is about 500 nm, which is green light, smack in the middle of visible spectrum. Clearly the animal vision evolved to be as sensitive as possible to the solar radiation. Luckily, it’s precisely this radiation that the atmosphere is transparent to. At the human body temperature (37oC) the maximum is at about 10 m so in far infrared. Only some snakes, bats and insects can see this radiation, we have to use a special infrared camera.

There is a helpful video that uses a thermal camera to look at very objects to see this effect. Figure 2. On a typical day, different amounts of heat are radiated from different objects in the same environment. The above pictures were taken with a normal camera (top) and an infrared camera (bottom) on a 24oC day in July. Objects emitting more radiation appear lighter and those emitting less radiation appear dark. Figure 3. The top picture was taken on a cool day where the people are much hotter than the environment and so stand out as being much lighter in colour in IR camera. The day the bottom picture was taken, the surface temperature of the person's clothing is ~29oC but the surface temperature of the concrete is 38oC and so the person appears darker than the surroundings.

3. The Stefan–Boltzmann Law

The integral of Planck’s law over all wavelengths of radiation and over the surface area of the blackbody gives us the the total emitted power of radiation (average power emitted per unit area of radiating body). This is written as:

where T is temperature in degrees Kelvin, σ is the Stefan-Boltzmann constant = 5.67 x 10-8 W/ m2K4, A is the surface area of the radiating object, and e is the object’s emissivity. The emissivity quantiﬁes how closely the body represents an idea black body radiator, with e=1 being a black body.

Emissivity is a material constant strongly dependent on the wavelength. Higher emissivity means that the material will emit or absorb more radiation at this wavelength. As illustrated in our video some materials have high emissivity in visible and low in infrared and vice versa. Shiny metallic surfaces have low emissivity both in visible and infrared. The human skin has emissivity of 0.98 in infrared, Sun has emissivity very close to 1.

An ideal winter paint for the house would have high emissivity in visible (most energy from the sun comes in visible) and low emissivity in infrared (the house emits infrared). For the summer we would like to have low emissivity in the visible and high in infrared to save on air conditioning. Unfortunately we do not have commercial paints like this and anyway we would be reluctant to repaint the house twice a year.

Question: As we talked about earlier, a person is also a blackbody radiator. Any object emitting thermal radiation is also absorbing radiation from its surroundings. Being warm blooded, we generally radiate more energy than we absorb.

Assuming the surface area of a person is A=1.5 m2 and their skin temperature = 33oC, calculate the net radiation emitted from them if they were sitting in a 20 oC room.

Answer: 730 Watts - 615 Watts = 115 Watts

So the net loss of energy is about 115 W. This is ignoring the effect on clothes but the order of magnitude is consistent with our energy input from food. In reality a dressed person looses about 50% of this heat to radiation and the rest to convection, evaporation and heat exchange due to breathing.

Balancing the energy in and out of a system will become a crucial part of our analysis in this course. The Earth balances the radiation it absorbs from the sun through its blackbody radiation. The energy balance of our houses and buildings also depend on this. Figure 4. The atmosphere is transparent to only a fraction of the total electromagnetic spectrum. Here is a zoom of the small window of wavelengths near visible light. It shows the wavelengths of the incoming light from the sun, which allows us to see, and the resulting blackbody radiation emitted in the infrared, which is responsible for heating us.

4. Summary

We’ve looked at two key concepts. First, all bodies emit radiation, and the spectrum of that radiation depends on the temperature of that object. Second, though this radiation bodies can absorb and lose energy. This idea that energy can balance through these processes will lead us into the idea of a steady state. Resources

1. http://phet.colorado.edu/en/simulation/blackbody-spectrum

2. http://c21.phas.ubc.ca/article/useful-numbers