Mechanics Physclips (preview above) is a set of multimedia tutorials on mechanics. Relativity Einstein Light: relativity in 10 minutes... or 10 hours is our contribution to the 100th anniversary of relativity Inertial frames Centripital force and inertial frames Twin paradox Thought experiments Mass defect "Mass dilation" Kinetic energy in relativity Relativity and space travel Space travel. See also the site Space for background. Forces during launch "Weightlessness" Gravitational potential energy Energy costs of space travel Comparing wavelengths for communication How fast does a bullet fall on Earth and on the moon? Astrophysics The electromagnetic spectrum Resolution and senstivity New generation telescopes Impact of astrophysics on society Parallax, resolution, Airy disc, parsecs, distances Cepheid variables: measuring longer distances The most distant visible objects The atom, photoelectric effect, energy levels, quanta, black body radiation, etc The electromagnetic spectrum Black body radiation curve and Wien's displacement law Photoelectric effect Electron waves Electron microscope Heisenberg's uncertainty principle Pauli's exclusion principle Zeeman effect Chadwick and Fermi The Bohr atom and the Hydrogen spectrum The spin quantum number Accelerators as probes of the nucleus Semiconductors, transistors, solar cells etc n-type and p-type semiconductors How diodes and transistors work Transistors as amplifiers and logic elements History of the invention of the transistor Transistors vs thermionic valves. Microchips and microprocessors More about transistors, computers Solar cells and the photovoltaic effect States of matter Solid, liquid, gas and plasma Bose-Einstein condensates Neutron stars Dark matter Exotic matter Superconductors What is a phonon? What causes the lattice distortions in a superconductor? Levitation of a magnet by a superconductor The Meissner effect Superconductors and computers Applications to magnetic fields, motors, power distributions, MRI

Motors and generators See also the multimedia site Motors and generators (example above). Environmental effects of AC generators Motor effect in a galvanometer Force between two wires Transformers and induction See also the site Transformers for background. Conservation of energy Transformers in electricity supply and the home Eddy currents in transformers Induction cooktops Electromagnetic braking Switching devices Eddy current braking Oscilloscope Drift velocity, current density, resitivity and Ohm's law Nuclear physics, radioisotopes, particle physics, neutrinos etc Medical applications of radioactivity and nuclear physics Isotopes in engineering and agriculture Particle physics and cosmology Neutrinos Beta decay Medical physics MRI, PET, CT, X-Ray. Scanning techniques in medical physics Medical physics: ultrasound, optical fibres and MRI Magnetic focusing of a charged beam Simple Harmonic Motion under gravity Miscellaneous questions in history and social studies Copenhagen Was there ever a debate between Einstein and Planck? Nobel prizes in physics Understanding matter Short answers suitable for the high school syllabus Glossary of terms and skills listed in the HSC syllabus Hints for exams Other useful links An essay about the history of ideas about light.

An inertial frame is one in which Newton's laws work: F = ma. The surface of the Earth is almost an inertial frame (not quite: it is rotating, but very slowly: one turn every 23.9 hours. See Foucault's pendulum for a discussion). Another frame moving at constant velocity to one inertial frame is also inertial. So a bus that is not accelerating (including not turning and not bumping) is an inertial frame. A bus going round a corner is not an inertial frame: a standing passenger seems to accelerate sideways, or must hold on to the bar just to stay 'in the same place'. These are investigations that you have already done.

At funfairs, there are often merry-go-rounds or more dangerous variants on this. I went on one at the Easter Show callled the 'gravitron': a big cylinder that spun--we were 'pinned to the wall'. On these, a ball does not travel in a vertical plane: it seems to turn corners. Kinematics seems seriously weird in this frame. Then the guy running the gadget shouted at us to stop throwing the ball and he was probably right to do so, because it was hard to predict where it was going to go. Inside the 'gravitron', or another fairground ride, many such simple experiments will show that Newton's laws seem to fail, and so that you are in a non-inertial frame. See the principle of relativity.

Let's imagine that the car is a convertible and I am watching you from above. Before the curve, the car is coasting and you and it are travelling in a straight line, with no horizontal forces acting on you or the car. Now the car turns to the left. It does this because the friction between the road and the tires produces a force acting towards the centre of the curve (a centripital force). I know this because, if I cover the road surface with ice so as to reduce that friction, the car continues in a straight line.

Now I look at you inside the car. At first, as the car swerves to the left, you continue in your straight line. The car seat slides under you: the car is curving and you are not. But this happens only for a short time. The sliding of the seat under you brings the raised edge of the seat against your thigh, and perhaps the seat belt against your chest. These lateral forces exerted by the seat edge and the seat belt push you around the corner: they provide the centripital force.

The observer above the car is in an inertial frame. For this observer, Newton's laws work: with an icy road and no horizontal force, the car travels in a straight line. No force, no acceleration. Add frictional force and there is an acceleration. However, if you measure all positions, velocities and accelerations inside the car, you are in a non-inertial frame. In this frame of reference, you appear to accelerate sideways towards the outside of the car. To deal with this, we could introduce fictitious forces. This is sometimes done. For instance, the surface of the Earth is rotating, but it is convenient to measure motion with respect to the surface of the Earth. When we do so, we have to introduced fictitious forces: both centrifugal force and the Coriolis force.

As a scientific experiment to test SR, this experiment is probably not ideal because of the GR complication and because the effect is small. However it does have the advantage of using macroscopic clocks. I think that its importance was rather at a person-in-the-street level. Imagine a converation like this.
- "You mean that if I get in a plane going East and let the Earth rotate underneath me, while my twin brother goes west at 2000 km/hr (speed of plane plus that of Earth), he will age less quickly than I do?"
- "No, the effect is less than a microsecond and biological aging is not nearly precise enough to see that, but we could do it with really precise atomic clocks."
- "So why haven't you done it?"

Much more impressive demonstrations of time dilation are in the life times of subatomic particles, particularly in cosmic rays, where the factors can be much greater than one.

But the round the world clock experiment is useful because, provided that one does the GR correction, it also answers the remaining supporters of the twin paradox.

Incidentally, I cannot resist recommending a lovely paper by Sam Drake: The equivalence principle as a stepping stone from special relativity to general relativity: A Socratic dialog.

You've read the famous equation, E = mc². This is usually described as the energy E that is created when a mass m is destroyed. But it works both ways. When you add energy E to something, you increase its mass by E/c². Because c² is large you don't notice this. If you supply say 100 kJ to a kettle full of water by heating it at 1 kW for 100 seconds, this energy increases the mass of the water by 100 kJ/c² = 1 ng. A nanogram is too small to measure on a balance that will hold your kettle, and in any case it is much less than the amount of water that evaporates while you heat it. However, the mass change is proportionally much more important when we consider nuclear energies.

To separate a helium atom into two atoms of deuterium (proton plus neutron plus electron), one must do work W against the attractive forces among the nucleons. Thus the mass of the two hydrogen atoms is greater than that of the helium atom, by W/c².

To dissemble any nucleus into protons and neutrons similarly requires doing work against the force (called the strong nuclear force, a very good name!) that holds the nucleus together. So the mass of any nucleus is less than the sum of the masses of its components. This difference is called the mass defect.

There is a mass defect when one rearranges molecules, too. The mass output by a fossil fuel power station is slightly less than the mass input. Indeed, the difference would be the same for two stations of equal power output. However, the fuel and the wastes of the fossil fuel station are so much greater than those of the nuclear station that we do not notice the chemical mass defect. See the explanation of mass defect on Einstein Light.

You can think of it as an accounting system that almost no-one uses anymore so you can forget about it and jump to the next topic. Or you can read on.

Today, we talk only about the mass m of an object, and we regard m as constant. If you are travelling in the same frame as the object, and if you apply a force F, it accelerates at F/m. If it is travelling at speed v with respect to you, it has a momentum

p = γmv
where γ is the usual relativistic factor, (1-v²/c²)^-1/2. Of course, if v is much less than c, γ is almost exactly one, so we regain the Newtonian expression. Now if we were to write the above equation as
p = (γm)v
then the term in parentheses could be regarded as a 'dilated mass', and that is what you will find in some old text books.

The reason why mass dilation is no longer used in discussing relativity is because it is very confusing.

In the original papers, Einstein himself did use the idea of a longitudinal mass that increased with speed, but he seems not to have used it after 1906, and indeed recommended against it. Some old text books use it (some use it in one chapter and discard it in another), some distinguish between longitudinal mass and lateral mass, but happily its use is disappearing. There's a nice discussion of the educational consequences of using it or not in On the abuse and use of relativistic mass.

Kinetic Energy in Relativity. While we are at it, let's talk about energy. The work we put into accelerating the mass up to v has been converted to kinetic energy. The difference is that relativistic kinetic energy is larger than ½mv². Indeed, I think that it was the expression

kinetic energy = (γ - 1)mc²
that led Einstein to identify mc² as the proper energy of the mass m. We then have
total energy = proper energy + kinetic energy = mc² + (γ - 1)mc².

The question is ambiguous: is |3W| the total force acting on her, or is it the force that the spacecraft exerts on her? Before takeoff, the astronaut is lying in her seat and is not accelerating (a = 0) so the total force on her (F = ma) is zero. Her weight is a downwards force W, where the magnitude of W is mg, so the seat is pushing her upwards with a force F_seat = −W, and the magnitude of F_seat is mg.



Now let's suppose that the spacecraft accelerates vertically upwards at acceleration a. Because she is strapped in, she accelerates with the space craft so the total force on her is ma. This force F is supplied by sum of the forces of the seat and her weight, so

F = F_seat + W = ma
where F_seat is upwards and W is downwards.

If |3W| is the total force, then we have

ma = |3W| = 3mg
so the spacecraft is accelerating upwards at an acceleration of a = 3g, about 30 m.s⁻². This is the situation in the diagram at right, in which the spaecraft exerts a force |4W| on her and so she feels 4 times heavier than usual.
If the total force that the spacecraft exerted on her were |3W|, then we would have
|3W| + W = ma
upwards at an acceleration of a = 2g, about 20 m.s⁻². This is the situation sketched in the middle.

Now we need to find out wheter the maximum accleration of a space shuttle is closer 2g or 3g. According to one NASA site, "the shuttle goes from standing still on the launch pad to more than 27,359 kilometers per hour (17,000 mph) in just over eight minutes". This gives an average acceleration that is a little under 2g. So over these eight minutes, the astronauts would feel two to three times heavier than normal. However, this is an average: the maximum may exceed this value. Also, the flight is not vertical for all of this time, and so the angles between the forces must be considered in adding them.

"Weightlessness"

While we're on the subject of space travel, let's talk about free fall and something that is misleadingly called "weightlessness"

As you are reading this, you can probably feel your chair pushing upwards on you with a force of several hundred newtons. If your feet are not touching the ground, this is an upwards force equal in magnitude to your weight (a downwards force). My weight is 680 N downwards, so I know that the force from the chair is about this much, upwards. You can also feel your abdominal muscles holding your abdominal organs in place. These forces and some others give you the sensation of having weight. You do not really sense your weight directly very much, because it is applied homogeneously over your whole body. When the forces from the chair or on your abdominal wall are reduced or zero, you may feel 'weightless'--the feeling you get when a lift starts to accelerated rapidly downwards, or when you go quickly over a peak on a roller coaster. I have put 'weightless' in inverted commas because in these situations, and in an orbiting spacecraft, your weight is virtually normal. Since the moon flights stopped, no human has been far enough from the Earth for his/her weight to be substantially reduced.

The three diagrams below show two situations that produce free fall. In an orbiting spacecraft, the spacecraft and the cosmonaut are both accelerating towards the centre of the earth at the same rate (their centripital acceleration is a_c = v²/r, where v is the orbital speed and r the radius). Their weight is what keeps them in orbit: W = ma_c. Because they are *both* accelerating towards the centre of the earth at the same rate, there is on average no force between the cosmonaut and the spacecraft. This absence of forces from seat, floor, abdominal wall etc is what is commonly but misleadingly called 'weightlessness': the cosmonauts in the space station are not without weight, in fact the have (almost) their usual weight. It's just that they don't feel the force of chairs on their bums and they don't feel their abdomens holding in their organs.

                                  
In the figure in the middle, the cable of a lift (elevator) has broken. Both the lift and its passenger are in free fall, accelerating downwards at g. The passenger no longer feels the force on his abdominal muscles, and so, like the cosmonaut, he might say that he feels 'weightless'. He is not, of course, without weight. His weight is still W = mg. Indeed it is his weight that will probably lead to his death, because his unopposed weight is continuously accelerating him.

In figure at right, a NASA airplane (nicknamed the 'vomit comet') cuts the power in its engines and, for about 25 seconds, travels in a trajectory that is nearly parabolic. Both the plane and its occupants accelerate towards the Earth at g: all are in free fall. Astronauts are thus exposed to free fall and obtain brief periods of experience in working in this condition.

Some physicists tend to use the word 'weightless' in scare quotes (as I have done here), to make it clear that they are not talking about a situation in which there is no weight. Many physicists prefer to avoid the word altogether and talk instead about free fall.

There are some similiarities between the passenger (mass m) in the lift (let's put it at the equator) and a cosmonaut (mass m) in low Earth orbit. The weight of each is about mg. Both accelerate towards the centre of the Earth at approximately g. The difference is that the spacecraft makes a circle around the Earth in about 90 minutes, whereas the lift makes a circle around the Earth in about 24 hours. The acceleration g is just enough to keep an object in low Earth orbit with a period of 90 minutes. It is far too great for the 'orbit' of the hapless passenger in the lift. If a satellite loses speed, it gradually spirals in towards the Earth. The horizontal speed of the passenger in the lift is so low that his 'spiral' towards the centre of the Earth is almost a straight line. (There have been a few 'approximately's and 'almost's in the above. If you are interested in the analysis of motion in the rotating frame of the Earth, have a look at the formal analysis of the motion of a pendulum at the Earth's surface.)

"Components" probably means the names for the different wavelength ranges, which are listed below. The atmosphere (including the water in it) absorbs many ranges, including nearly all of the high energy photons from ultraviolet to Xray. Water also absorbs in the infra-red. (UNSW Astronomers are currently testing sites at high altitude in Antartica: above most of the atmosphere, and in the driest atmosphere on Earth.)

For the names, wavelengths, frequencies and uses of the different bands, see the electromagnetic spectrum.

Australia is part of the international consortium for the Gemini telescopes. The Gemini home page is a good place to start. Go to the "Public Information" pages.

This is a very broad (and deep) question. Here are some starting points:
• it helps us understand the really big questions (e.g., how we got here, the long term future for life in the universe, are there likely to be other examples of life in the universe, is there any meaning to existence, etc).
• it helps us realise the possible threats to the existence of our civilisation (e.g., asteroid/comet collisions, gamma ray burst annihilation, random black holes colliding with the planet, the Sun running out of fuel, collision with the Andromeda galaxy, etc). It may even give clues as to how to avoid some of these events.
• there are technological spin-offs (e.g., the development of photography was accelerated through astronomical research; CCD cameras have been advanced through astronomy; the mathematics behind medical imaging techniques was developed by radio astronomers; the reason that new-generation Nokia phones don't have external aerials is due to a radio astronomer)
• having a rational understanding of the universe rather than a mystical one is probably a good thing for the long-term peaceful coexistence of humans
• knowing the insignificance of the earth and its inhabitants with respect to the universe is good for our perspective on life. Seeing an image of the earth from a distant spacecraft makes many people appreciate the fragility of the planet, and the fact that all humans on it have (in Carl Sagan's words) an important role in its stewardship. This should make us less likely to pollute or otherwise damage the environment.
(This answer contributed by Prof Michael Ashley.)

Cepheid variables---measuring longer distances

How can one tell how far away a star is? For close stars, you can use parallax. But because the Earth's orbit is small compared to interstellar distances, this doesn't work for most stars, or for galaxies.

Fortunately for astronomers and cosmoligists, there is a class of stars called Cepheid variables. These stars have brightness that varies periodically over time. Further, the period T of the oscillation in brightness is related to the total output power of the star. For a large, high power Cepheid variable, the period may be longer than a month. For a small, low power star, it may be days.

Once we know the relation between the period T and the light power P, we can determine how much light power P it puts out by measuring the variation in its brightness. If we know how much light it puts out and how bright it appears viewed from Earth, we can work out how far away it is. Here's how it works:

First, we look at Cepheid variables whose distances are known. Some of them are close enough to allow us to determine their distance r from parallax. From the intensity (I = power per unit area) of light received on Earth, we work out their power P from the inverse square law. Consider a sphere centred on the Cepheid variable. The area of the sphere is 4πr². All of the power P passes through this area, so the power per unit area is:

I = P/(4 π r²).
From these measurements on relatively close Cepheid variables, we construct the calibration curve of P as a function of T. Once we have this curve, then whenever we find a distant Cepheid, we use T to get P. Then, using the same inverse square law, we can work out the distance r.

Cepheid variables were first studied in the first decade of the twentieth century by Henrietta Leavitt, one of the first women to become famous in astronomy. She studied cepheid variables in one of the clouds of Magellan. There is little proportional difference in the distance from Earth to the stars in this galaxy, so she knew that the different apparent brightness was determined only by the power output. At the time, she could only use the cepheid variables for relative distances, because the parallax method was not sufficiently accurate.

We now know that the Cepheid variable cycle involves thermal feedback produced by the different ionization states of helium, which is relatively abundant in older stars. Doubly ionised helium is more opaque than singly ionised. If the star is hot enough to produce doubly ionised helium, this opaque layer insulates the star, making it hotter still. As the temperature rises it expands, but this expansion cools it, so the helium captures an electron and becomes less opaque, which continues the cooling. Cooling causes it to contract, which raises the temperature, and the cycle continues.

The Cepheid variable method works for distant stars in our own galaxy, and it also works for 'close' galaxies such as the Magellanic clouds. However, for distant galaxies, we can no longer distinguish individual stars. In these cases, various other methods are used. For instance, we can estimate the power of the whole galaxy (eg the brightest galaxy in a cluster) and use that to infer the distance from the inverse square law. One type of supernova (the exploding white dwarf star) provide another method, because the stage at which they explode depends on their size, and so they do not vary much in intrinsic brightness.

THE EXPERIMENT. Black body radiation is (by definition) radiation in thermal equilibrium with its container. So let us think of a box with hot walls giving off and receiving photons at an equal rate. If we open a small window in the box, the radiation coming out will have the same spectrum, and this is what we measure with our spectrometer to determine the black body radiation spectrum. The spectral radiancy is given by Planck's radiation law, which was initially empirical.



On the left is a sketch of the experimental apparatus used to observe black body radiation. An object of controlled temperature T contains a cavity, joined to the outside by a small hole. If the hole is very small, the radiation in the cavity comes to equilibrium with the walls. The hole allows a small fraction of the radiation to pass to a spectrometer. On the right is a plot of Planck's radiation law for two temperatures. Note that the wavelength for maximum emission becomes shorter (higher frequency) for higher temperature. Note also the strong dependence on temperature of the total emission. The radiancy is the power emitted per unit area per increment of wavelength and so has units of W.m^-3.

Note that the peak of the curve moves to the left as the temperature increases: hotter objects output a larger fraction of their electromagnetic radiation at shorter wavelengths. This displacement of the peak of the curve is called Wien's displacement law. After taking the derivative of Planck's radiation law and setting it to zero, one finds an expression for the wavelength λ_max at which the radiation is a maximum. It is related to the temperature T of the black body by the simple equation

λ_max = (2.9 x 10^-3 mK)/T.
THE QUESTION. What we want to know is the distribution of energy among the photons in the box. To do this properly would require a bit of quantum mechanics and statistical mechanics. But your question is simpler: why does the distribution go from zero at very low frequency to a peak at some frequency and then back to zero at high frequency?

AT LOW FREQUENCIES, the wavelengths are long. There are relatively few ways in which standing waves can fit into the box if their wavelengths are long. So, if the energy is shared among the different possible standing waves, we should expect the radiation intensity to go to zero at low frequency and to increase with increasing frequency. This is the classical result.

But now let's add the quantum hypothesis, that radiation energy is quantised and comes in photons that have an energy hf. In thermal equilibrium, the atoms and electrons of the wall have thermal motion. At sufficiently high frequency, few atoms or electrons will have enough energy to emit a photon with energy hf. Note how this depends on the quantisation hypothesis: if you could have radiation with frequency f and an energy less than hf, then the walls would emit high frequencies like crazy. They don't because they have to emit only whole photons.

So, at low frequency, the long wavelength limits the number of possible photons. At high frequency, the high energy makes emission unlikely. So the distribution goes to zero at both f = 0 and f = infinity, and has a maximum in between.

Incidentally, the frequency for maximum energy is proportional to the temperature of the walls. Wien's Law. The typical energy of any motion is kT/2, where k is Boltzmann's constant and T is the temperature. (k = R/N_A where R is the gas constant and N_A is Avagadro's number.) Here are some technical links from Uni of Virginia and the Heriot Watt University. There's a nice essay by by G. Pattison, a high school student, at Black body radiation. Some pictures taken with a thermal imaging camera are at Thermal physics and how clothes work.

See also The electromagnetic spectrum.

Max Planck used quantisation to explain black body radiation. His biography, courtesy of the Nobel Foundation, has some of the details.

The photoelectric effect refers to the emission of electrons from the surface of a conductor (usually a metal) by incoming electromagnetic radiation.



A clean metal surface, often in vacuum, is exposed to light whose wavelength or frequency may be controlled. A nearby electrode can receive the emitted electrons and so allow a current, which may be measured. A variable potential difference may be applied to stop the current. For a given metal, there is a minimum frequency f_o (ie a maximum wavelength) of light that will cause electrons to be emitted. For light (or UV radiation) with higher frequency, the stopping voltage increases linearly with frequency f, as sketched. More reactive metals have lower f_o.

This phenomenon was investigated experimentally by Philipp Lenard and then later and more precisely by Robert Millikan. Both received Nobel prizes for the work. The photoelectric effect was explained by Albert Einstein in work for which he received the Nobel prize in 1921.

This would be stretching the usual meaning of the phrase 'photoelectric effect', which usually refers to the interaction of a photon and an electron in a metal to produce an electron no longer bound by the metal. Probably it is a misprint for 'photovoltaic effect', which is used in breathalysers, solar cells and photocells.

Photocells come in different types. Some are photovoltaic cells, some are phototransistors. In a phototransistor, the base of the transistor is exposed to light. Because photons can produce electrons in this region, the input light effectively replaces the base current (the input current in a normal transistor). So the output current of the transistor is determined by the light input. (See the photovoltaic effect and transistors.)

One type of breathalyser uses a chemical reaction involving alcohol to change the colour in an indicator. (The only one that I've ever been able to examine worked that way.) Some use the infrared spectrum of alcohol. They work like this:

Light of known spectrum passes through the breath and then through optical filters that respond to different wavelengths onto a photocell. Thus (part of) the spectrum of chemicals in the breath, including that of alcohol, if present is measured.

The photovoltaic effect is involved in the photocell. Go to Solar cells and the photovoltaic effect

As explained above, this may be a confusion between 'photoelectric' and 'photovoltaic'. Solar cells are usually semiconductors. A photon of light interacts with an electron and transfers it to a state with a higher electric potential. This creates an emf: the electron with higher potential can flow back to its original state via the external circuit and thus do work. Go to Solar cells and the photovoltaic effect.

By thinking that electrons behave like waves, how does it help to explain that the accelerating particle does not give out energy?

In the 'solar system' model of the atom, a particle-like electron travels in a circular orbit around the atom. There are different circles for orbits with different energy. Travelling in a circle, it would be accelerating (centripital acceleration) and so would radiate. In quantum mechanics, the atom has an electron wave. The wave is a bit like a standing wave in a string (see waves and strings), except that it is three dimensional. It is going nowhere. No acceleration, so no radiation. Different energy orbits have different waves, most of them have nodes, just like the waves in a string, except that in three dimensions nodes are surfaces, not points.

Now in a string, the wave is in the displacement of the string. What is it that waves in an electron wave? The quantity is called ψ, the Greek letter psi. If you take magnitude of ψ at any place and square it, you get the probability of interacting with the electron at that point. So the atomic nucleus is pictured (especially in chemistry text books) with clouds of 'electron probability' around it in different orbitals.

The interpretation of ψ as a function of position and time is a subtle question. In the case of the atom, ψ is a standing wave. In other cases, like the ψ for electrons in a cathode ray tube, ψ has the form of a travelling wave.

Is an electron a particle or a wave? Another subtle question. It can have wave properties (eg a wavelength) and particle properties (eg a position). However, it cannot be a 'good' wave and a 'good' particle at the same time. Because of Heisenberg's Uncertainty Principle, a precise measurement of the wavelength implies a poor measurement of position, and vice versa. So an experiment in which wavelength is controlled precisely will give you wave effects (such as interference), but the electrons will not be localised in space. Conversely, if you constrain the position, you have an uncertainty in the momentum and wavelength. Some people say that little things like electrons are 'wavicles'. I prefer to say that they are neither wave nor particle, and that these macroscopic ideas are misleading when applied to electrons.

A magnetic field exerts a force on a moving charge. Hence, magnetic fields can be used to bend a beam of electrons. This bending is called electron refraction. Suitable geometry can be used to focus the beam.

We usually talk about the resolution of microscopes, and the resolving power of diffraction gratings and spectrometers. The resolution of a microscope is limited by the wavelenght it uses. The wavelength of light is typically half a micron, so the optical microscope cannot do much better than this size (though clever techniques can improve it noticeably, eg confocal microscopes.)

The wavelength of the electron beam is inversely proportional to the momentum of the electrons, and therefore inversely proportional to the square root of the accelerating voltage. By using high voltages, one can make the wavelength of atomic size or even smaller. Electron microscopes use this principle. Here is a link to some cool pics taken with a scanning electron microscope. It also has links to how it works.

(See also the separate page on this topic.) Heisenberg's Uncertainty Principle follows from a classical result, which is at least as old as Fourier. I prefer to introduce it as the Musician's Uncertainty Principle. When musicians tune up, we listen to the note for a long time so that we can adjust the frequency precisely. We tune by removing beats. (See What are interference beats?) If the frequency difference is one Hz, then you hear an interference beat every second. So, roughly speaking, if the frequencies differ by Δf, then you need a time of 1/Δf to notice. In other words:
Δf.Δt > ~ 1

(time taken to measure f) times (error in f) is about one or greater.

Musicians know this: if the chord is short, or if you are playing a percussive instrument, the tuning is not as critical. In a long sustained chord, you have to get the tuning accurate.

Now in quantum mechanics and atomic physics, the energies of photons are hf, where h is Planck's constant. So in order to know the energy, you have to take a certain time to measure the frequency. Mutliplying our previous inequality by h on both sides gives us

Δ(hf).Δt > ~ h

(uncertainty in energy) times (uncertainty in time) is greater than about h.

which is one example of the Heisenberg uncertainty principle. Applying it to spatial frequency (number of cycles per unit distance) rather than temporal frequency (number of cycles per unit time) gives
Δp.Δx > ~ h
(uncertainty in momentum) times (uncertainty in position) is greater than about h.

Werner Heisenberg won the Nobel prize in 1932.

Because h is so small (6.63 10^-34 Js), the consequences of the uncertainty principle are usually only important for photons, fundamental particles and phonons. There are, however, many physical processes whose evolution with time depends sensitively on the initial conditions. (Sensitivity to intial conditions is fashionably called chaos.) The uncertainty principle prohibits exact knowledge of initial conditions, and therefore repeated performances of such processes will diverge. (Physicists will also tell you that one cannot have exact knowledge anyway, for a variety of practical reasons, including the fact that you don't have enough memory to record the infinite number of significant figures required to record an exact measurement.)

Some philosophers regard the consequences of the uncertainty principle as having a more fundamental importance. The argument goes like this: if one could know exactly the position, velocity and other details, one could, in principle, compute the complete future of the universe. Since one cannot know the position and momentum of even one particle with complete precision, this calculation is impossible, even in principle. Most scientists find this a trivial argument. A memory capable of storing all this information would be as complex as the universe, and then the contents of that memory would have to be included in the calculation, and that would make the amount of information greater, and that information would have to be stored..... We rather point out that all of that information is actually contained in the universe which, as an analogue computer, is computing its own future already.

(See also Heisenberg's uncertainty principle and the musician's uncertainty principle, which has some demonstrations.)

Return to top of page and menu

Pauli's exclusion principle states that electrons (and other fermions, ie other particles having half integer spins) cannot have the same set of quantum numbers.

The important consequence of this is that 'electron shells' in the atom become filled. Once the innermost orbital is filled by two electrons having the same quantum number except for spin, then no further electrons can have that same state. So higher energy orbitals become occupied. This gives rise to chemistry.

Wolfgang Pauli won the Nobel prize in 1945.

To deal with the Zeeman effect properly requires a level of quantum physics that very few high school students will have. I'm saying this because what I shall do next is discuss it improperly, using a picture rather like Bohr's and Sommerfeld's. Strictly, it is not appropropriate to think of an electron orbiting a nucleus, because this involves us imagining it as localised in space and having a position that is a function of time. This ideas we now know to be misleading and false. However, they are a nice picture, and contain some of the essential ideas, as does the Bohr-Sommerfeld picture.



Imagine a charge in a circular orbit around a charge of the opposite sign. This moving charge can be considered as a current, so we now have a little electromagnet, with a magnetic dipole moment μ, as shown in the diagram. When we calculate its energy, we would put in an electrostatic term and a kinetic term, as in the Sommerfeld-Bohr picture. (See notes on Bohr and the hydrogen spectrum.) These terms do not depend upon the direction of the motion and, in the absence of an external magnetic field, there is no term for magnetic energy. However, if there is an external magnetic field, then there is another energy term: the magnetic potential energy will be lower if our electromagnetic is aligned with the field, and higher if antiparallel. So this last term has different sign for charges moving in clockwise or anti-clockwise direction. Physicists say that the energy level is *split*. New energy levels means a new spectrum. This splitting is explained by the simple classical model given here, and is also explained by the quantum mechanical model. However, the Rutherford atom, which did not have either electron orbits or wave functions, does not explain it. See the links on the Zeeman effect from CERN, from Thomson Learning, and from John Hopkins University.

How did Chadwick and Fermi's work change our understanding of the atom?

I am hoping that a historian will visit this site and help out with this side of things. Until he does, here is some inexpert help from a physicist who is not a historian.

Chadwick (see links below) is credited with discovering the neutron. Without neutrons, hydrogen would be the only stable element, which would simplify chemistry considerably, but simultaneously eliminate chemists. The neutron is not only important for holding the nucleus together, but also as a projectile for smashing into the nucleus (it is not deflected by electrostatic forces). It can be used as a probe of the nucleus. Neutron bombardment is important in chain reactions.

See Chadwick and Fermi courtesy of the Nobel foundation.

Fermi and Dirac devised the statistics to describe particles (like the electron) that are subject to Pauli's exclusion principle: they cannot have the same quantum numbers. Such particles are called fermions (cf bosons). In the atom, these statistics explain the distribution of electrons among different states or orbitals.

These statistics are also fundamental in solid state physics and thus to its applications in the electronics and computing industry.

The Bohr-de Broglie-Sommerfeld atom and the Hydrogen spectrum

Can anyone could explain to me how Bohr's postulates led to the development of a mathematical model to account for the existence of the hydrogen spectrum.

Here is short explanation of the Bohr-de Broglie-Sommerfeld model and the expression for the lines of the hydrogen spectrum. I repeat my caveat: I don't know much about the history of who did what when and am hoping to obtain assistance from a historian. However, the derivation below is worth reading because it is a good example of physics. It uses a few physical postulates, which are based on generalisations of experiments. It is mathematical and quantitative. It delivers quantitative predictions that may be readily compared with new experiments. This derivation requires only simple algebra, so it can be followed by high school students.

In the 19th century, a purely empirical formula was known for the hydrogen spectrum: it gives the frequency f of wavelength λ of EM radiation either absorbed or emitted by the hydrogen atom. It is

f = c/λ = const(1/n² − 1/N²)
where the constant was an empirical constant, and n and N are integers.

Bohr's name is associated with the 'solar system' model: the proton is so massive that its motion is neglected (cf sun) and the electron 'orbits in circles' (cf planet). I have put the scare quotes on 'orbits in circles' because we now know that this phrase is not merely untrue, but doesn't have a meaning for an electron.

In this classical model, the electron energy E is kinetic plus electric potential

E = (1/2)mv² − ke²/r
where m is the electron mass, v its (tangential) velocity, k is Coulomb's constant, and r the 'orbit' radius. But the centripital force F is provided by the electrostatic attraction
mv²/r = F = ke²/r²    (1)
so substitution gives the classical result (exactly analogous to planetary mechanics):
E = − (1/2)ke²/r.    (2)
Now we introduce de Broglie's contribution. In the 19th century, classical electromagnetism (Maxwell's equations) gave the momentum p of light as
p = E/c.
Using Einstein's quantisation of energy E = hf, we get the momentum of a photon
p = h/λ
where λ is the wavelength of the light. de Broglie speculated that this formula could hold for an electron also. Now the electrons in the H atom have sufficiently low energy that we may neglect relativistic effects, so de Broglie's speculation gives us for the electron:
mv = h/λ.
Now if the electron wave is to give constructive interference in a circular orbit, one requires that an integral number n of wavelengths make up a circumference, so
2.π.r = nλ = nh/mv,    whence

v = nh/(2.π.r.m)    (3)

Let's now solve (1) and (3) for r, and substitute in (2) to get the energy of the electron. Combining (1) and (3) gives
1/r = (4.π².k²e²m)/(n²h²)
and substituting in (2) gives
E = − (2.π².k²e⁴m/h²)(1/n²)
All the messy first factor is now seen to be the empirical constant. So the energy E of an electron in an orbit that has n waves around the circumference is
E(n) = −constant/n²
Now in this model, orbits with non-integral values of n are forbidden, because they correspond to destructive interference of electrons. So energy can only be absorbed or given to photons whose energy is E(n) − E(N) where n and N are both integers. So we can write the wavelengths λ of the emitted or absorbed spectrum as
1/λ = ΔE/hc = − R(1/n² − 1/N²)
where R, called Rydberg's constant, has the value of 100 fm⁻¹.

The sketch at left shows the relative radii of the 'orbits' for the first four values of the integer n. The diagram at right shows some of the possible transitions from higher energy level to lower.



Now if N = 1, the set of lines with different n are called the Lyman series. N = 2 gives the Balmer series and N = 3 the Paschen series. (Our historian will explain why.) For example, the lowest energy (longest wavelength) in the Paschen series has

1/λ = − R(1/4² − 1/3²).
So, from simple postulates established under other circumstances, and with a bit of logic and mathematics, we have derived a key result that may be checked by the very precise science of spectroscopy. This is what physics is about: using physical and mathematical arguments to explain the world in a quantitative way.

n-type semiconductors are 'doped' with a small percentage of atoms that have an extra valence electron. These extra electrons can be considered to provide most of the charges available for conduction. The electrons, being negatively charged, move in the direction opposite that of the electric field (see drift velocity).

p-type semiconductors are 'doped' with a small percentage of atoms that have the capacity to accept an extra valence electron. This can be thought of as an electron hole. Further, the hole can move: if an electron from a neighbouring atom enters the hole, it leaves a hole next door, so the hole appears to have moved. And since it was a negatively charged electron that moved to fill it, the movement of the hole is effectively the movement of a positive charge. In p-type semiconductors, one can think of the current being carried by positively charged electron holes, moving in the direction of the electric field (see drift velocity).

This is convenient as a way of thinking, although one can say that it is really the electrons that are moving. Here is a good analogy: take a sealed bottle of water and invert it. You will see the bubble of air move upwards through the water. Now of course you know that what is really happening is that water is flowing down into the bubble and leaving a hole in the water where it has come from, so really what you are watching is water motion. But because there is lots of water and only a small bubble, it is easier to think of a moving 'hole' in the water than to consider the motion of the water.

Amplification. The circuit below left allows you to apply a small, variable voltage or current to the base of npn transistor via the (large) resistor R_base. Raising this voltage or current (by decreasing R_base) turns the transistor 'on', i.e. it allows more current to flow from collector to emitter (= more electrons from emitter to collector). A bigger current in the collector means a bigger voltage drop across the (smaller) resistance R_collector. So a small increase in the input voltage (voltmeter at left) causes a large decrease in the output voltage (voltmeter at right). Thus this is a (very simple) inverting amplifier.

Circuit diagrams for a simple amplifier (left) and a logic gate that performs the NOR operation.

Logic operations. In the ciruit at right, let's consider digital signals, ie voltages that we consider only as 'high' (1) or 'low' (0). If either A or B is high, there will be a high base current, so the transistor will be on, lots of current will flow through the load resistor at right, so the output voltage will be low. A and B both high gives the same result. The only way to turn the transistor off (and so obtain low collector current and high output voltage) is if neither A NOR B is on. For that reason, this circuit is a NOR gate. Its output is 1 if neither A NOR B is 1, and 0 otherwise.

One can also use transistors and resistors to construct simple gates for the logical operations NOT, AND, OR, NAND (= not and) and XOR (exclusive or). Alternatively, one can construct any of these from a combination of NOR gates. By putting together combinations of gates, one can easily construct memory, arithmetic and more complicated operations. Although this material is usually taught in a laboratory course normally taught in second year, it is possible to do some experimental work yourself, if you are keen. The School of Physics at UNSW has an interactive course in digital electronics set up as a series of panels in the corridor. By manipulating knobs and switches you can do a series of experiments, starting with resistors and transistors and ending with the elements of a computer--it is a small, self-contained experimental course in digital electronics. Small groups would be welcome to visit by arrangement (info@phys.unsw.edu.au).

For versions of the history of the invention of the transistor, see American Institute of Physics version or the Bell Labs' version and Time Magazine article on Shockley.

Simple logic circuits can be made with resistors and diodes, but for complicated circuits one needs an amplifying component. Which generally means valves or transistors.
• Thermionic valves are much bigger than transistors. The volume of a valve varies from several ml to litres. They are usually expensive and unreliable compared to transistors. Part of their unreliability was due to temperature: to eject electrons, the electrodes had to be heated, and thermal deterioration is difficult to avoid. Another part was due to the need to maintain a vacuum, usually inside a glass capsule sealed to a metal base. (Large size, cost and the need to replace from time to time is not such a problem in very high power applications, and so valves still have uses in such things as radio transmitters.) Computers (Univac, CSIRAC etc) using thermionic valves were very large, slow, expensive and unreliable by modern standards. (The distortion produced by valve amplifiers is different to that prodcued by transistor amplifiers. Some rock and roll musicians, who like distortion, prefer the distortion of valves. Further, valve amplifiers make an electrical noise when you hit them hard enough. So there is still a market for valve amplifiers.)
• Transistors are usually small. When you buy a single transistor, the package and wires are usually much bigger than the device. (This is often because a big package and wires are necessary to get rid of the heat produced in the transistor.) Physically big transistors (ie with packages more than several mm across) are used for applications where substantial amounts of power (several watts or more) is required. Transistors are not good for very high power applications, in part due to thermal runaway. The conductivity of semiconductors increases with temperature, this causes them to conduct more current, which produces more heat, which raises their temperature.... Unless you can get rid of this heat, it leads to ex-transistors.
• Microchips and microprocessors are impossible without transistors (though in the future they may have optical or quantum components instead). In logic circuits, one needs hardly any power, and the minimum size for transistors may be determined by the lithography that is used to make them. (Ultimately it is determined by quantum effects and thermal noise.) So it is possible to put many millions of them together on a single wafer of silicon to make a chip. So transistors made microchips, microprocessors and personal computers possible. For the impact of microchips and microprocessors on society, look around you.

The US PBS has a website associated with their documentary Transistorized. It has a good basic review of the science involved, but much historical detail (including a lot of interesting things about the various personalities involved).

Another possibility is the Nobel prize website. The 2000 prize in Physics was awarded to Alferov, Kroemer and Kilby for contributions to the foundations of much of modern electronics and infomation technologies. The resources on this site range from the somewhat technical (each awardee gets to write a scientific article for reviews of modern physics on their work), through to their acceptance speeches and so forth, right down to the basic including graphics-heavy descriptions of their work, and even an online game for kids to learn about the various prizes (see , which relates to Kirby's work on the integrated circuit).

The more recent the prize, the more public stuff they have on the website, but you can go right back to Bardeen, Brattain and Shockleys prize in '56 for the transistor, and some of the greats like Einstein, Heisenberg and co.

A photon of light interacts with an electron via the photovoltaic effect and transfers it to a state with a higher electric potential. This creates an emf: the electron with higher potential can flow back to its original state via the external circuit and thus do work. Solar cells are usually semiconductors. In solar cells, a photon interacts with an electron, but does not remove it from the material (as it does in the photoelectric effect). Rather, the energy hf of the photon increases the potential energy of the electron by eV, and so produces a potential difference which has an upper limit of hf/e (which cannot be reached in practice because of the thermal effects). The electron with higher potential energy (lower voltage) can then flow through the external circuit and do electrical work (e.g. charge a battery). This is explained in more detail on an introductory page on the photovoltaic effect from the UNSW Centre for Photovoltaics, which produces the world's most efficient silicon solar cells.

This question is mainly about the taxonomy of matter. Taxonomy is sometimes rather arbitrary, but let's put in some physical insight. We begin with the four commonly defined states of matter, which, ranked by increasing temperature, are:
solid, liquid, gas, plasma.
We can differentiate these by considering the thermal energy---a measure of the typical energy of a molecule or mole due to its thermal motion. In molecular terms, the thermal energy is kT (where k is Boltzmanns constant and T is the aboslute temperature) and in molar terms it is RT (where R is the gas constant). Let's compare this with the energies of intermolecular interaction (U) and the energy of ionisation or work function (W).

Roughly speaking, molecules with lots of thermal energy escape from their neighbours and evaporate. Those will little energy stay close to the same neighbours (solid). Those with intermediate energy can exchange neighbours but not escape from them (liquid). Finally, if molecules or atoms have enough thermal energy, their collisions may remove electrons and form a plasma. So:

solid: kT <~U
liquid: kT ~ U
gas: kT >~U
plasma: kT >~W
One could further subdivide solids into crystals (solids with regular structure), glasses (solids with much less regular structure) etc. Do these count as different states of matter, or are they different forms of solids? It's a semantic question.

The ranking above neglects pressure (and only considers entropy implicitly). If we make the pressure low, we can get gases at very low temperature. Solids will sublime at low pressure (CO2 does at atmospheric pressure). Very wide ranges of pressure and temperature can deliver some exotic states of matter:

Bose-Einstein condensates. At very low temperatures and pressures, Bose-Einstein condensation is sometimes possible. This occurs when the wave properties of atoms or molecules become important. The Heisenberg's uncertainty principle imposes a constraint here: we need the thermal momentum so slow that the error in position of the atom/molecule is large enough for several of them to be superposed. What happens then depends on the overall spin of the atom or molecule. If they are bosons, then one may have several or many atoms all with the same quantum numbers and all confined to the same region in space. They are indistinguishable. There's a question about them below.

I think that this is best introduced by an analogous concept: it's what surfers might call a 'set' of waves. Suppose a surfer's 'set' has 5 waves, with wavelength of 30 metres. This means that that particular 'set' will not interfere with another 'set' that is more than 150 metres away. (Physicists would refer to a wave train, or wave packet and its coherence length--and they would object to the very simplified arm-waving explanation I'm giving here!)

Now the wavelength of a massive particle (the de Broglie wavelength) is inversely proportional to its speed. And the speed of an atom goes as the square root of its temperature. At room temperature, the wavelengths of atoms are so small (smaller than the atom itself) that interference effects are negligible. When you cool them enough, however, their speeds become slower and their wavelengths longer. And, if you can confine a number of similar atoms in a small volume, they can begin to interfere.

This (roughly speaking) is what happens in a condensate. If you have a collection of particles with the same quantum numbers, and if they are 'in the same place', then they are indistinguishable. This indistinguishability leads to some peculiar statistics, first worked out by Bose and Einstein.

We don't notice this at normal temperatures, because atoms are not 'in the same place', which here means being close enough and having long enough de Broglie waves for interference to occur.

news item on BECs
NIST site on BECs

They certainly have properties that are very different from those of solids, liquids, gases & plasmas. Is that enough to be called a new state of matter? This is a semantic question.

Neutron stars. At exceedingly high pressures and temperatures, electrons and protons can combine to form neutrons. This happens in some massive stars when they cool enough for their gravity to squeeze the atoms into ultra high density. A whole, star bigger than the sun, becomes something rather like a gigantic atomic nucleus, several km in diameter, containing only neutrons.

Dark matter. Most of the matter in the universe is not in stars, as was thought until recently. Because it's not in stars, this matter doesn't shine, whence 'dark matter'. Planets, comets and dust are all dark matter, but these are thought to be much less massive than their stars, and so negligible in cosmology. Many cosmologists believe that most of the matter (or perhaps energy) in the universe exists in large 'clouds' around galaxies called MACHOs (MAssive Compact Halo Objects). Many others believe it is in undetected particles called WIMPS (Weakly Interacting Massive Particles.). Do the WIMPS count as a new state of matter? Probably not, but there maybe news on this subject soon.

Exotic matter. There are a number of types of matter so unusual that they are only encountered in high energy laboratories, or perhaps in extreme conditions that may exist or have existed in the universe. Antimatter is in principle very much like ordinary matter. Antihydrogen consists of a positive antielectron (a positron) and a negative antiproton. As far as we know, it has the same spectrum as hydrogen. It is very much harder, but in principle possible, to make antimatter versions of heavier elements. Antimatter has the disagreeable habit of annihilating ordinary matter to make lots of E=mc² energy in the form of gamma rays.

It is also possible to make 'atoms' (often very short lived) using other combinations of positive and negative particles, such as a proton and a muon or an electron and a positron. Further, whereas the neutron and proton are made only of up and down quarks, it is in principle possible to make much more massive hadrons using less common quarks. When we think of matter, we usually think of lots of atoms together, rather than a single briefly existing entitiy. Are there places in the universe hot enough to make such exotic matter in quantity? I don't know, but if there are, it is nice to be a long way away from them!

A conducting solid, such as a metal, contains electrons which can move relatively freely through the background of positively charged ion cores. As an electron moves it exerts a Coulomb attraction on neighbouring ions and will distort the lattice structure locally. This is referred to as 'electron-phonon interaction'. In some materials, at sufficiently low temperatures, this effect can lead to a dynamic pairing between electrons. This is believed to the mechanism behind superconductivity in most, but not all, superconductors.

(The answers in the phonon and lattice distortion section were supplied by Prof Jaan Oitmaa.)

The magnetic field does not penetrate into a superconductor. This is called the Meissner effect. This effect together with conservation of the magnetic flux provides the levitation of a magnetic object above a superconductor or a superconductor above the magnet.

When a magnet is brought near a superconductor, this exclusion of the magnetic field distorts the field lines, as shown in the diagram below.



There are some further subtleties in magnetic levitation discussed in www.calpoly.edu/~rbrown/levitation.html.

In order to be quantitative about it, we observe that the smaller the gap between the object and the superconductor, the larger the magnetic field in the gap: the field is compressed in this gap. We can calculate the magnetic pressure (a force per unit area), which at any point is equal to the energy density (energy per unit volume) of the magnetic field. This is given by:

p = B²/2μ_o.
where B is the field strength and μ_o is the magnetic permeability of space. As we bring the magnet closer to the superconductor (middle picture), the field lines must be closer together, so the field is more intense, so its energy density and magnetic pressure p rise. Mechanical equilibrium (levitation) is achieved when the force due to this magnetic pressure is equal to the weight of the object.

To estimate the size of this effect, let us consider a cubic iron bar magnet with side a = 1 cm. Let the mass of the magnet be 0.01 kg, and let the field near the pole, in the absence of any superconductor (picture at left), be B_o ~ 0.01 Tesla. Now let us put the cube at distance x from the superconductor. To do the geometry properly is a little difficult, but we can make some approximations.

In the diagram at left, the field from the pole diverges over a distance comparable with the size of the magnet, ie over ~a. In the middle diagram, it diverges over the distance x, so the field lines have been concentrated by a factor of about a/x, so the field between the magnet and the superconductor has a field strength

B ~ B_oa/x
For mechanical equilibrium, we want the force of the magnetic pressure acting on the pole of the magnet to equal the weight, so
mg = pa² = a²B²/2μ_o.
Rearranging gives an estimate for x, the levitation height:
You probably don't have liquid nitrogen to do the experiment, but you can make a similar measurement using two similar magnets and by taking advantage of symmetry. If you keep the magnets symmetrical while bringing similar poles towards each other, then (because the magnets are equally strong) no field lines cross the plane of symmetry. So the top half of the field diagram looks just like that of the levitation case.


So the obvious question is why you cannot levitate in the second case. Well, the problem is that the symmetry of the two magnets is unstable and you will need to supply horizonatl forces to keep them upright. With some ingenuity, you may be able to supply such horizontal forces without much vertical force. If you do, you can measure the distance at which the upper magnet is supported, and this is roughly twice the distance at which it would levitate over a superconductor. It is also possible to supply the horizontal forces using an array of magnets, and some 'executive' toys use this principle to levitate permanent magnets.

By the way, this trick of using two similar magnets in symmetry would also be an easy way to calculate the forces, rather than to estimate them as we have done above. We treat the superconductor as a 'magnetic mirror' and calculate the 'image forces' due to the mirror image of the magnet.

The two following properties are specific for superconductors:
• 1) Zero resistance to electric current;
• 2) Repulsion of the magnetic field (Meissner effect).
Actually a normal metal (non superconductor) also has zero resistance at zero absolute temperature. However a normal metal never manifests the Meissner effect. Thus the Meissner effect is the most important property of a superconductor.

The Meissner effect is a complex quantum phenomenon. It is due to the fact that electrons in the superconductor are in a quantum condensate described by a collective wave function for all electrons. For an introduction to quantum condensates, see Bose Einstein condensates.

This wave function has a very special property of rigidity: it requires a lot of applied energy to change the 'shape' of the wave function. (To discuss this correctly, we really do need complex mathematical operations, so this simplified discussion is not quite correct. and the shape we are talking about is a shape in Hilbert space.) As a result of this rigidity, the curl of the electric current density is proportional to the magnetic field. (The curl is a mathematical way of representing properties of the shapes of lines in a vector field, the current density in this case. It is a measure of the amount of twist in the lines.) Let us assume now that the magnetic field penetrates inside volume of the superconductor. Hence, because of the property mentioned above, the field induces currents inside the volume of the superconductor. Any current is related to some internal movement and hence to the kinetic energy related to this movement. Therefore such a state would have a very high energy. To minimize the energy the superconductor develops currents on the surface in such a way that they exactly compensate the magnetic field inside. In this case current flows only within thin 10^-8m surface layer and therefore energy of the system is relatively low. This explains the mechanism of the Meissner effect.

The Meissner effect is very important for condensed matter physics and it is equally important for elementary particle physics. The masses of the particles arise due to the effect that is very similar to the Meissner one because the physical vacuum is in the state similar to the state of a superconductor (Meissner state). However at early stages of the history the Universe was very hot, the vacuum was in the "normal" (non "superconducting") state, and all the particles were massless.

(The answers on levitation and the Meissner effect were provided by Prof Oleg Sushkov.)

There are a few different effects.

First, if you have higher conductivity, you can make thinner conducting elements. That makes chips physically smaller, signals travel smaller distances and that improves speed.

Second, with higher conductivity, you produce less heat. Getting rid of heat limits the size and sometimes the clock speed of processors. If you reduce that problem, you can pack large arrays of circuits, again making distances shorter. Instead of having two dimensional circuits (all the elements located at or very near a single, plane surface of semiconductor), you could have three dimensional ones (many parallel interfaces in a block), and that would much reduce the distances and increase the maximum number of components.

Then there are RC charging times. The junctions of a transistor have a capacitance C that is charged via the conductors (with a resistance R) leading to them. To charge a capacitor takes typically a time RC (ohm*farad = second). So low R means a faster switching time. See RC filters for more information.

By the way, a current set up in a superconductor circuit keeps flowing until you do something to stop it. This could in principle be used as a memory element.

Currently, these are almost entirely potential applications, so do not expect to find much information. For potential applications to computers, see the previous section.

There are also potential applications to motors and generators. The stationary magnets or stators in these devices are often electromagnets, in order to save weight or initial expense. (See Motors and generators for details.) If we could easily make these electromagnets superconducting, then we would save on the electrical power required to keep current flowing in them to maintain the field. This would make them more efficient. However, the insulation required to retain the liquid helium necessary for high current super conductors makes such systems large, heavy and expensive.

Power engineers dream of using superconducting cables for the transmission of electricity through power grids. About two thirds of the power generated by power stations is lost in the distribution network, including ohmic heating of the transmission cables. However, the prospect of cooling the distribution network is daunting. 'High' temperature superconductors (those that superconduct at liquid nitrogen temperatures) are not (yet) suitable for transmission cables because they cannot transmit high current density, they are usually not ductile or flexible.

(A little parenthesis about comparing electric cars with petrol cars. Although fossil fuelled power station generators are much more efficient than motor car engines, the distribution of electrical energy is much less efficient than the distribution of petrol, which tends to cancel out the efficiency of generation. However, power stations are less polluting than cars, and electric cars are still much more efficient than petrol cars. This is because electric cars are designed rationally for intelligent driving. Petrol cars are constrained by the need to protect the fragile egos of some male drivers and so are almost always vastly overpowered. In principle one could design efficient petrol cars but, because of the temporarily low price of oil, there is little incentive to do so.)

There are however some actual applications of superconductors. One relatively common example is in the large electromagnet used for the constant field in Magnetic Resonance Imaging (see MRI). Liquid helium cools the wire coils to allow the large currents required to maintain the large, uniform magnetic field efficiently. An interesting problem arises when one wishes to turn off the field and bring the magnet back to room temperature. The magnet is a large inductance (see AC circuits) with value L, carrying a large current i, and thefore storing an energy Li²/2. When the temperature starts to rise, resistance appears in the coils and the ohmic (i²R) heating would quickly dissipate all this magnetic energy as heat. Despite the presence of the liquid helium, there are difficulties in disposing of this energy safely, so the current must be gradually reduced to zero before the coils can be warmed.

In medicine, hard (high energy) X-rays are used to treat cancer. Because the cancerous cells divide more quickly than normal ones, they are more vulnerable, so the radiation kills the cancer cells preferentially. Sometimes the photons are delivered directly from an X-ray beam (e.g. breast cancer). In other cases, radioactive sources may be used.

In industry, radioisotopes are sometimes used as 'tracers': label a chemical by making one of its atoms radioactive, and you can trace where that chemical goes. They are also sometimes used to measure the composition of materials by measuring the amount of radioactivity absorbed. Radioactive tracers are used to identify organs and pathways for different chemicals. Positron Emission Tomography is also used to identify the distribution of different substances. (See also Medical physics.)

What are the links between high energy particle physics and cosmology?

Much astronomical evidence (particularly the galactic red shifts for which Hubble is famous and the ubiquitous 3 Kelvin background microwave radiation in the sky (Penzias and Wilson, more recently the cosmic background explorer COBE)) point to a 'Big Bang' about 13 billion years ago. At that time, the universe was extremely dense and hot. Let's imagine going backwards in time towards t =0.

As the temperature gets higher, the typical energy due to thermal motion (kT, where T is the absolute temperature and k is Boltzmann's constant) gets greater. When it becomes comparable with the ionization energy of hydrogen (kT ~ 10 eV), nuclei can no longer hold on to the electrons and so there are no longer atoms, just a plasma.

Keep going back in time, hotter and hotter. Eventually the protons are colliding with each other with the same sorts of energies produced in the big atom smashers (kT ~ GeV), and so all of the weird particles recorded at e.g. CERN are now present in the tiny, dense universe. To understand cosmological evolution at this stage, one needs to understand about these particles.

Hotter and hotter, and the difference between the different forces ceases to be evident: first electromagnetism and the weak force, then the strong force all become one (in the language of cosmology, they have not yet 'frozen out'). Some theoreticians think that in the early universe quantum gravity was as strong as the other forces.

Finally one gets back to the Planck length ((Gh/c³)^1/2 ~ 10⁻³⁵m) and the Planck time ((Gh/c⁵)^1/2 ~10⁻⁴³s), over which the spontaneous creation of transient black holes dominates proceedings. On this scale time and space cease to have a separate meaning, or even a meaning at all. Without time and space, cause and effect cease to have a meaning and so one of the Big Questions (How did it all start?) disappears. (The last sentence may seem glib, but I think that it is actually quite important and profound.)