Question Video: Analyzing a Leslie Cube Physics

A Leslie cube is a hollow watertight aluminum cube that has vertical faces, each with a particular color and reflectivity, as shown in the table. How well each face of the cube emits infrared radiation can be tested by filling the cube with boiling water and aligning an infrared sensor with each face of the cube, each the same distance from the cube, several centimeters away from it. Thermometers are placed in contact with each face. Which of the following options correctly describes the thermometers’ temperature readings? [A] All the thermometers read different temperatures. [B] All the thermometers read the same temperature. [C] The thermometers on faces A and B read the same temperature. [D] The thermometers on faces A and C read the same temperature. Which face of the cube produces the highest reading on an infrared sensor? Which face of the cube produces the lowest reading on an infrared sensor?

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Video Transcript

A Leslie cube is a hollow watertight aluminum cube that has vertical faces, each with a particular color and reflectivity, as shown in the table. How well each face of the cube emits infrared radiation can be tested by filling the cube with boiling water and aligning an infrared sensor with each face of the cube, each the same distance from the cube, several centimeters away from it. Thermometers are placed in contact with each face. Which of the following options correctly describes the thermometers’ temperature readings? (A) All the thermometers read different temperatures. (B) All the thermometers read the same temperature. (C) The thermometers on faces A and B read the same temperature. (D) The thermometers on faces A and C read the same temperature.

Let’s start by clearing some space at the top of our screen, and we’ll sketch out this object called a Leslie cube. The idea with a Leslie cube is that its four vertical faces — this face here, this face here, this one that we can’t see, and that one that we can’t see — have different colors as well as different reflectivities that affect the thermal properties of the cube. In our table, we’re told that these four faces of the cube are called A, B, C, and D. Let’s call this face face A. The face back here is B. The other face we can’t see from this perspective is C. And we’ll say this face on the front right is D. Our table tells us that faces A and B are colored black, while faces C and D are colored white.

Another way the faces are different is in what is called their reflectivity. This is a measure of how strongly these faces reflect incident light. To our eyes, a highly reflective surface might look shiny and smooth. On the other hand, a face with low reflectivity might appear dull and even a bit rough. The four faces of our Leslie cube have these different properties. When we fill the Leslie cube up with boiling hot water, the cube, because it’s made of metal, quickly heats up to the temperature of the water. If we then attach thermometers to each of these four vertical faces, we want to choose the answer option that tells us most accurately what these thermometers will read.

Let’s assume that the only differences between our four cube faces is in their color and in their reflectivity. In all other ways, these faces, we’ll say, are the same. All four faces are equally in contact with a boiling hot water, and they’re all made of the same material, so they respond to that water thermally in the same way. Therefore, even though some of the faces are colored black and some are colored white and some have high reflectivity and some have low, that will not have an effect on the measured temperature of each face. Recall that our thermometers are in direct physical contact with each of the faces A, B, C, and D. They’re not measuring radiation at a distance from these faces but rather through direct contact, their temperature.

These four faces of the Leslie cube are in fact all at the same temperature. Their differences in color and reflectivity affect how they emit infrared radiation, but they don’t affect the actual temperature of that face. We choose option (B).

Let’s look now at the second part of this question.

Which face of the cube produces the highest reading on an infrared sensor?

The way we make a reading with an infrared sensor is different than how we did it with a thermometer. An infrared sensor is a device that can look like a camera that we place a small distance, perhaps a few centimeters, away from one of the faces on our cube. The sensor takes in infrared radiation from that face. In this part of our question, we imagine doing this for all four of our faces and then comparing those four readings.

Let’s start by remembering that all four of our faces have the exact same temperature, in this case about 100 degrees Celsius. Any difference in the infrared radiation they give off then has to do with their color and their reflectivity. At a given temperature, darker colors tend to give off infrared radiation more than lighter colors. It’s often possible to confirm this on a hot day when the sun is high in the sky. If we hold our hand near the surface of a light-colored sidewalk and then move and place our hand near the surface of dark-colored asphalt, we may well be able to feel that the asphalt seems warmer. Even when the temperatures of these two different surfaces are the same, this effect will still take place.

Whichever face of our cube produces the highest reading then, it will be one of the ones with a darker color. It won’t be face D or face C. Comparing the faces A and B, we see that one has high reflectivity and one has low reflectivity. It’s the face with low reflectivity that actually will tend to emit more infrared radiation than one with high reflectivity. A highly reflective surface is less likely to absorb incident radiation. And emitted infrared radiation is due to some absorption process going on beforehand. A surface that is better, we could say, at absorbing one with low reflectivity will tend to emit more infrared radiation. Of all four vertical faces in our cube, it’s actually face B that produces the highest reading on an infrared sensor.

Let’s look now at the last part of our question.

Which face of the cube produces the lowest reading on an infrared sensor?

Just as we found that a dark-colored, low-reflectivity face produces a high reading on an infrared sensor, to find the face creating the lowest reading, we’ll look for the opposite properties. We’ll look for a face that has a light color, those are faces C and D, and of these two the face that has the highest reflectivity. At a given temperature, a face with these properties will emit the least infrared radiation. Therefore, when we put an infrared sensor in front of all four faces, it’s the reading in front of face C that is the lowest.

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