The Milk-in-Water Sky: When More Scattering Ruins the Blue
For a science exhibition a student fills a rectangular glass tank with clean water and sends a narrow beam from a white LED torch through it, viewing the beam from the side against a black cloth. With pure water the beam is practically invisible from the side, and the spot on the far wall is white. He adds three drops of milk and stirs: the side view now shows a faint bluish track, while the spot on the wall stays white. A visitor insists the blue is simply the colour of the milk. Wanting a stronger effect, the student pours in about half a bottle of milk. The side-scattered light now looks a dull white-grey and the spot on the far wall has turned orange-red. The student expected the blue to become brighter and thinks he has ruined the demonstration. The tank is 60 cm long and the torch beam is 5 mm wide.
1. Explain the faint blue track seen with three drops, and show why the visitor's 'blue milk' explanation cannot be right.
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Three drops in a tank of water produce a very dilute suspension of fat and protein particles roughly 0.1–1 µm across - not a dissolved blue dye. These particles scatter the beam sideways, which is why the track becomes visible at all.
For particles much smaller than the wavelength, scattered intensity varies roughly as 1/λ⁴. Taking blue as 450 nm and red as 650 nm, blue is scattered (650/450)⁴ ≈ 4.4 times more strongly. Side-scattered light reaching the eye is therefore rich in blue, and the directly transmitted beam is slightly depleted of blue.
The visitor's claim fails on two counts. Bulk milk looks white, not blue, because at high concentration light is scattered many times by all wavelengths and the eye receives a mixture. And if the milk were simply blue, the transmitted spot on the wall would look blue-tinted too, whereas it stays white - blue has been removed from the beam sideways, not added to it.
2. Explain the dull white-grey side view and the orange-red spot that appear after half a bottle of milk is added.
Read the worked answer
With a very large number of particles, a photon entering the tank is scattered many times before it escapes. Each scattering event redirects it, and after several events the light emerging sideways is a mixture of all wavelengths - the eye sees white-grey, not blue. Larger clumps also make the scattering less wavelength-selective.
Meanwhile the beam travelling straight down the 60 cm tank is being stripped of its short wavelengths near the entrance, because blue is scattered out first and most strongly. By the far end the surviving beam is dominated by red and orange, so the spot on the wall turns orange-red.
This is exactly the sunset mechanism: a long path through a scattering medium removes blue from the direct beam. The student has not ruined the demonstration - he has reproduced the second half of it. Restoring the blue is simply a matter of diluting back to a few drops.
3. Predict what would be seen if a red laser (650 nm) replaced the white torch in the three-drop tank, and estimate how the brightness of the side track compares with the blue in the white-light case.
Read the worked answer
The side track would be far fainter, and the spot on the wall would be bright red.
Using the 1/λ⁴ dependence, scattering of 650 nm light is weaker than that of 450 nm light by (650/450)⁴ ≈ 4.4. Since the blue component of white light is the one that produced the visible track, the red laser should give a track roughly four to five times weaker - probably invisible in a lit room, though it could be seen in a darkened room or with more milk drops.
The wall spot stays bright red because very little of the red beam is scattered out over 60 cm with only three drops; the beam arrives almost undiminished. A further practical point is that the laser beam is narrow and collimated, so the track is a thin line rather than the diffuse glow from the torch, which makes it harder, not easier, to see.