Interactive Lecture Demonstration 11 – Radiometer

A radiometer is a pinwheel with paddles that are painted either black or white. All of the white sides face in one direction, all of the black sides face in the other direction. The pinwheel sits inside an evacuated glass bulb. When light shines on the radiometer, the pinwheel rotates due to the force of the light hitting the paddles.

The goal of this problem is to determine the force of sunlight on the radiometer.

Let's assume we are given the following information:

• The total power radiated by the Sun in the form of light is 3.92 10²⁶ Watts.
• The distance from the Sun to Earth is 1.5 10¹¹ meters.
• The Sun emits light of different wavelengths, but more light is emitted at a wavelength of 500 nm (green) than at any other wavelength. We will assume for this problem that all of the light emitted by the Sun has the same wavelength of 500 nm.

Rather than approaching this problem using the concept of light as a wave, we will use the concept of photons (light as a particle). We can think of a wave of light as being composed of many individual photons. Photons have the following properties:

• The energy carried by a single photon is given by E = h f where h = 6.63 10^-34 J s is Planck's constant and f is the frequency of the light.
• Note that we can also write this as E = h c/l since the product of a wave's (or a photon's) frequency and wavelength is equal to the speed of the wave: c = f l.
• The momentum of a photon is related to its energy by p = E/c.

We can calculate the force exerted on the radiometer paddle by using Newton's Second Law in the form F_net = Dp/Dt. Here, F_net is the net force on an object. Dp is the change in the object's momentum as a result of the force on the object. Dt is the amount of time during which the force acts on the object. In the following, we will apply this Law to the collision of a photon with the surface of the radiometer.

1. Imagine what will happen to the radiometer when it is hit by a single photon of sunlight. If the photon hits a black paddle, it will be absorbed  (a black surface appears black because it absorbs most of the light that hits the surface). If the photon hits a white paddle, it will be reflected (a white surface appears white because it reflects most of the light that hits the surface). In which situation – absorption or reflection – is a greater force exerted by the light on the surface ?

2. Suppose a single photon of green light hits a surface and is absorbed. What is the change in momentum of the photon ?

In order to find the force of the paddle on the light that hits it (which, by Newton's Third Law, is equal but opposite to the force that the light exerts on the paddle), we need to multiply the change in momentum of one photon that hits the paddle by the number of photons per second that hit the paddle. We can find that number using the information given in the beginning of this problem. Our strategy will be to find the total number of photons leaving the Sun every second, then find the number of those photons that actually strike the paddle of the radiometer.

3. Use the total power emitted by the Sun to find the number of photons emitted by the Sun every second, assuming that all of the photons are in the form of green light. Remember that power is P = DE/Dt.

4. The photons emitted by the Sun travel away from the Sun in random directions. At a given distance from the Sun, we can imagine all the photons to be streaming through a large sphere centered on the Sun. Use your result from #3 to find the number of photons per second per square meter that are streaming past the Earth. Hint: the surface area of a sphere of radius R is A = 4 p R² .

5. To find the number of photons emitted by the Sun every second that hit the radiometer, think of the radiometer paddle as being a small part of the area of the very large sphere centered on the Sun. Since the photons are emitted in random directions, we can find what we want by multiplying the number of photons per second per square meter of area of the whole sphere by the area of the paddle. Estimate the area of one of the radiometer paddles, then calculate how many photons hit the paddle every second.

Note that we are simplifying things here somewhat. Some of the photons streaming past the Earth get absorbed or reflected by clouds and other material in the atmosphere before they get to the radiometer.

6. Finally, combine your result from #5 with your result from #2 to find the total change in momentum per second of photons that are absorbed by a black paddle of the radiometer. Since F_net = Dp/Dt, this is the force that the paddle exerts on the photons (which, by Newton’s Third Law, is equal but opposite to the force that the photons exert on the paddle). The force exerted on a white paddle would be twice this number, since the change in momentum of each photon that gets reflected is twice the change in momentum of a photon that is absorbed.

7. Compare your answer in #6 with the number the book gives for the force per square meter of direct sunlight on a surface (see p. 707). You will need to divide your answer by the area of the paddle in order to get the same units that the book uses.

8. Let's do another thing with our calculation. Since the energy of a photon is related to its momentum by E = cp, the change in energy of the photon when it gets absorbed is DE = cDp (when it gets absorbed, the photon disappears and its energy is given to the paddle). Since power is P = DE/Dt = cDp/Dt, we can find the total power of sunlight that is absorbed by a black paddle by multiplying our result in #6 by c. Do that now, then compare your answer with the number the book gives on p. 708 for the power per area delivered by sunlight to Earth’s surface (again, you’ll have to divide out the area of the paddle to get the same units that the book uses). The power per square meter of a wave is called the intensity of the wave.

9. One last thing. In this problem we have been talking about light as being composed of streams of photons. We can also think of light as being a wave of electric and magnetic fields. Use your result from #8 to calculate the maximum value of the electric field from sunlight that strikes the Earth's surface. You’ll need a formula from p. 707.