It is often said that “mathematics is the language of physics“. But what does this mean? In this post I’m going to try to explain, by using one of my favourite proofs as an example.
The problem is this:
A ball is placed atop a sphere and released. At what angle to the vertical does the ball lose contact with the sphere?
You would be excused for thinking that the answer is 45° or 90°, but the correct answer is more complicated than that.
Close your eyes. What colour do you see? Black? Look again.
Eigengrau (“intrinsic grey”), also called “dark light” or “brain grey” is the dark grey colour “seen” in the absence of light. Eigengrau appears lighter than a black object viewed in normal light because the brain prioritises contrast over true colour representation. For example, in the diagram below the two circular dots are the same colour, but appear to be different because of the way that they contrast with their backgrounds.
Eigengrau is produced when rhodopsin molecules undergoing a process of spontaneous isomerisation, indistinguishable from the process that would occur if the rhodopsin molecule had been struck by an incoming photon of light. In individual rod cells these events occur only about once every 100 seconds, but as there are about 125 million rod cells in each human eye this level of background signal is enough to produce eigengrau.
Like most physicists, I have a soft spot for Galileo thermometers.
A Galileo thermometer* works because the density of water changes as its temperature changes. The mass of a substance remains constant as it is heated (because the number of atoms doesn’t change) but because those atoms move faster the volume increases and therefore the density decreases, as shown in the graph for water below.
As the water inside the thermometer is heated by its environment its density decreases and the more dense bubbles, that represent lower temperatures, are then more dense than the surrounding fluid and therefore sink. The Galileo thermometer is read by reading the temperature tag of the bubble closest to the middle of the cylinder.
The density of the bubbles inside the thermometer is set by altering the size of the metal tags attached to them: the bubbles that represent higher temperatures have smaller tags and therefore lower densities. (The overall density of the bubble does not change as it is heated because the overall density of the bubble depends only on its overall mass and overall volume, and as the liquid inside a bubble expands it merely compresses the air inside that bubble.)
* Not actually invented by Galileo, but by a group that included one of his students.
There are many circumstances in which a system will emit light.
During incandescence objects emit light because of their temperature. Everything above absolute zero emits electromagnetic radiation due to its temperature as the electrons in the object vibrate back and forth due to the motion of the atoms that make up the object. The type and amount of EM radiation emitted depends on the temperature – this is how infrared thermometers work, by measuring the intensity and wavelength of the IR radiation emitted.
A red-hot piece of metal demonstrating incandescence.
Luminescence is the emission of light not due to temperature, and can be broken up into many sub-processes, listed below.
Fluorescence is one of the most familiar of these processes. An object fluoresces when it absorbs electromagnetic energy of one sort and subsequently emits another, usually longer-wavelength energy. This is how hidden “ultraviolet ink” works and why clothes sometimes “glow” under UV illumination. Fluorescence is a subset of photoluminescence, in which light emission is the result of absorption of photons, with the other photoluminescent process being phosphorescence, a much slower process than fluorescence in which the emission of photons is highly delayed. It is phosphorescence which is responsible for the light produced by glow-in-the-dark materials that are “charged” by light.
A selection of fluorescent minerals.
A phosphorescent statue.
Chemiluminescence is the process by which light is emitted during a chemical reaction, such as the reaction which occurs in glow sticks. Bioluminescence is a subset of this, when the process occurs in living organisms like fireflies. The other subset of chemiluminescence, electrochemiluminescence occurs when an voltage is applied to a solution; this is how LEDs operate. Cathodoluminescence is itself a subset of electroluminescence, occurring when electrons strike a material such as a phosphor, causing the electron’s energy to be converted to light. This is how old-fashioned cathode ray tube (CRT) televisions operate.
A solution of luminol demonstrating chemiluminescence.
L-R: A squid and a firefly demonstrating bioluminescence.
A selection of electroluminescent blue LEDs.
Crystalloluminescence is the process by which light is emitted during crystallisation and fractoluminescence when the bonds in crystals are broken.
Fractoluminescence is a subset of mechanoluminescence in which light is emitted as a result of forces acting on a solid. Other mechanoluminescent processes include triboluminescence in which the action of friction causes light to be emitted as chemical bonds in a substance are broken; piezoluminescence in which the action of pressure on a solid causes light to be emitted as electrons and holes recombine; and sonoluminescence is which bubbles in liquids excited by sound waves collapse, emitting light in the process. The exact process that causes sonoluminescence is unknown, though many suggestions including bremsstrahlung radiation, coronal discharge and proton tunnelling have been suggested.
Radioluminescence occurs when light is emitted as the result of bombardment by ionising radiation. It is radioluminescence that was previously used in glow-in-the-dark materials (in particular radium dials) and which is responsible for the glow produced by tritium illumination.
A radioluminescent tritium light source.
Finally, thermoluminescence occurs when certain crystalline materials emit energy they had previously absorbed in the form of EM radiation or via bombardment of ionising radiation as a result of being heated.
Imagine that you’re on holiday, trying to photograph a famous landmark. There are sure to be other tourists around, messing up your photographs. But what if there were a way to automatically remove these interlopers from your photographs?
Here are eight photographs of the street outside a local car park, taken from the car park’s roof. In each of the photographs there is some sort of foreign object present – either a pedestrian or a car.
Below is a copy of the image, but with all of those foreign objects removed. This isn’t the result of hours of painstaking manipulation – it’s the result of running one special filter, a median layer blend, on the collection of images.
The median layer blend works by taking the colour values for the same pixel in each photograph and then using the median value as the value used in the output image.
For example, if the red values for the first pixel in each image were 234, 234, 197, 251, 222, 193 and 218 then the median would be 218, as it falls in the middle when they are arranged in order (193, 197, 213, 218, 222, 234, 234, 251). Because each foreign object is in a different position in each frame, the RGB values for the pixels that make them up will lie at either end of the scale, and those values will be eliminated when the median layer blend filter is applied.
It is very important that whilst taking your images that the camera remains in a fixed position; if the camera is allowed to move you end up with a blurry and oddly smooth image. The leaves in the output photograph above are slightly blurred because they were moved by the wind as the original photographs were being taken.
This technique is also very useful when taking photographs with a high ISO setting in low light. Images taken in low light are prone to noise, but because this noise is different in every image, a median layer blend filter does a very good job of removing this noise.
Here is a boring image of a London Tube network map, taken at ISO 3200 in poor light.
If we look closely, the image is very noisy.
But after running ten of these images through a median layer blend filter, the noise is very satisfactorily removed.
L-R: The original noisy image and the resulting “de-noised” processed image.
I used the GIMP image processing software with the G’MIC plugin to create the images above, but I’m pretty sure similar tools are available for other packages (e.g. Photoshop).