Thermal Radiation and Kirchhoff's Law

Heat that needs no medium

Stand in the sun on a cold winter morning in Delhi. Your face feels warm at once, although the air around you is cold. The heat did not come through the air. It came from the Sun, 150 million kilometres away, across empty space where there is no air at all. Neither conduction nor convection can cross empty space, because both need matter to carry the heat.

This third way is thermal radiation. A hot body emits energy as electromagnetic waves, the same family of waves as light. The waves need no medium, and in empty space they travel at the speed of light, \(3 \times 10^8\,\text{m/s}\). The Sun's heat takes about eight minutes to reach the Earth. The waves warm a body only when they are absorbed by it.

Every body radiates, not only glowing ones. Your own body, a cup of chai, the walls of your room and the sand at night all radiate. At everyday temperatures the waves are infrared, longer than red light, so the eye cannot see them. A body glows visibly only above about 800 K. Night-vision cameras and the thermometer guns used at airports see the infrared that warm bodies emit.

Absorbed, reflected or passed on

When radiation falls on a body, three things can happen to it. Part of it is absorbed and warms the body. Part is reflected back from the surface. The rest passes through the body and out the other side. The three parts together must make up all of the radiation that arrived, because energy is conserved.

Divide each part by the energy that arrived. The fraction absorbed is the absorptance \(a\), also called the absorptive power. The fraction reflected is the reflectance \(r\), and the fraction transmitted is the transmittance \(t\). In Fig 1.1, \(r = 0.25\) and \(t = 0.15\), so \(a = 1 - 0.25 - 0.15 = 0.60\).

Most solids are opaque, so \(t = 0\) and \(a + r = 1\). A polished silver surface reflects about 97 % of the radiation on it, so it absorbs only 3 %. Soot or dull black paint reflects almost nothing and absorbs about 97 %. A body that absorbs all the radiation falling on it, with \(a = 1\) and \(r = t = 0\), is called a perfect black body.

The exchange never stops

A cup of water left in a room reaches the room's temperature and stays there. Does it then stop radiating? In 1791 the Swiss physicist Pierre Prevost answered no. The water still radiates, because its temperature is above absolute zero. It also absorbs the radiation that the walls send to it. At the same temperature, the two rates are exactly equal, so the net flow is zero.

So thermal equilibrium by radiation is a dynamic balance, not a pause. A hot body cools because it emits more than it absorbs. A cold body warms because it absorbs more than it emits. The net flow depends on both temperatures, the body's own and that of whatever it faces.

Now the desert night is clear. After sunset the sand radiates upward all night, at a rate set by its own temperature. What comes back down depends on the sky. A clear, dry sky sends back little, as if it were a body at about 250 K, far below freezing. A cloud is made of water drops at about 280 K, and it sends back much more. Under the cloud the net loss of the sand is about half as large, so the ground stays warmer.

Good absorbers are good emitters

Measure how much a surface radiates by its emissive power \(E\): the energy radiated per second from each square metre, in W/m². A perfect black body at the same temperature has emissive power \(E_b\). Now put two plates, one dull black and one polished, inside a closed oven at a steady temperature. Each must end at the oven's temperature, so each must emit exactly what it absorbs.

The same radiation falls on both plates. The black plate absorbs most of it, so to stay at the oven's temperature it must also emit a lot. The polished plate absorbs little, so it may emit only a little. If it emitted more than it absorbed, it would cool below the oven by itself, which never happens. So a surface's emission must be tied to its absorption.

Two cans hold water at 80 °C: one is painted dull black and the other is polished. The black can absorbs well, so by Kirchhoff's law it also emits well, and it cools faster. Fill both with cold water and stand them in sunlight, and the black can warms faster too. A good absorber is a good emitter, and a good reflector is a poor emitter.

The blackest thing is a hole

No painted surface absorbs every wavelength perfectly. The best black body is not a surface at all. Take a hollow box with blackened inner walls and make a small hole in one side. Radiation that enters through the hole strikes the inner wall. Part is absorbed, and the rest is reflected to another part of the wall.

Fig 1.3 follows one ray. After each bounce only half is left: 50 %, 25 %, 12.5 %, and so on. The chance of the ray finding the small hole again is tiny, so almost nothing comes back out. The hole therefore absorbs nearly everything that enters it, with \(a\) very close to 1. This is why the open door of a room seen from far away, or the pupil of your eye, looks black.

Now heat the box. By Kirchhoff's law a perfect absorber is also a perfect emitter. So the radiation that comes out of the hole is black-body radiation. Its amount and its spread over wavelengths depend only on the temperature of the walls, not on what the box is made of. Physicists measure black-body radiation with exactly such a heated cavity.