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Last Updated: Friday, 8 October, 2004, 13:44 GMT 14:44 UK
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 Einstein makes the list
You could almost call it Equation Idol - readers of Physics World have voted for their favourite equations of all time. But what do they mean?

Deputy editor of Physics World, Dr Matin Durrani, offers an idiot's guide to the top five equations of all time.

1. (JOINT 1st) CLERK MAXWELL'S ELECTROMAGNETISM THEORY

∇.D=p

∇.B=0

∇xE=-∂B/∂t

∇xH= ∂D/∂t+j

Where D is the displacement field, E is the electric field, B is the magnetic-flux density, H is the magnetic-field strength, p is the free charge density and j is the free current density.

These were written down by the great Scottish physicist James Clerk Maxwell in 1873. They describe how an electromagnetic wave - like a light beam, an X-ray or a microwave - varies with time and position in space.

What is interesting about the equations is that they showed that electricity and magnetism - two forces that scientists previously thought were unrelated - are actually linked to one another. Since then, physicists have also gone on to link electromagnetism with two of nature's other forces - the "weak" and "strong" forces that act inside the nucleus of an atom.

The resulting theory is known as the Standard Model of particle physics. The big challenge is now to find out how nature's fourth fundamental force - gravity - is linked to this model. So Maxwell was essentially the first physicist to start unifying the forces of nature into a single theoretical framework.

What good is it to me? Maxwell's equations are used throughout the telecoms industry - for example, to design the antenna on your mobile phone

1. (JOINT 1ST) EULER'S EQUATION

ei p + 1 = 0

This was joint top with Maxwell's equations and was discovered by the Swiss mathematician Leonhard Euler in the 18th Century. Physicists like this equation because it has nine basic concepts of mathematics - once and only once - in a single equation.

These include, p which is the circumference of a circle divided by its diameter; i, which is the square root of minus one; and e, which is the number 2.71828. The other six concepts are: multiplication; plus; equals; one; zero; and the "exponent operation". The exponent operation describes what you do when you multiply a number by itself a certain number of times: two squared, for example, means 2x2, while two cubed means 2x2x2.

What good is it to me? None. Euler's equation is a purely mathematical construct with no obvious practical relevance, although it is what some physicists might call "beautiful".

3. NEWTON'S SECOND LAW

F=ma

This describes the fact that if you give a force, F, to an object with a mass, m, it will have an acceleration, a. It was derived by Isaac Newton in the late 17th Century and forms the basis of his second law of motion.

What good is it to me? Newton's second law could be used to work out how fast your flashy new Mini Cooper will accelerate from 0 to 60mph.

4. PYTHAGORAS'S THEOREM

a²+b²=c²

A classroom favourite, Pythagoras's theorem explains how the lengths of the sides of a right-angle triangle are related. If a and b are the lengths of the two shorter sides and c is the length of the long side, then all you need to do work out c is to add up the squares of the other two sides and take the square root of the answer. It was devised by the Greek scientist Pythagoras in the 6th Century BC.

What good is it to me? Pythagoras's equation helps in the process of "triangulation", which can pinpoint the location of someone using a mobile phone simply by bouncing signals off three different phone masts.

5. SCHRÖDINGER'S EQUATION

HΨ=EΨ

This was devised by the Austrian physicist Erwin Schrödinger in the mid-1920s. It describes how tiny sub-atomic particles like electrons behave and forms part of the theory known as "quantum mechanics".

With particles like electrons, it's impossible to say where exactly they are in space or how fast they're moving; all you can do is give them a probability of being in a certain place at a certain time. The symbol in the equation is called the "wave function" - it describes the probability of the particle being at different points in space.

What good is it to me? Schrödinger's equation has applications in electronics: it was, for example, used by a Cambridge firm called Quantum Beam to build a laser-based system to connect home computers to the internet without wires.

6. EINSTEIN'S EQUATION

E=mc²

This is Einstein's famous equation that shows that mass and energy are not separate but are actually related. What the equation says is that an object with a mass m has an energy E = mc², where c is the speed of light. Since c is so big - light moves at 300 million metres a second - even a tiny mass has a huge energy.

Equally, energy also has mass. You can expect to hear a lot more about the equation in 2005, which marks the 100th anniversary of its discovery by Einstein as part of his special theory of relativity. Events will be held around the world as part of what has been dubbed by the United Nations as the "International Year of Physics".

What good is it to me? E=mc² determines how much energy is generated when atoms are split in your local nuclear-power station.

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