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towards James Clerk Maxwell 6: Newton’s universal law of gravitation and G

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Newton’s law of gravity goes like this:

{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}

where F is the force of gravitational attraction, G is the constant of proportionality or gravitational constant, m is an entity, particle or object with a particular mass, and r is the distance between the centres of mass of the two entities, particles or objects.

What’s the relation between all this and Maxwell’s electromagnetic work? Good question – to me, it’s about putting physics on a mathematical footing. Newton set us on this path more than anyone. The task I’ve set myself is to understand all this from the beginning, with little or no mathematical expertise.

The law of gravity, in its un-mathematical form, says that every object of mass attracts every other massive object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

It seems often to be put about that Newton was revolutionary because he was the first to wonder why objects fell to the ground. This is unlikely, and Newton wasn’t the first to infer an inverse square law in relation to such falling. Two Italian experimenters, Francesco Grimaldi and Giovanni Riccioli, investigated the free fall (where no force acts besides gravity) of objects between 1640 and 1650, and noted that the distance of the fall was proportional to the time taken. Galileo had previously conducted free fall experiments and found that objects fell with uniform acceleration – an acceleration that is proportional to the square of the elapsed time. Nor was he the first to find a time-squared relationship. The point of all this is that science doesn’t proceed via revolutions proceeding from one brilliant person (which shouldn’t diminish Galileo or Newton’s genius). The more you find out about it, the more incremental and fascinatingly collaborative and confirmative over time it is.

Galileo used the geometry of his time to present his time squared law, but algebraic notation, invented principally by Descartes, superseded this approach in the seventeenth century.

What about the gravitational constant? This appears to be a long and complicated mathematical story. I think it tries to answer the question – why do objects fall to Earth at such and such a rate of acceleration? But I’m not sure. The rate of acceleration would have been easy enough to measure – it’s approximately 9.8 m/sec2. This rate would appear to be caused by the mass of the Earth. The Moon has a fraction of Earth’s mass, and I believe the gravitational force it exerts is approximately one sixth that of Earth. It has been measured as 1.62 m/sec² (for Mars it’s 3.71).

It’s frustratingly difficult to get an explanation online of what the gravitational constant (G) is or really means – without very quickly getting into complex (for me) mathematics. Tantalisingly, Wikipedia tells us that the aforementioned Grimaldi and Riccioli ‘made [an attempted] calculation of the gravitational constant by recording the oscillations of a pendulum’, which means nothing to me. Clearly though, there must be some relationship between G and the mass of the Earth, though how this can be ascertained via pendulums is beyond me. Anyway, on with the struggle.

We do have a number for G, or ‘Big G’, as it’s called (explanation to come), and it’s a very very small number, indicating that, considering that the multiplied masses divided by the square of the distance between them then get multiplied by G, gravitation is mostly a very small force, and only comes into play when we’re talking about Big Stuff, like stars and planets, and presumably whole galaxies. Anyway here’s the actual number:

G = 0.0000000000667408, or 6.67408 × 10-11

I got the number from this useful video, though of course it’s easily available on the net. Now, my guess is that this ‘Big G’ is specific to the mass of the Earth, whereas small g is variable depending on which mass you’re referring to. In other words, G is one of the set of numbers in g. We’ll see if that’s true.

Now, looking again at the original equation, F stands for force, measured in newtons, m for mass, measured in kilograms, and and r for distance in metres (these are the SI units for mass and distance). The above-mentioned video ‘explains’ that the newtons on one side of the equation are not equivalent to the metres and kilograms squared on the other side, and G is introduced to somehow get newtons onto both sides of the equation. This has thrown me into confusion again. The video goes on to explain how G was used by Einstein in relativity and by Max Planck to calculate the Planck length (the smallest possible measure of length). Eek, I’m hoping I’m just experiencing the storm before the calm of comprehension.

So, to persist. This G value above isn’t, and apparently cannot be, precise. That number is ‘the average of the upper and lower limit’, so it has an uncertainty of plus or minus 0.00031 x 10-11, which is apparently a seriously high level of uncertainty for physicists. The reason for this uncertainty, apparently, is that gravitational attraction is everywhere, existing between every particle of mass, so there’s a signal/noise problem in trying to isolate any two particles from all the others. It also can’t be calculated precisely through indirect relation to the other forces (electromagnetism, the strong nuclear force and the weak nuclear force), because no relationship, or compatibility, has been found between gravity and those other three forces.

The video ends frustratingly, but providing me with a touch of enlightenment. G is described as a ‘fundamental value’, which means we don’t know why it has the value it does. It is just a value ‘found experimentally’. This at least tells me it has nothing to do with the mass of the Earth, and I was quite wrong about Big G and small g – it’s the other way round, which makes sense, Big G being the universal gravitational constant, small g pertaining to the Earth’s gravitational force-field.

Newton himself didn’t try to measure G, but this quote from Wikipedia is sort of informative:

In the Principia, Newton considered the possibility of measuring gravity’s strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable. Nevertheless, he estimated the order of magnitude of the constant when he surmised that “the mean density of the earth might be five or six times as great as the density of water”

Pendulums again. I don’t quite get it, but the reference to the density of the Earth, which of course relates to its mass, means that the mass of the Earth comes back into question when considering this constant. The struggle continues.

I’ll finish by considering a famous experiment conducted in 1798 by arguably the most eccentric scientist in history, the brilliant Henry Cavendish (hugely admired, by the way, by Maxwell). I’m hoping it will further enlighten me. For Cavendish’s eccentricities, go to any online biography, but I’ll just focus here on the experiment. First, here’s a simplification of Newton’s law: F = GMm/R2, in which M is the larger mass (e.g. the Earth), and m the smaller mass, e.g a person. What Cavendish was trying to ascertain was nothing less than the mass and density of the Earth. In doing so, he came very close – within 1% – of the value for G. Essentially, all that has followed are minor adjustments to that value.

The essential item in Cavendish’s experiment was a torsion balance, a wooden bar suspended horizontally at its centre by a wire or length of fibre. The experimental design was that of a colleague, John Michell, who died before carrying out the experiment. Two small lead balls were suspended, one from each end of the bar. Two larger lead balls were suspended separately at a specific distance – about 23cms – from the smaller balls. The idea was to measure the faint gravitational attraction between the smaller balls and the larger ones.

the ‘simple’ Michell/Cavendish device for measuring the mass/density of the Earth – Science!

Wikipedia does a far better job than I could in explaining the process:

The two large balls were positioned on alternate sides of the horizontal wooden arm of the balance. Their mutual attraction to the small balls caused the arm to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the large and small lead spheres. By measuring the angle of the rod and knowing the twisting force (torque) of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the Earth to be calculated, using Newton’s law of gravitation.

To fully understand this, I’d have to understand more about torque, and how it’s measured. Clearly this weak interaction is too small to be measured directly – the key is in the torque. Unfortunately I’m still a way from fully comprehending this experiment, and so much else, but I will persist.


Go to youtube for a number of useful videos on the gravitational constant

Written by stewart henderson

July 14, 2019 at 4:51 pm

towards James Clerk Maxwell 5: a bit about light

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Canto: Our last piece in this Maxwell series dealt with the apparently irrelevant matter of Newton’s laws of motion…

Jacinta: But not irrelevant in that Newton was so seminal to the foundation of, and mathematisation of, modern physics, and he set the course…

Canto: Yes and we’ll have to go back to his work on gravity to really get a feel for the maths side of things I think.

Jacinta: No doubt, but in keeping with our disorganised approach to out topic I’m going to fast forward to give a partial account of Maxwell himself, before he did his major work. Maxwell was clearly indefatigably curious about the physical world even in childhood. He was conducting various chemical, electrical and magnetic experiments at home and later at the University of Edinburgh, from his early teens, and writing papers – the first at the age of fourteen – which were accepted by the Royal Society of Edinburgh, though he was considered too young to present them himself.

Canto: But we’re going to focus here on his focus on light, since we’ve been going on mostly about electricity and magnetism thus far. Light, and its wave properties, is something we’re going to have to get our heads around as we approach Maxwell’s work from various angles, and it’s horribly mathematical.

Jacinta: Yes, the properties of polarised light were among Maxwell’s earliest and most abiding areas of interest and research, so we need to understand what that’s all about.

Canto: Okay, here’s a simple definition of the term ‘polarisation’. It’s ‘a property applying to transverse waves that specifies the geometrical orientation of the oscillations’. That’s from Wikipedia.

Jacinta: That’s not simple. Do you understand that?

Canto: No, not yet. So waves are generally of two types, transverse and longitudinal. A moving wave oscillates. That’s the up-and-down movement you might see on a graph. In a transverse wave, the oscillations are at right angles to the movement of the wave. Light waves are transverse waves apparently, as opposed to sound waves, which are longitudinal – in which the wave oscillates, or vibrates, in the direction of propagation. That doesn’t make immediate sense to me, but for now we’ll focus on transverse light waves and polarisation. A light wave, we now know, is an electromagnetic wave, but don’t worry about that for now. Let me try to explain unpolarised light. The light from the sun is unpolarised, as is your bedroom light or that from a struck match. The light waves from these sources are vibrating in a multitude of directions – every direction, you might say. Polarised light is light that we can get to vibrate on a single plane, or in some other specific way..- .

Jacinta: So how do we polarise light is presumably the question. And why do we call it polarised?

Canto: I don’t know why it’s called polarised, but it’s light that’s controlled in a specific way, for example by filtration. The filter might be a horizontal grid or a vertical grid. Let me quote two sentences from one of many explaining sites, and we’ll drill down into them:

Natural sunlight and almost every other form of artificial illumination transmits light waves whose electric field vectors vibrate in all perpendicular planes with respect to the direction of propagation. When the electric field vectors are restricted to a single plane by filtration, then the light is said to be polarized with respect to the direction of propagation and all waves vibrate in the same plane.

So electric field vectors (and we know that vectors have something to do with directionality, I think) are directions of a field, maybe. And a ‘field’ here is an area of electric charge – the area in which that charge has an influence, say on other charges. It was Michael Faraday who apparently came up with this idea of an electric field, which weakens in proportion to distance, in the same manner as gravity. A field is not actually a force, but more a region of potential force.

Jacinta: It seems we might have to start at the beginning with light, which is a huge fundamental force or energy, which has been speculated on and researched for millennia. I’ve just been exploring the tip of that particular iceberg, and it makes me think about how particular forces or phenomena, which are kind of universal with regard to humans on our modern earth, are taken for granted until they aren’t. Think for example of gravity, which wasn’t even a thing before Newton came along, it was just ‘natural’ that things fell down to earth. And think of air, which many people still think of as ’empty’. Light is another of those phenomena, but it’s been explored for longer than the others because it’s much more variable and multi-faceted, at least at first glance haha.. Darkness, half-light, firelight, shadow effects, the behaviour of light in water, rainbows and other tricks of light would’ve challenged the curious from the beginning, so it’s not surprising that theories of light and optics go back such a long way.

Canto: Yes and the horror of it – for some – is that mathematics is key to understanding so much of it – especially trigonometry. But returning to those electric field vectors – and maybe we’ll go back to the beginning with light in the future, – in a light wave, the oscillations are the electric and magnetic fields, pointing in all directions perpendicular to the wave’s propagation.

Jacinta: Yes, I get that, and polarised light limits all those perpendicular directions, or perpendicular planes, to one, by filtration, or maybe some other means.

Canto: Right, but notice I spoke of electric and magnetic fields, which is why light is described as an electromagnetic wave. It should also be pointed out that we tend to call them light waves only in the part of the spectrum visible to humans, but physics deals with all electromagnetic waves. Our eyes, and it’s different for many other species, detect light from a very small part of the entire wave, or electromagnetic, spectrum. Wavelengths of less than about 380 nanometres (even less when we’re young) at the ‘ultraviolet’ end, and of more than about 750 nm at the long ‘infrared’ end, form the visible spectrum for humans. Beyond UV light we have x-rays and then gamma rays, and beyond the infrared we have microwaves and then radio waves.

Jacinta: I wonder if Maxwell knew about all this in his day.

Canto: We’ll no doubt find out…


Written by stewart henderson

June 29, 2019 at 10:55 am

Towards James Clerk Maxwell 2 – Francis Hauksbee’s experiments

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an electrostatic generator – one of Hauksbee’s many ingenious experimental devices

Canto: So we’ve witnessed electricity since we’ve had the wit to witness, in lightning. And through our attempts to understand and harness those scary bursts of energy we’ve transformed our world.

Jacinta: We’ve written about lightning before, but the info we presented there was accumulated over centuries. Now we’re going to travel back to the early years of the Royal Society in England, the early 1700s, a mere 300 years ago, to reflect on the first experiments with electricity – remembering that there was no electric power and light in those days, that gods were in the air and much was mysterious.

Canto: Electricity from the start was much sexier, and scarier, than magnetism – lightning very very frightning was the most obvious physical manifestation, and its power was easily recognised. It could kill at a stroke, while magnetism seemed all about metals getting stuck together, and needles pointing north. Interesting, but hardly earth-shattering.

Jacinta: Lightning was all about gigantic sparks shattering the sky, and the ancients, who spent so much of their time in darkness, must have seen other, less impressive and dangerous sparks, the sparks of static electricity, and wondered.

Canto: In the recent BBC documentary The story of electricity, narrator Jim Al-Khalili begins by describing Francis Hauksbee‘s experiments with static electricity and electroluminescence in the early 1700s, which dazzled visitors to the Royal Society. These were the first properly documented experiments with the mysterious force, and a collection of his papers describing these experiments was widely read by the 18th century cognoscenti – including Joe Priestley and Ben Franklin. He employed the newly-invented air pump (simply a pump for pushing out air, as in a common bike pump), popularised in England by Robert Hooke some decades before. Hauksbee made his own improvements, enabling the pump to create a vacuum.

Jacinta: Yes Hauksbee was a more interesting figure than The story of electricity presents. He didn’t ‘lose interest’ but worked on his experiments and reflected on them until his final illness in 1713 – and I’m thinking that illness, since he was only in his late forties – may have been due to mercury poisoning. Hauksbee was ‘lower class’ so few details of his life are documented. However, in these experiments he wasn’t thinking so much of electricity as of ‘attractive forces’. Also as an experimenter who must always have seen himself as an underling (in his book he mentions his ‘want of a learned education’), he doubtless felt obliged to follow the guidance of his Royal Society ‘master’, Newton, which took him into different fields of research….

Canto: The term ‘electricity’ was possibly not in common use then? You’re right, though, about Hauksbee, who rose from obscurity to become a member of the Royal Society, probably under the auspices of Newton. In late 1705, as a result of some spectacular effects displayed to the Society he became intrigued by ‘mercurial phosphorus’. The fact that mercury, in a vacuum, glowed when shaken, had already been noted by Jean Picard, a 17th century French astronomer, and the Swiss mathematician Johann Bernoulli.

Jacinta: And this has to do with electricity?

Canto: We shall see. Hauksbee wanted to work out the conditions under which this mercurial light was produced. He found that the more air in the container, the weaker the light. Also the light’s intensity depended on the movement of the mercury. He concluded that the friction of the mercury against the glass was the major cause. But was it only mercury that had this property, and was it only glass that brought it out? He experimented with other materials, finding a means of rubbing them together in a section of his air pump, Amber rubbed with wool produced a light, brightened in the absence of air. By contrast metal on flint only produced sparks when air was present. Remember, oxygen wasn’t known about at the time. In late 1705 Hauksbee presented one of his most spectacular experiments for the Society. Ingenious instrument-maker that he was, he created a glass globe, from which air could be pumped in and out, on a rotating spindle. The spinning globe came into contact with woollen cloth, and the contact created a weird purple light inside the evacuated globe, which reduced as air was let in. It was a fantastic mystery.

Jacinta: I’m hoping you can solve it.

Canto: Great expectations indeed. He experimented further, and found that when he pressed his own hands against a spinning evacuated globe, the same bright purple glow was produced, which again faded when air was let in to the globe.

Jacinta: Okay, what Hauksbee was exploring in these experiments are what we now call triboelectric effects. I remember playing around with this in schooldays by rubbing a plastic pen along the sleeve of my jersey and watching the fibres stand on end as the pen passed, and hearing the prickling sound of static electricity. The pen was then capable of lifting scraps of paper from the desk, for a time. But I didn’t see any purple lights and I’m not sure how the presence or absence of air relates to it all.

Canto: Yes, triboelectricity is about the exchange of electric charge between different materials – the build-up and discharge of electrical energy. It seems that some materials have a more or less positive charge and some have a more or less negative or opposite charge (because positive and negative are really arbitrary terms, the key point is their relation to each other), and we know that like charges repel and opposite charges attract.

Jacinta: You’ve brought up the word ‘charge’ here, and I’m wondering if that’s just an arbitrary word too – like degree of positive charge just means degree of being repulsed by its opposite, negative charge. In other words, different materials are attracted to or repulsed by each other to varying degrees under various conditions, and that degree or ‘amount’ of attraction or repulsion is referred to as ‘charge’. So ‘charge’ is a relational term…

Canto: Ummm. Maybe. In any case, if you take these different materials down to the atomic level, and I’m not sure how you take plastic and wool down to that level – I mean I know plastic is a petrochemical product, but wool, which I’ve just looked up, has a very complex chemistry – but the fact that the plastic pen, after some rubbing, pulls the fibres of your woollen sleeve towards it is because there’s an attractive force operating between opposite charges. In fact there’s a movement of electrons between the materials, from the wool to the plastic. This electron transfer leaves those woollen fibres with a net positive charge, which is attracted to the now negatively charged plastic due to the electron flow. I think.

Jacinta: Mmm. None of this was understood in the early eighteenth century, obviously. But before we go back there, we’ll stay with this concept of charge, which is nowadays calculated as a fundamental or base unit, based on the electron or its opposite, charge-wise, the proton. These elementary particles have the same but opposite charge, though they’re very different in mass (something which seems suspect to me). Anyway, taking things on trust, a unit of charge is ‘defined’ in standard macro terms as a coulomb, named for the 18th century French physicist Charles-Augustin de Coulomb. One coulomb equals approximately 6.24 x 1018 protons (or electrons). We’ll come back to this later, no doubt. Returning to Hauksbee’s experiments, he soon realised that it was the glass, not the mercury inside it, that was the agent of electrical effects. His experiments with glass globes were written down in great detail, a boon to later researchers.

Canto: Interestingly, I’ve discovered that, more or less exactly at the same time, one Pierre Polinière was conducting and presenting experiments on electroluminescence in Paris:

A closer examination of these experiments reveals not only that Polinière had personally presented them before the French Academy of Sciences, but that Polinière and Hauksbee, starting from a common interest in the ‘mercurial phosphor’, had conducted similar investigations and had in fact simultaneously announced their independent discoveries of the luminescence of evacuated glass containers.

Pierre Polinière, Francis Hauksbee and electroluminescence: a case of simultaneous discovery.
David Corson, 1968.

Jacinta: So we might finish by trying to explain our current understanding of electroluminescence (EL) and its applications. It’s a sort of combo of electricity and light, as you can imagine, or electrons and photons on the level of particles. For example, semiconductors emit light when subjected to a strong electric field or current….

Canto: Is that the basis of LED lighting?

Jacinta: Absolutely. Electrons in the semiconductor material recombine with electron holes, emitting energy in the form of photons. So it has taken us three centuries to really effectively harness the electroluminescent effects demonstrated by Hauksbee in the early days of the Royal Society.

Canto: What are electron holes? I’m thinking not ‘holes in electrons’ but holes left by electrons as they’re displaced in an electric current?

Jacinta: Yes, or almost. It’s like the lack of an electron where you might expect an electron to be. These holes where you might expect an electrically charged particle (an electron) to be, act like positively charged particles – a positron, say – and move through a lattice like an electron does. We could get into very complicated electronics here, if we had the wherewithal, but these holes are examples of quasiparticles, which mostly exist within solids. Fluid movement within solids (not apparently a contradiction in terms) is extremely complicated. Who would’ve thunk it? This movement of electrons and protons within solids is ‘regulated’ by Coulomb’s Law. Remember him? We’ll be getting to that law very soon, as it’s essential to the field of electromagnetism. And that’s our topic don’t forget.