Why do hot things glow? (black-body radiation)
Every object gives out light because its tiny particles jiggle. The hotter it is, the more light it gives and the bluer the strongest colour becomes. An iron rod goes dull red, then orange, then white as you heat it.
What is a black body?
A black body is an ideal object that takes in all the light that falls on it and, when hot, gives out light in the most perfect way. Its colour curve depends only on its temperature, not on what it is made of. A furnace with a tiny hole, and a star, are close to it. Do not think it must look black: when hot it glows brightly.
Wien's law: the strongest wavelength is λmax = 2.898 × 10⁻³ m K ÷ T. Hotter object, shorter wavelength.
The puzzle: what went wrong with the old idea
Before 1900, scientists thought a hot object could give out any amount of energy at any colour. Their sums said that blue and ultraviolet light should get brighter and brighter without limit. That would make every furnace give out deadly ultraviolet light. Real furnaces do not. This failure is called the ultraviolet catastrophe. In the 3D, the red see-through bars are the old guess and the coloured bars are the truth.
Planck's hypothesis: energy comes in packets
In 1900 Max Planck made a bold guess. The tiny vibrating particles in a hot object can give or take energy only in whole packets, never in small pieces. For light of frequency f, one packet is
E = h f, and the energy given out is always E = n h f with n = 1, 2, 3, …
Here h = 6.63 × 10⁻³⁴ J s is Planck's constant. This is called quantisation of energy: "quantum" means a fixed small amount.
Why it fixes the puzzle: a violet packet is big, so a warm object seldom has enough energy to make one. Very few violet packets come out, and the glow falls at high frequency, exactly as real objects behave.
Why do we never notice the steps?
h is extremely small. For a swinging pendulum or a hot cup the steps are so tiny that the staircase looks like a smooth ramp.
Photons: packets of light
In 1905 Albert Einstein went further: light itself travels as packets. One packet is a photon. Its energy is E = hf = hc/λ. A photon has no mass and always moves at the speed of light, c = 3 × 10⁸ m/s in vacuum.
Bright light means more photons, not bigger photons. Violet light has more energy per photon than red light. Energy of atoms is often written in electron-volts: 1 eV = 1.6 × 10⁻¹⁹ J.
Try it
Go to step 2 in the 3D and move the Energy slider slowly. Watch the ramp ball and the staircase ball. Where does the stair ball jump? Now go to step 3 and slide f up. Predict first: will the stair ball jump more often or less often? Then check. At home: watch a stove ring or a toaster glow as it warms, and write down the colour changes.
Key formulas and definitions
- E = h f (one photon)
- E = n h f (n = 1, 2, 3 …)
- c = f λ, so E = h c / λ
- h = 6.63 × 10⁻³⁴ J s; c = 3 × 10⁸ m/s
- 1 eV = 1.6 × 10⁻¹⁹ J
- Wien: λmax × T = 2.898 × 10⁻³ m K
Worked examples
1. Red light has frequency 4.5 × 10¹⁴ Hz. Find the energy of one photon.
E = hf = 6.63 × 10⁻³⁴ × 4.5 × 10¹⁴ = 2.98 × 10⁻¹⁹ J. In eV this is 2.98 × 10⁻¹⁹ ÷ 1.6 × 10⁻¹⁹ ≈ 1.9 eV.
2. Green light has wavelength 500 nm. Find its frequency and photon energy.
f = c/λ = 3 × 10⁸ / 5 × 10⁻⁷ = 6 × 10¹⁴ Hz. E = hf = 6.63 × 10⁻³⁴ × 6 × 10¹⁴ = 3.98 × 10⁻¹⁹ J ≈ 2.5 eV.
3. The Sun's surface is at about 5800 K. At what wavelength does it shine strongest?
λmax = 2.898 × 10⁻³ / 5800 = 5.0 × 10⁻⁷ m = 500 nm, which is green-yellow light.
4. A star shines strongest at 290 nm. Estimate its temperature.
T = 2.898 × 10⁻³ / 2.9 × 10⁻⁷ ≈ 10 000 K. It is much hotter than the Sun, so it looks blue-white.
5. A lamp gives out 2 W of green light of photon energy 3.98 × 10⁻¹⁹ J. How many photons leave each second?
Number per second = power ÷ energy of one photon = 2 / 3.98 × 10⁻¹⁹ ≈ 5.0 × 10¹⁸ photons per second.
6. How many red photons (4 × 10¹⁴ Hz) carry as much energy as one violet photon (8 × 10¹⁴ Hz)?
Energy is proportional to f. The violet photon has twice the frequency, so it has twice the energy. It equals 2 red photons.
Common mistakes
- Thinking a black body must look black. When hot, it glows brightly.
- Thinking brighter light means bigger photons. Brighter means more photons; photon size depends only on frequency.
- Mixing up f and λ. Higher frequency means SHORTER wavelength and more energy per photon.
- Forgetting to convert nm to m (1 nm = 10⁻⁹ m) and eV to J before using E = hf.