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## A cylinder rolling down a slope

The other week I was asked to explain how a cylinder (or ball) rolling down a slope differs from e.g. a ball being dropped vertically. It is an interesting question, because it illustrates some things which are not immediately obvious. We all know that, if you drop two balls, say a tennis ball and a cannon ball, they will hit the ground at the same time. This is despite their having very different masses (weights). Galileo supposedly showed this idea by dropping objects of different weights from the tower of Pisa (although he probably never did this, see our book Ten Physicists Who Transformed Our Understanding of Reality).

With a tennis ball and a cannon ball, they clearly have very different masses (weights), but will fall to the ground at the same rate. This fact, contrary to the teachings of Aristotle, was one of the key breakthroughs which Galileo made in our understanding of motion. But, what about if we roll the two balls down a slope? If we build a track to keep them going straight, will a tennis ball roll down a slope at the same rate as a cannon ball? The answer is no, and I will explain why.

## Rolling rather than dropping

When a ball rolls down a slope, it starts off at the top of the slope with gravitational potential energy. When it starts rolling down the slope, this gravitational potential energy gets converted to kinetic energy. This is the same as when the ball drops vertically. But, in the case of the ball dropping vertically, the kinetic energy is all in the form of linear kinetic energy, given by

$\text{ linear kinetic energy } = \frac{ 1 }{ 2 } mv^{2}$

where $m$ is the mass of the ball and $v$ is its velocity (which is increasing all the time as it falls and speeds up). The gravitational potential energy is converted to linear kinetic energy as the ball drops; by the time the ball hits the bottom of its fall all of the PE has been converted to KE.

If, instead, we roll a ball down a slope, the kinetic energy is in two forms, linear kinetic energy but also rotational kinetic energy, which is given by

$\text{ rotational kinetic energy } = \frac{ 1 }{ 2 } I \omega^{2}$

where $I$ is the ball’s moment of inertia, and $\omega$ is the ball’s angular velocity, usually measured in radians per second. The key point is that the the moment of inertia for the two balls in this example (a tennis ball and a cannon ball) have a different value, because the distribution of the mass in the two balls is different. For the tennis ball it is all concentrated in the layer of the rubber near the ball’s surface, with a hollow interior. For the cannon ball, the mass is distributed throughout the ball.

## Two cylinders rolling down a slope

Let us, instead, consider the case of two cylinders rolling down a slope. One is a solid cylinder, the other is a hollow one with all of its mass concentrated near the surface. We will make the two cylinders have the same mass; this can be done by making the material from which the hollow cylinder is made denser than the material for the solid cylinder. So, even though the material of the hollow cylinder is all concentrated near the surface of the cylinder, and there is a lot less of it, if it is denser it can have equal mass.

A solid cylinder on an inclined plane. We will make the mass of this solid cylinder the same as that of the hollow cylinder, by making it of less dense material. Although it will have the same mass $m$ and the same radius $R$, it will not have the same moment of inertia $I$.

We will start both cylinders from rest near the top of the slope, and let them roll down. We will observe what happens.

A hollow rolling down an inclined plane. We will make the hollow cylinder denser than the solid one, so that they both have the same mass $m$ and the same outer radius $R$. But, they will not have the same moment of inertia $I$.

When things are dropped, the rate at which they fall is independent of the mass, but when they roll the rate at which they roll is not indpendent of the moment of inertia. In particular, it is not independent of the distribution of mass in the rolling object. As this video shows, the solid cylinder rolls down the slope faster than the hollow one!

But, why??

## Why does the solid cylinder roll down quicker?

The reason that the solid cylinder rolls down faster than the hollow cylinder has to do with the way that the potential energy (PE) is converted to kinetic energy. Because the cylinder is rolling, some of the PE is converted to rotational kinetic energy (RKE), not just to linear kinetic energy (LKE). The only way that a cylinder can roll down a slope is if there is friction between the cylinder and the slope, if the slope were perfectly smooth the cylinder would slide and not roll.

The torque (rotational force) $\tau$ is related to the angular acceleration $\alpha$ in a similar way that the linear force $F$ is related to linear accelerate $a$. From Newton’s second law we know that $F = ma$ where $m$ is the mass of the object. The rotational equivalent of this law is

$\tau = I \alpha$

where $I$ is the moment of inertia. The moment of inertia $I$ is different for a hollow cylinder and a solid cylinder. For the solid cylinder it is given by

$I_{sc} = \frac{ 1 }{ 2 } mR^{2} = 0.5mR^{2}$

where $m$ is the mass of the cylinder and $R$ is the radius of the cylinder. For the hollow cylinder, the moment of inertia is given by

$I_{hc} = \frac{ 1 }{ 2 }m(R_{2}^{2} + R_{1}^{2})$

where $R_{2} \text{ and } R_{1}$ are the outer and inner radii of the annulus of the cylinder. We are going to make the hollow cylinder such that the inner 80% is hollow, so that $R_{1} = 0.8R_{2} = 0.8R$. We will make the mass $m$ of the two cylinders the same.

Thus, for the hollow cylinder, we can now write

$I_{hc} = \frac{ 1 }{ 2 }m(R^{2} + (0.8R)^{2}) = \frac{ 1 }{ 2 }mR^{2}(1+0.64) = \frac{ 1 }{ 2 }mR^{2}(1.64) = 0.82 mR^{2}$

The cylinder accelerates down the slope due to the component of its weight which acts down the slope. This component is $mg sin(\theta)$ where $g$ is the acceleration due to gravity and $\theta$ is the angle of the slope from the horizontal. To make the maths easier, we are going to set $\theta = 30^{\circ}$, as $sin(30) =0.5$.

Friction always acts in the opposite direction to the direction of motion, and in this case the friction $F_{f}$ is related to the torque $\tau$ via the equation

$\tau = F_{f}R \text{ (1)}$

so we can write

$F_{f}R = \tau = I \alpha \text{ (2)}$

where $\alpha$ is the rotational acceleration. Re-arranging this to give $F_{f}$, we have

$F_{f} = \frac{I \alpha}{ R }$

The force down the slope, $F (=ma)$ is just the component of the weight down the slope minus the frictional force $F_{f}$ acting up the slope.

$ma = mg\sin(30) - F_{f} = 0.5mg - \frac{ I \alpha }{ R } \text{ (3)}$

The angular acceleration $\alpha$ is given by $\alpha = a/R$ where $a$ is the linear acceleration. So, we can re-write Eq. (3) as

$ma = 0.5mg - \frac{ Ia }{ R^{2} } \text{ (4)}$

Now we will put in the moments of inertia for the solid cylinder and the hollow cylinder. For the solid cylinder, we can write

$ma = 0.5mg - \frac{ 0.5mR^{2}a }{ R^{2} } = 0.5mg - 0.5ma$

The mass $m$ can be cancelled out, and assuming $g=9.8 \text{ m/s/s}$, we have

$a = 0.5g - 0.5a \rightarrow 1.5a = 4.9 \rightarrow \boxed{ a = 3.27 \text{ m/s/s (5)} }$

Notice that Equation (5) does not have the mass $m$ in it, as this cancels out. It also does not have the radius $R$ of the cylinder in it; the acceleration of the cylinder as it rolls down the slope is independent of both the mass and the radius of the cylinder.

For the hollow cylinder, again using Eq. (4), we have

$ma = 0.5mg - \frac{ 0.82maR^{2} }{ R^{2} } = 0.5mg - 0.82ma$

This simplifies to

$a = 4.9 - 0.82a \rightarrow 1.82a = 4.9 \rightarrow \boxed{ a = 2.69 \text{ m/s/s} (6)}$

As with Equation (5), Equation (6) is independent of both mass and radius.

So, as we can see, the linear acceleration $a$ for the hollow cylinder is 2.69 m/s/s, less than the linear acceleration for the solid cylinder, which was 3.27 m/s/s. This is why the solid cylinder rolls down the slope quicker than the hollow cylinder! And, the result is independent of both the mass and the radius of either cylinder. Therefore, a less massive solid cylinder will roll down a slope faster than a more massive hollow one, which may seem contradictory.

## Summary

All objects falling vertically fall at the same rate, but this is not true for objects which roll down a slope. We have shown above that a solid cylinder will roll down a slope quicker than a hollow one. This is because their moments of inertia are different, it requires a greater force to get the hollow cylinder turning than it does the solid cylinder. Remember, the meaning of the word ‘inertia’ is a reluctance to change velocity, so in this case a reluctance to start rolling from being stationary. A larger moment of inertia means a greater reluctance to start rolling.

The solid cylinder will start turning more quickly from being stationary than the hollow cylinder, and this means that it will roll down the slope quicker. This result is independent of the masses (and radii) of the two cylinders; even a less massive solid cylinder will roll down a slope quicker than a more massive hollow one, which may be counter-intuitive.

## Five top facts about Jupiter – no. 4

Continuing my blogs on the five top facts about Jupiter which were posted as a tweet during my appearance on BBC Radio 5’s morning show a few weeks ago, at number 4 in my list was –

It [Jupiter] has more than 60 moons, four of which were discovered by Galileo in 1610

The four Galilean moons were the first moons to be discovered orbiting another planet, back in January 1610 when Galileo first turned his newly-fashioned telescope to look at Jupiter. Initially he thought the four bright dots he could see near Jupiter were background stars, but as he observed Jupiter over a period of several weeks he saw that not only did these bright dots follow Jupiter as Jupiter moved against the background stars, but they appeared to “dance” around it. He realised quite quickly that he was seeing moons orbiting another planet.

The actual sketches that Galileo made in January 1610 in his notebook of the moons of Jupiter

The four Galilean moons are called (in order of distance from Jupiter) Io, Europa, Ganymede and Callisto. The usual pneumonic for remembering this is I Eat Green Carrots, or In Every Good Class 🙂 I will do a series of blogs about each of these moons, as each one is fascinating and have been studied in detail by the Galileo space probe in the 1990s. But, one thing I will mention here is that you can see this four moons with just a pair of binoculars or a low powered telescope, you do not need any highly sophisticated equipment. If you are trying to see them with binoculars then the trick is to steady your elbows on something like a wall or the roof of a car, and lean against something to reduce any wobbling.

How Jupiter appears through a small telescope. On a good night with steady air the bands should be visible, and the Galilean moons are easy to spot.

Another thing I will mention in this blog is that Io only takes about 2 days to orbit Jupiter, and so in the matter of just a few hours you can see a change in its position. If you look at Jupiter at e.g. 8pm and then again at e.g. 2am, or even midnight, you will see that Io has moved. The best way to know which moons are where is to go online and do a search for “jupiter moons positions” or something similar, and you should be able to find a chart which shows which moons are where (east of west of Jupiter, or behind or in front of it) on which nights.

Jupiter has many other moons, and more are being discovered, but they are all tiny compared to the Galilean moons. The Galilean moons and Jupiter form what is, in many ways, a mini Solar System, and I will talk more about that when discuss Io and Europa in more detail in future blogs.

## Inertia and Newton’s 1st law

This weekend I was helping my youngest daughter revise for her Christmas physics exam. She tells me she enjoys physics, but I’m not sure whether she’s just saying this to please me! Her brother has just started his physics degree this last September, and her elder sister seriously thought of doing physics for A-level before deciding against it; but I suspect my youngest is more on the languages and creative arts side than a scientist. We shall see, she is only 13.

The material we were going over was the basics of motion, or mechanics. She has been learning about forces, pressure, velocity and resistances to motion (friction and air resistance). I asked her if she had been learning Newton’s three laws of motion, and from her answer I wasn’t sure whether she had or not!

I distinctly remember my own first encounter with Newton’s three laws of motion. I was about my daughter’s present age, and due to a Horizon programme about cosmology and particle physics (that I mention in this blog), I had already decided I wanted to study astronomy and physics. It therefore came as a bit of a shock to me when we were presented with Newton’s laws of motion, and I found I could not remember them.

Despite seeming to have a very good ability to remember poems and song lyrics, I am terrible at remembering ‘facts’, and our physics teacher presented Newton’s three laws to us as a series of facts. After several days of trying and failing to remember them in the words he had used (or which the text book had used), I was beginning to have serious doubts that I could go into physics at all.

And then I had an epiphany. I realised that if I understood Newton’s laws of motion, I did not need to remember them. If I could understand them, I could just state them in my own words; and within less than an hour I felt I had understood them thoroughly (although I’d like to think I have a deeper understanding of them now than I did at 13!). I often say to my students that the only things they need to remember in physics are the things they do not understand. That, certainly, has been my own experience.

## Inertia

The concept of inertia is fundamental to our ideas of motion, and yet it is not the easiest concept to understand or explain. But, I will have a go! Galileo was the first person to think of the concept of what we now call ‘inertia’. He realised that a stationary object wants to stay stationary, and you have to do something to it (push it, pull it, or drop it) to get it moving.

Galileo was the first person to outline the concept of inertia.

He also realised that objects which are moving want to carry on moving. This is not obvious, as we all know if we give an object a push it may start to move but will slow down and stop. Galileo realised that objects which were moving stopped because of resistive forces like friction or air resistance, and in the absence of these an object would carry on moving. This is, essentially, the idea of inertia. The tendency a body has to remain at rest or to carry on moving.

## Newton’s 1st law of motion

Newton’s 1st law of motion is essentially a statement of the concept of inertia, and is sometimes called the ‘law of inertia’. If someone asks me to state Newton’s 1st law the wording I use will probably change slightly each time, but the key idea I make sure I try to get across is the concept of inertia. So, a way to state Newton’s 1st law is

A body will maintain a constant velocity unless a force acts upon it

This is, more or less, the most succinct way I can express his 1st law. But, by stating it so succinctly, there are hidden complications. The first is to realise that the term ‘velocity’ has a very precise meaning in physics. I was trying to explain this to my daughter over the weekend (except we were doing it in Welsh, so using the Welsh word ‘buanedd’).

For a physicist, velocity is not the same as speed, even though we may use them interchangeably in everyday language. Speed (‘cyflymder’ in Welsh 😛 ) is a measurement of how quickly an object is moving, but velocity also includes the object’s direction of motion. A stationary object has zero velocity.

Therefore, a more long-winded (but no less correct) way to state Newton’s 1st law is

A body will remain at rest, or carry on moving with a constant speed in a straight line, unless acted upon by a force

The last thing I should mention in relation to the wording of Newton’s 1st law is that, strictly speaking, I should say ‘an external resultant force’, because an object can maintain a constant velocity with more than one force act upon it, as long as those forces are equal in size and opposite in direction. A good every-day example of this is driving a car at a constant speed in a straight line. The car does not continue to do this if we take our foot off of the accelerator, because air resistance and friction will slow the car down. When it is moving at a constant velocity the resistive forces are balanced by the force the engine is transferring to the tyres on the road. An ice puck, once pushed, will take a long time to slow down. This is because the friction on ice is much less than on a rougher surface, so the puck comes closer to acting like Newton’s 1st law says objects should act.

I don’t think my daughter has learnt about Newton’s 2nd law yet, which I often tell my students contains the most important equation in physics. His second law tells us what happens to an object if there is a (resultant external) force acting up on it. I will blog about that next week.

## Measuring the distances to stars

For many centuries it was assumed that the stars resided in a heavenly sphere surrounding the Earth, which was thought to lay at the centre of the Universe. When Galileo found evidence that not all heavenly bodies orbited the Earth, he gave his support to the Copernicus model that placed the Sun, not the Earth, at the centre. But, amongst the objections to this heliocentric model was the one that the stars did not appear to move when viewed from different parts of Earth’s suggested orbit.

## Parallax

When a foreground object is viewed from two different positions it appears to move against the background. For example, in the diagram below the foreground tree at C is viewed from two different positions, A and B. From position A the tree at C appears to lie in front of a gap between two background trees. But, from position B the tree at C appears to lie in front of a hut. This effect is known as parallax.

If we observe a foreground object from two different positions, it appears to move against the background. Here the tree at C appears to be in front of the hut as seen from point B, but between two background trees as seen from point A.

The effect of parallax can even be seen by looking through each eye alternatively. In addition, you can verify that the more distant the object the smaller the parallactic effect.

## The parallax of stars

As the Earth orbits the Sun, we should notice nearer stars appearing to move against the background of more distant stars. The average distance from the Earth to the Sun (called the Astronomical Unit (AU), was determined in the mid 1700s using the Transit of Venus. I will blog about this history in the near future, but I went to Mongolia in June 2012 to see the last Transit until December 2117.

If we can measure the angle $2p$ between the position of the star in, say, January and its position in, say, July (6 months apart), then a simple bit of trigonometry will give us the distance $d$ of the star.

$\tan (p) = \frac{ 1 AU }{ d } \text{ so } d = \frac{ 1 AU }{ \tan (p) }$

A nearby star will appear to move against the background stars if we view it 6 months apart at opposite sides of our orbit about the Sun.

## The definition of the parsec

The unit astronomers use for measuring distances within the Solar System is the Astronomical Unit (AU), as it is a convenient size compared to using kilometres or miles. The value of the AU is 149.6 million km, or $1.496 \times 10^{11} \text{m}$. For example, Jupiter is about 5 AU from the Sun, Saturn about 10 AU and Uranus about 20 AU.

But, the distances of stars is so vast that even the Astronomical Unit is too small. Instead, astronomers use a unit called the parsec. The word is actually a contraction of parallax and second (par + sec), and comes from its definition. A star would be at a distance of 1 parsec if the parallax it exhibited were 1 second of arc (remember there are 60 seconds of arc in an arc minute, and 60 arc minutes in a degree). So, using our definition of stellar parallax we can write

$\boxed{ d \text{ (in parsecs) } = \frac{ 1 }{ p \text{ (measured in arc seconds) } } }$

From this definition we can easily see that, if a star exhibits a parallax $p \text{ of e.g. } 2^{\prime \prime}$ then the distance would be

$d = \frac{ 1 }{ 2 } = 0.5 \text{ pc }$

and if the parallax $p \text{ were } 0.5^{\prime \prime}$ then the distance would be

$d = \frac{ 1 }{ 0.5 } = 2 \text{ pc }$.

To calculate the value of 1 pc in metres we can write

$1 \text{pc} = \frac{ 1.49 \times 10^{11} }{ \tan(1^{\prime \prime}) } = \frac{ 1.49 \times 10^{11} }{ 4.848 \times 10^{-6} } = 3.0857 \times 10^{16} \text{ m }$

## The relationship between a parsec and a light year

Measuring stellar parallaxes is the only direct method we have of measuring the distances to the stars. All other methods are based on this initial method as our “yard stick”.

The reason astronomers work in parsecs is because that is the unit most easily calculated when we measure stellar parallaxes. If we measure a parallax angle of $p = 0.5^{\prime \prime}$ then we know the star is at 2 parsecs, if we measure an angle of $p = 0.1^{\prime \prime}$ then the star is at 10 parsecs. Notice in my blog on the supernova in Messier 82, I quoted the distance to M82 in Mega parsecs (Mpc), and in my blog on the most distant galaxy yet discovered, I quoted Hubble’s constant in km/s per Mpc.

Because of the unfamiliarity of the concept of stellar parallax, when astronomers talk to the public we tend to use light years, the distance light travels in a year. This is a much easier concept to understand. We can readily determine the relationship between the two, as to calculate a light year we use the speed of light $c = 3 \times 10^{8} \text{ m/s }$, and the number of seconds in a year, which is $60 \times 60 \times 24 \times 365.25 = 3.15576 \times 10^{7} \text{ s }$, and so the distance light travels in one year is

$1 \text{ ly } = (3 \times 10^{8})(3.15576 \times 10^{7}) = 9.467 \times 10^{15} \text{ m }$

.
We see that this is less than the size of a parsec, in fact

$\boxed{ 1 \text{ pc } = 3.26 \text{ ly } }$

## The measurement of the first stellar parallax

As I mentioned above, the lack of observed stellar parallax was used as an objection to the idea that the Earth orbited the Sun. The first astronomer to try to measure stellar parallax was James Bradley, in 1729, but he was unsuccessful. The first successful attempt was not until 1838, when Friedrich Bessel measured the parallax of the star 61 Cygni. He found the parallax to be $p = 0.3136^{\prime \prime}$, less than a third of an arcsecond. This is why people had not been able to measure stellar parallax before, the angles for even nearby stars are tiny. Even the closest star to us, Proxima Centauri, has a parallax of only $p = 0.769^{\prime \prime}$, placing it at more than 1 pc away. An angle of $0.769^{\prime \prime}$ corresponds approximately to the angle subtended by an object 2cm in diameter as seen from 5.3km!

The blurring effects of the Earth’s atmosphere (“seeing” as astronomers call it) limits our ability to measure these tiny angles. As a consequence, only a few hundred stars have parallax angles large enough to be able to measure from terrestrial telescopes. By the end of the 19th Century, about 60 stellar parallaxes had been obtained, and the smallest angles measured corresponded to distances considerably less than a kiloparsec.

This made the technique useless for measuring distances on the Galactic scale. Fortunately, in 1914, Henrietta Leavitt discovered the period-luminosity-relationship for Cepheid variables. I mentioned this relationship in passing in my blog on how Edwin Hubble was able to show that the Andromeda Nebula was in fact a galaxy beyond our Milky Way galaxy. I will blog about Leavitt’s discovery in more detail in the future, as it formed the next step in what is often referred to as “the cosmological distance ladder”.

## Hipparcos and Gaia

In 1989 the European Space Agency (ESA) launched a satellite called Hipparcos, the main purpose of which was to measure stellar motions (due to the rotation of the disk of the Milky Way), and also to measure parallaxes from space, beyond the blurring effects of the Earth’s atmosphere. Hipparcos orbited the Earth at an altitude of between 507 and 35,888 km (an eccentric elliptical orbit with the centre of the Earth at one of the foci). Hipparcos was able to measure parallaxes down to $p=0.002^{\prime \prime}$, or 2 milliarcseconds (or 2,000 microarcseconds). This meant it could measure the parallax of stars out to a distance of 500pc, and by the end of its 3.5 year mission it had measured the parallax of over 100,000 stars.

In December 2013 ESA launched Gaia, which will go into an orbit about the second Lagrangian point (L2) rather than orbiting the Earth. It has a planned mission length of 5 years, and will be able to measure parallaxes down to 0.0001 arcseconds (10 microarcseconds), which corresponds to a distance of 100,000 pc (100 kpc) or about 320,000 light years. In comparison, the Milky Way galaxy is about 35 kpc in diameter and our Sun lies about 8 kpc from the centre. It is hoped that Gaia will measure the stellar parallax of over 1 billion stars!

## The 10 best physicists – no. 5 – James Clerk Maxwell

At number 5 in The Guardian’sten best physicists” is the Scottish theoretical physicist James Clerk Maxwell.

As the caption above says, James Clerk Maxwell is probably one of the least known of the physicists in this list, and yet one of the most important. It was through his theoretical work that a full understanding of electromagnetism became possible. His four equations, known as Maxwell’s equations, are as important to understanding electromagnetism as Newton’s laws of motion are to understanding mechanics. In some ways, Maxwell was to Faraday what Newton was to Galileo. Neither Galileo nor Faraday had the mathematical ability to write in mathematics the experimental results they obtained in mechanics and electricity and magnetism respectively. Newton wrote the mathematical equations to explain Galileo’s experiments in mechanics; and similarly Maxwell wrote the mathematical equations to explain the experiments Faraday had been doing on electricity and magnetism.

## Maxwell’s brief biography

James Clerk was born in 1831 in Edinburgh, Scotland. He was born into a wealthy family, and later added Maxwell to his name when he inherited an estate as part of his wider family’s wealth. His mother died when Maxwell was eight years old, and at ten years of age his father sent him to the prestigious Edinburgh Academy.

At the age of 16, in 1847, Maxwell left Edinburgh Academy and started attending classes at Edinburgh University. His ability in mathematics quickly became apparent. At the age of only 18 he contributed two papers to the Transactions of the Royal Society of Edinburgh, and in 1850, at the age of 19, he transferred from Edinburgh University to Cambridge. Initially he attended Peterhouse College, but transferred to the richer and better known Trinity College. He graduated from Trinity College Cambridge in 1854 with a degree in mathematics. He graduated second in his class.

In 1855 he was made a fellow of Trinity College, allowing him to get on with his research. But in November 1856 he left Cambridge, being appointed Professor of Natural Philosophy at the University of Aberdeen in Scotland. He was 25 years of age, and head of the Natural Philosophy department, some 15 years younger than any of his colleagues there.

Maxwell left Aberdeen in 1860 to become Professor of Natural Philosophy at Kings College, University of London. He resigned in 1865 to return to his estate in Scotland, and for the next 6 years he got on with his research, and due to being independently wealthy he didn’t need a formal paying position at a university. But in 1871 he was tempted back into university life, he was appointed the first Cavendish Professor of Physics at Cambridge University, and given the role of establishing the Cavendish Laboratory, which opened in 1875.

Maxwell died in 1879 of abdominal cancer, and is buried in Galloway, Scotland, near where he grew up.

## Maxwell’s scientific legacy

Maxwell is best known amongst physicists for two main areas of work, his work on electromagnetism and his work on the kinetic theory of gases. As I mentioned above, his work on electromagnetism can be likened to a certain extent to Newton’s work on mechanics. Newton was able to take the experimental results Galileo had found on the behaviour of moving bodies, and find the underlying laws of mechanics to explain them, and to express these in mathematical form. This is essentially what Maxwell did for the work of Faraday, he was able to find the laws which explained the observations Faraday had been making, and was a sufficiently skilled mathematician to write these laws in equation form.

Maxwell first presented his thoughts on electromagnetism in a paper in 1855, when he was 24 years old. In this he reduced all the known knowledge of electromagnetism as it existed in 1855 into a set of differential equations with 20 equations and 20 variables. In 1862 he showed that the speed of propagation of an electromagnetic field was approximately the same as the speed of light, something Maxwell thought was unlikely to be a coincidence. This was the first indication that light was a form of electromagnetism, but a form to which are eyes are senstive.

In 1873 Maxwell reduced the 20 differential equations of his 1855 paper into the four differential equations which are familiar to every undergraduate physics student. These equations, expressed in their differential form are
$\boxed{ \begin{array}{lcll} \nabla \cdot \vec{D} & = & \rho & (1) \\ & & & \\ \nabla \cdot \vec{B} & = & 0 & (2) \\ & & & \\ \nabla \times \vec{E} & = & - \frac{\partial \vec{B}}{\partial t} & (3) \\ & & & \\ \nabla \times \vec{H} & = & - \frac{\partial \vec{D}}{\partial t} + \vec{J} & (4) \end{array} }$

The symbol $\nabla$ is known as the vector differential operator, or del, and I have explained del in this blog.

The four equations can also be written in integral form, which many people find easier to understand. In integral form, the equations become

$\boxed{ \begin{array}{lcll} \iint_{\partial \Omega} \vec{D} \cdot d\vec{S}& = & Q_{f}(V) & (5) \\ & & & \\ \iint_{\partial \Omega} \vec{B} \cdot d\vec{S} & = & 0 & (6) \\ & & & \\ \oint_{\partial \Sigma} \vec{E} \cdot d\vec{\l} & = - & \iint_{\Sigma} \frac{\partial \vec{B} }{\partial t} \cdot d\vec{S} & (7) \\ & & & \\ \oint_{\partial \Sigma} \vec{H} \cdot d\vec{l} & = & I_{f} + \iint_{\Sigma} \frac{\partial \vec{D} }{\partial t} \cdot d\vec{S} & (8) \end{array} }$

## Kinetic theory of gases

Maxwell also did important work on one of the other main areas of physics research in the 19th Century – the kinetic theory of gases, a branch of thermodynamics. Maxwell and the Austrian physicist Ludwig Boltzmann independently came up with an expression which describes the distribution of speeds of the atoms or molecules in a gas. The speed depends on the temperature, the higher the temperature the faster the atoms or molecules move. But Maxwell and Boltzmann showed that not all the atoms will be moving at this speed; some will be moving faster and some will be moving slower. The distribution of speeds is known as the Maxwell-Boltzmann distribution.

## The theory of colour vision

Less well known to most physicists is the work Maxwell did on optics, and on colour vision. He was the first person to show how a colour image could be created from combining three images taken through red, green and blue filters. This lies at the foundation of practical colour photography, and is also how our colour television sets work.

I think it is true to say that most physicists would place Maxwell in their top four physicists. His contributions to physics are immense, creating the formal framework for our understanding of electromagnetism, without which the modern world would not be possible. That he is so unknown outside of the world of physics is strange, but whereas Newton had his $F=ma$ and Einstein his $E=mc^{2}$, Maxwell’s equations are far too complicated to have permeated into the consciousness of the public. But they are every bit as important as Newton and Einstein’s.

You can read more about James Clerk Maxwell and the other physicists in this “10 best” list in our book 10 Physicists Who Transformed Our Understanding of the UniverseClick here for more details and to read some reviews.

Ten Physicists Who Transformed Our Understanding of Reality is available now. Follow this link to order

## The 10 best Physicists?

In an interesting exercise, The Guardian newspaper recently drew up a list of the “10 best physicists”. I don’t think the list they compiled is in any particular order, but here it is.

1. Isaac Newton (1643-1727)
2. Niels Bohr (1885-1962)
3. Galileo Galilei (1564-1642)
4. Albert Einstein (1879-1955)
5. James Clerk Maxwell (1831-1879)
7. Marie Curie (1867-1934)
8. Richard Feynman (1918-1988)
9. Ernest Rutherford (1871-1937)
10. Paul Dirac (1902-1984)

How many of these names do you recognise? Whilst some are “household names”, others are maybe only known to physicists.

Over the next several months I will post a blog about each of these entries, giving more details of what their contribution(s) to physics were. Any such list is, of course, bound to promote discussion and disagreement, and I can also see that “The Guardian” have also allowed readers to nominate their own names.

You can read more about the physicists in this “10 best” list in our book 10 Physicists Who Transformed Our Understanding of the UniverseClick here for more details and to read some reviews.

Ten Physicists Who Transformed Our Understanding of Reality is available now. Follow this link to order

## Galilean Relativity and Electrodynamics

Quite a few months ago now I derived the so-called Galilean transformations, which allow us to relate one frame of reference to another in the case of Galilean Relativity.

$\boxed {\begin{array}{lcl} x^{\prime} & = & x + vt \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & t \end{array} }$

It had been shown that for experiments involving mechanics, the Galilean transformations seemed to be valid. To put it another way, mechanical experiments were invariant under a Galiean transformation. However, with the development of electromagnetism in the 19th Century, it was thought that maybe results in electrodynamics would not be invariant under the Galilean transformation.

## The electrostatic force between two charges

If we have two charges which are stationary, they experience a force between them which is given by Coulomb’s law.

$\vec{F}_{C} = \frac{ Q^{2} }{ 4\pi\epsilon_{0}\vec{r}^{2} }$ where $Q$ is the charge of each charge, $r$ is the distance between their centres, and $\epsilon_{0}$ is the permittivity of free space, which determines the strength of the force between two charges which have a charge of 1 Coulomb and are separated by 1 metre.

Coulomb’s law gives us the force between two charges. If the charges are the same sign the force is repulsive, if the charges are opposite in sign the force is attractive.

## Moving charges produce a magnetic field

If charges are moving we have an electric current. An electric current produces a magnetic field. The strength of this field is given by Ampère’s law

$\oint \vec{B} \cdot d\vec{\l} = \mu_{0}I$ where $d\vec{l}$ is the length of the wire, $\vec{B}$ is the magnetic field, $\mu_{0}$ is the permeability of free space and $I$ is the current. So, if the two charges are moving, each will be surrounded by its own magnetic field.

A wire carrying a current produces a magnetic field as given by Ampère’s law.

## The Lorentz force

If the two charges are moving and hence producing magnetic fields around each of them then there will be an additional force between the two charges due to the magnetic field each is producing. This force is called the Lorentz force and is given by the equation

$\vec{F}_{L} = Q\vec{v}\times\vec{B}$. If $r$ is the distance between the two wires, and they are carrying currents $I_{1}$ and $I_{2}$ respectively, and are separated by a distance $r$, we can write $B=\frac{\mu_{0}I}{2\pi r}$ which then gives us that the Lorentz force $F_{L} = \frac{ I_{1} \Delta L \mu_{0} I_{2} }{2 \pi r }$ and so the Lorentz force per unit length due to the magnetic field in the other wire that each wire feels is given by $\boxed{ \frac{ F_{L} }{\Delta L} = \frac{ \mu_{0} I_{1} I_{2} }{ 2 \pi r} }$. Writing the currents in terms of the rate of motion of the charges, we can write this as

$F_{L} = \frac{ \mu_{0} Q_{1} Q_{2} }{ 4\pi r^{2} } v^{2}$

The Lorentz force is the force on a wire due to the magnetic field produced in the other wire from the current flowing in it.

## Putting it all together

Let us suppose the two charges are sitting on a table in a moving train. This would mean that someone on the train moving with the charges would measure a different force between the two charges (just the electrostatic force) compared to someone who was on the ground as the train went past (the electrostatic force plus the Lorentz force).

The force measured on one of the charges by the person on the train, for whom the charges are stationary, which we shall call $F$ will be

$F = \frac{ Q_{1}Q_{2} }{ 4 \pi \epsilon_{0}r^{2} }$.

The force measured on one of the charges by the person on the ground, for whom the charges are moving with a velocity $v$, which we shall call $F^{\prime}$ will be

$F^{\prime} = \frac{ Q_{1}Q_{2} }{4 \pi \epsilon_{0}r^{2} } + \frac{ \mu_{0} Q_{1} Q_{2} }{ 4\pi r^{2} } v^{2}$.

These two forces are clearly different, and so it would seem that the laws of Electrodymanics are not invariant under a Galilean transformation, or to put it another way that one would be able to measure the force between the two charges to see if one were at rest or moving with uniform motion because the forces differ in the two cases.

As I will explain in a future post, Einstein was not happy with this idea. He believed that no experiment, be it mechanical or electrodynamical, should be able to distinguish between a state of rest or of uniform motion. His solution to this problem, On the Electrodynamics of Moving Bodies, was published in 1905, and led to what we now call his Special Theory of Relativity. This theory revolutionised our whole understanding of space and time.