One of the topics we teach in second semester physics is blackbody radiation. The typical kind of scenario students would be asked about is, given the temperature of a star and information about the size and orbit of a planet, determine how much energy arrives on the planet each second. One of the main difficulties students have is deciding how to use the relationship that intensity = power/ area. There are lots of different energies, areas, and intensities to consider, so students who are used to plug-n-chug can easily fall apart here. Since we introduced the topic two weeks ago, I’ve been starting each day with various discussion (clicker) questions asking student to think about intensity, energy, and power qualitatively. We’ve had lots of good days of discussion stemming from this and progress is certainly being made, but students’ handle on the ideas seem to be quite elusive and fleeting with lots of side-steps and backslides, even for the students who don’t usually struggle.  On Thursday, I asked the following question to start our day, which pulled us into a really good discussion that lasted 15-20 minutes or so:

Assume you know how much energy is emitted from a star each second, Es. You want to find the intensity of the light arriving at a planet. Which calculation should you use? The question included a diagram that showed three distancse: Rs, the radius of the star, Rp, the radius o the planet, and Rsp, the distance between planet and the sun. The four options where.

A. Es/ 4πRp²

B. Es/ πRp²

C. Es/ 4πRsp²

D. Es/ 4πRs²

Students thought to themselves, voted, and then talk in groups. When students re-voted, we were split between B and C, with a few unsure whether it was A or B. We’ve been getting used to these kinds of discussions, so I asked a few students to explain why those chose B.  The basic line of reasoning was that we were interested in the intensity at the planet, so the relevant area had to be the area of the planet, because the planet that was catching the energy with its cross-sectional area.

Instead of letting people voice an argument for C, I said that those who picked C had to explain what they was wrong with B without explaining why they thought C was correct. I motivated this by talking about why so many hot button issues arguments are unproductive, such as abortion rights, whereas everyone just keeps repeating their arguments without listening to the other side.

One really nice argument, which ended up being convincing to most in the class, was this:

– Es/ πRp² says in words that you are taking all the energy from the sun and spreading it over the area of the planet. This can’t be right because not all the energy from the sun gets to the planet. In fact, most of energy misses the planet because it goes off in other directions.

I made sure at least one other student could repeat the argument, and then another argument was made: This argument was about how we could actually “correct” the equation so it did give the intensity at the planet. The argument was that if the “area” you want to divide by is the area of the planet, than the numerator has to be energy arriving at the planet Ep, not Es. Intensity *is* an energy divided by an area, but to get the intensity of the planet using the area of the planet, you have consider the actual energy arriving at the planet, so it would be Ep/πRp².