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the geometry of his time to present his time squared law, but algebraic notation, invented principally by Descartes, superseded this approach in the seventeenth century.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

References

https://en.wikipedia.org/wiki/Newton’s_law_of_universal_gravitation

https://en.wikipedia.org/wiki/Gravitational_constant

https://energyeducation.ca/encyclopedia/Gravitational_constant

https://en.wikipedia.org/wiki/Cavendish_experiment

https://www.school-for-champions.com/science/gravitation_cavendish_experiment.htm#.XSrCrS3L1QI

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…

References

https://en.wikipedia.org/wiki/Transverse_wave

https://www.physicsclassroom.com/class/light/Lesson-1/Polarization

https://en.wikipedia.org/wiki/History_of_optics

https://www.britannica.com/biography/James-Clerk-Maxwell

Written by stewart henderson

June 29, 2019 at 10:55 am

towards James Clerk Maxwell 3 – Benjamin Franklin and Coulomb’s Law

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Coulomb’s law – attraction and repulsion

Canto: So we’ve been looking at electricity and magnetism historically, as researchers, scientists, thinkers, experimenters and so on have managed to piece these processes together and combine them into the one thing, electromagnetism, culminating in J C Maxwell’s equations…

Jacinta: Or going beyond those equations into the implications. But of course we’re novices regarding the science and maths of it all, so we should recommend that real students of this stuff should go to the Khan academy lectures, or Matt Anderson’s lectures for the real expert low-down. As will we. But we need to point out, if only to ourselves, that what we’re trying to get our heads around is really fundamental stuff about existence. Light, which is obviously fundamental to our existence, is an electromagnetic wave. So, think magnetism, think electricity, and think light.

Canto: Right, so we’re going back to the eighteenth century, and whatever happens after Hauksbee and Polinière.

Jacinta: Well, scientists – or shall we say physical scientists, the predecessors of modern physicists – were much influenced throughout the eighteenth century by Newton, in particular his inverse square law of gravity:

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

Newton saw gravity as a force (F), and formulated the theory that this force acted between any two objects (m1 and m2 – indicating their masses) in a direct line between their respective centres of mass (r being the length of that line, or the distance between those centres of mass). This force is directly proportional to the product of the two masses and inversely proportional to the distance. As to G, the gravitational constant, that’s something I don’t get, as yet. Anyway, the success of Newton’s theory, especially the central insight that a force diminishes, in a precise way, with distance, affected the thinking of a number of early physical scientists. Could a similar theory, or law (they didn’t think in terms of theory then) apply to electrical forces? Among those who suspected as much were the mathematician Daniel Bernoulli, who made major contributions to fluid dynamics and probability, and Alessandro Volta, who worked on electrical capacitance and storage, the earliest batteries.

Canto: And Joseph Priestley actually proposed an inverse square law for electricity, but didn’t work out the details. Franz Aepinus and Benjamin Franklin were also important 18th century figures in trying to nut out how this force worked. All of this post-Newtonian activity was putting physical science on a more rigorous and mathematical footing. But before we get to Coulomb and his law, what was a Leyden Jar?

Jacinta: Leyden jars were the first capacitors. They were made of glass. This takes us back to the days of Matthias Bose earlier in the 18th century, and even back to Hauksbee. Bose, a professor of natural philosophy at the University of Wittenberg, worked with and improved Hauksbee’s revolving glass-globe machine to experiment with static electricity. He added a metal ‘prime conductor’ which accumulated a higher level of static charge, and gave spectacular public demonstrations of the sparks he created, using them to set alcohol alight and to create ‘beatification’ effects on a woman wearing a metal helmet. All great japes, but it promoted interest in electricity on the continent. The trick with alcohol inspired another experimenter, Jurgen von Kleist, to invent his Leyden jar, named for Kleist’s university. It was a glass container filled with alcohol (or water) into which was suspended a metal rod or wire, connected to a prime conductor. The fluid collected a great deal of electric charge, which turned out to be very shocking to anyone who touched the metal rod. Later Leyden jars used metal foil instead of liquid. These early capacitors could store many thousands of volts of electricity.

Canto: At this time, in the mid-eighteenth century, nobody was thinking much about a use for electricity, though I suppose the powerful shocks experienced by the tinkerers with Leyden jars might’ve been light-bulb moments, so to speak.

Jacinta: Well, take Ben Franklin. He wasn’t of course the first to notice that electrostatic sparks were like lightning, but he was possibly the first to conduct experiments to prove the connection. And of course he knew the power of lightning, how it could burn down houses. Franklin invented the lightning rod – his proudest invention – to minimise this damage.

Canto: They’re made of metal aren’t they? How do they work? How did Franklin know they would work?

Jacinta: Although the details weren’t well understood, it was known in Franklin’s time that some materials, particularly metals (copper and aluminium are among the best), were conductors of electricity, while others, such as glass, were insulators. He speculated that a pointed metal rod, fixed on top of buildings, would provide a focal point for the electrical charge in the clouds. As he wrote: “The electrical fire would, I think, be drawn out of a cloud silently, before it could come near enough to strike….” He also had at least an inkling of what we now call ‘grounding’, as per this quote about the design, which should use “upright Rods of Iron, made sharp as a Needle and gilt to prevent Rusting, and from the Foot of those Rods a Wire down the outside of the Building into the Ground”. He was also, apparently the inventor of the terms negative and positive for different kinds of charge.

Canto: There are different kinds of charge? I didn’t know that.

Jacinta: Well you know of course that a molecule is positively charged if it has more protons than electrons, and vice versa for negative charge, but this molecular understanding came much later. In the eighteenth century electricity was generally considered in terms of the flow of a fluid. Franklin posited that objects with an excess of fluid (though he called it ‘electrical fire’) were positively charged, and those with a deficit were negatively charged. And those terms have stuck.

Canto: As have other other electrical terms first used by Franklin, such as battery, conductor, charge and discharge.

Jacinta: So let’s move on to Charles-Augustin De Coulomb (1736-1806), who was of course one of many scientists and engineers of the late eighteenth century who were progressing our understanding and application of electricity, but the most important one in leading to the theories of Maxwell. Coulomb was both brilliant and rich, at least initially, so that he was afforded the best education available, particularly in mathematics…

Canto: Let me write down Coulomb’s Law before you go on, because of its interesting similarity to Newton’s inverse-square gravity law. It even has one of those mysterious ‘constants’:

{\displaystyle F=k_{e}{\frac {q_{1}q_{2}}{r^{2}}},}

where F is the electrostatic force, the qs are particular magnitudes of charges, and r is the distance between those charges.

Jacinta: Yes, the Coulomb constant, ke, or k, is a constant of proportionality, as is the gravitational constant. Hopefully we’ll get to that. Coulomb had a varied, peripatetic existence, including a period of wise retirement to his country estate during the French revolution. Much of his work involved applied engineering and mechanics, but in the 1780s he wrote a number of breakthrough papers, including three ‘reports on electricity and magnetism’. He was interested in the effect that distance might have on electrostatic force or charge, but it’s interesting that these papers placed electricity and magnetism together. His experiments led him to conclude that an inverse square law applied to both.

Canto: I imagine that these constants required a lot of experimentation and calculation to work out?

Jacinta: This is where I really get lost, but I don’t think Coulomb worked out the constant of proportionality, he simply found by experimentation that there was a general law, which he more or less stated as follows:

The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.
The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive.

It seems the constants of proportionality are just about units of measurement, which of course were different in the days of Coulomb and Newton. So it’s just about measuring stuff in modern SI units using these laws. It’s about conventions used in everyday engineering, basically. I think.

Canto: Equations like these have scalar and vector forms. What does that mean?

Jacinta: Basically, vector quantities have both magnitude and direction, while scalar quantities have magnitude only. The usual example is speed v velocity. Velocity has magnitude and direction, speed only has magnitude. Or more generally, a scalar quantity has only one ‘dimension’ or feature to it in an equation – say, mass, or temperature. A vector quantity has more than one.

Canto: So are we ready to tackle Maxwell now?

Jacinta: Hell, no. We have a long way to go, with names like Gauss, Cavendish and Faraday to hopefully help us along the path to semi-enlightenment. And I think we need to pursue a few of these excellent online courses before we go much further.

References

Khan academy physics (160 lectures)

Matt Anderson physics (191 lectures)

https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation

https://www.britannica.com/technology/Leyden-jar

http://www.americaslibrary.gov/aa/franklinb/aa_franklinb_electric_1.html

http://www.revolutionary-war-and-beyond.com/benjamin-franklin-and-electricity-letters.html

https://en.wikipedia.org/wiki/Coulomb_constant

https://www.britannica.com/biography/Charles-Augustin-de-Coulomb

https://www.britannica.com/science/Coulombs-law

https://en.wikipedia.org/wiki/Coulomb%27s_law

Written by stewart henderson

May 18, 2019 at 6:04 pm