Archive for the ‘physics’ Category
an interminable conversation 10: more basic physics – integrals
that’s for sure
Jacinta: So I watched episode 3 of the crash course physics videos, on integrals, and found it so overwhelming I had to immediately go and take a nap. I’m surely too old and thick for this stuff, but I must soldier on.
Canto: We can do battle together – so this episode is about integrals, the inverse of derivatives. I’m not sure we gleaned much from the episode about derivatives, but if we combine this with exercises from Brilliant and some other practical application-type videos and websites we might make some more progress before we die.
Jacinta: Okay so equations, at least some of them, can be plotted on graphs with x-y axes, and the integral of the equation is the area between the curve and the horizontal x axis. Dr Somara is going to teach us some shortcuts for calculating these integrals, which sounds ominous.
Canto: I didn’t really understand the stuff about derivatives, but I’ll keep going, hoping for a light-bulb moment. So integrals help us to understand how things move, she says, which in itself sounds weird. And then she mentions the displacement curve, which I’ve forgotten.
Jacinta: Looking elsewhere, I’ve found a simple video showing displacement-time graphs. Displacement, which is simply movement from one position to another, is shown on the y (vertical) axis, and time on the x axis. A graph showing a straight horizontal line would mean no displacement, therefore zero velocity. A graph showing a vertical straight line would mean displacement in zero time, which would indicate something impossible – infinite velocity? Anyway, a straight line between the horizontal and the vertical would indicate a fixed velocity – neither accelerating nor decelerating. A curve would indicate positive or negative shifts in velocity. I think. Sorry – the terms used are constant velocity and variable velocity. That’s much neater. Oh and there’s also negative velocity, but that’s a weird one.
Canto: Thanks, that’s useful. Need to point out though that ‘curve’ is just the term for representing the data on a graph – in that sense the curve could be a straight line, or whatever. So Dr Somara starts with a gravity problem. You want to know the distance between your bedroom window and the ground, in a multi-storey building. You have a ball, a stop-watch and a knowledge of gravity. The ball will fall at g, 9.81 m/sec². What is the distance? According to the Doctor, discussion so far about motion has involved three aspects, position, velocity and acceleration, and has focussed on velocity as the derivative of position, and acceleration as the derivative of velocity. The connection has to be reversed to work out the distance problem. So velocity is the integral of acceleration.
Jacinta: And of course position is the integral of velocity. And this is important – velocity is the area under the acceleration curve, and position the area under the velocity curve. Which might be difficult to calculate. Areas within polygons, without too many sides, is easy enough, sort of, but under complex curvy stuff, not so much. And when we talk about ‘under’ here, it’s the area, often, to the axis, which represents some sort of zero condition, I think. Anyway, one method of calculating this area is to treat it as a series of rectangles, growing more or less infinitely smaller. Imagine dividing a circle into squares to determine its area – a big one extending from four equidistant points on the circumference, which would account for most of the area, and then progressively smaller squares in the interstices. You get the idea?
Canto: Yes, with that method you’d get infinitely closer to the precise area… Anyway, Dr Somara shows us a curve which is apparently the graphic representation of a formula, x⁴- 3x² + 1, and shows us how to find the integral, or at least shows us how we can divide the curve and its connection to the x axis by dividing it into rectangles. But what’s a more practical way of doing it? Well I’ll follow her precisely here. ‘If you know that your velocity is equal to twice time (v = 2t), then you know this is the derivative of position. So to find the equation for position, you have to look for an equation whose derivative is 2t, as for example, x = t². So x = 2t is the integral of v = t²
Jacinta: Yeah I can barely follow that. But the good doctor assures us that integral calculation is a bit messy. But apparently we can use the ‘power rule’ which we used for derivatives, and reverse it, in some instances. To quote: ‘Basically, you add one to the exponent, then divide the variable by that number’. Here’s an example: with v = 2t, x = 2/2t¹+¹, so x = 2/2t², so x = t². x = t² is the integral of v = 2t. She shows another more complex example, but I can’t do the notation for it with my limited keyboard skills. It involves some division. Anyway, with these mathematical methods we can look at trigonometric derivatives and do them backwards, e.g. the integral of cos(x) is sin(x).
Canto: We need to look at a variety of explanations of all this to bed it down methinks. I can only say I know a little more than I did, and that’s progress. And next we get onto constants. So what’s a constant (c)? It’s a number, and can literally be any number, positive, negative, fractional, whatever. It can be a placeholder, as for gravity, g (here on Earth), or presumably the speed of light, or ye olde cosmological constant, which is apparently still alive and well. Anyway, the derivative of a constant is always 0. That’s because a derivative’s a rate of change, and constants don’t change, by definition.
Jacinta: The derivative of t² is 2t, which presumably works by the power rule. Add any number and the derivative will always be 2t. That’s to say, the derivative of t² +/- (any number) is 2t. So, ‘if you’re looking for the integral of an equation like x = 2t, you have infinite choices, all of which are equally correct’. It could be t² or perhaps t² -18 or t² + 0.456. But I’m not clear on what this has to do with constants.
Canto: We’re flying blind, but it’s not too dangerous. The idea seems to be that the integral of x = 2t is t² + c. And here, if not back there, is where it gets tricky. With a bit of practice, we might know what the graph of the integral would look like, but not so much where it will lie vis-a-vis the vertical axis. For that we need to know more about the constant, ‘in order to know where to start drawing its integral. Whatever the constant is equal to, that’s where the curve will intersect with the vertical axis’. If it’s just t², it will intersect at zero, if it’s t² – 10, it’ll intersect at minus 10, etc. To avoid this infinity of integrals, the practice is to add c at the end of the integral, to stand for all the possible constants. So, saying that the integral of x = 2t is t² + c covers all the infinite options for c.
Jacinta: Well, Dr Somara next talks about the ‘initial value’, which you can apparently use to work out ‘where your integral is supposed to be on the y-axis’ without knowing the value for c – I think. For a graph of position, the initial value would be your starting position – where it intersects the vertical axis. This is the c value.
Canto: So returning to the bedroom window and the ball, the ball is dropped from the window sill at the same time the stopwatch starts. It hits the ground at 1.7 secs. So we know the time and the acceleration, 9.81m/sec². We need an equation for the ball’s position. We do this by finding its velocity, working out the integral of its acceleration. If you have a graph with the y-axis representing acceleration, which in this case is constant, and the x-axis representing time, the uniformly accelerating ball would be represented as a flat line, making the area under the ‘curve’ – between it and the x-axis – fairly easy to calculate. The area would be rectangular, and would be calculated by base x height. The base is t, the amount of time the ball took from release to hitting the ground, and the height is a, the acceleration, so it’s just a matter of a x t. The integral is at plus c, the constant. We need this constant, according to Dr Somara, because we can see that the velocity graph will be diagonal, a line ‘slanted in such a way that, every second, it rises by an amount equal to the acceleration’.
Jacinta: But, where to put it the line on the vertical axis? We’re looking for the integral of the acceleration, so we may use the power rule, which I still don’t get. So I’ll quote the doctor, for safety: ‘The acceleration, a, is a constant, but we could also say that it’s a x tº (t to the power zero)’. And anything ‘to the power zero’ is always 1. So, according to this mysterious power rule, the integral of acceleration (i.e the velocity) would be equal to the acceleration multiplied by the time – plus c. The c is added because we didn’t know where to place it on the y axis when time, on the x axis, is zero.
Canto: Yeah right. Let’s continue to quote the doctor – ‘Now here’s where the initial value [??] comes in. The velocity graph tells you what the velocity is for each moment in time. But we had to add the c, because we didn’t know…’ the initial value, being the velocity at time zero. ‘So the integral of the acceleration could have just been a x t, or at’. Or at plus or minus whatever. The c in the integral represents these options. But if we can work out the velocity at time zero (v0) we won’t need c. So, according to our Doctor, ‘if we write our equation with that v0 in it, as a placeholder for the velocity when time equals zero, we end up with the full equation for velocity, v = at + v0′. This is the kinematic equation, the definition of velocity. So the equation tells us that the final velocity of out tennis ball is at, that’s to say 9.8m/sec x 1.7 secs, that’s to say, 16.7m/sec (down, towards the Earth’s centre of gravity).
Jacinta: All of which seems to complicate something not quite so complicated. Anyhow, the pain isn’t over yet – we need to link acceleration with position, and this requires further integration, apparently. So, according to the power rule, which we should have learned, the integral of at is half at squared, and to get the integral of v0 you multiply it by t. Get it?
Canto: No. Let me quote from a highlighted comment: ‘i cant imagine how the avg viewer with no prior knowledge of calculus would actually understand calculus just by watching this video’.
Jacinta: Hmmm. Maybe we’ll try Khan Academy next. Anyway, if you put these integrals together you’ll get something that looks like the kinematic equation, the displacement curve. So for our example, we can work out the height from which the ball was dropped, or the distance the ball has travelled, using the initial position and time (zero and zero) plus half at squared. a was 9.81 m/sec², and t was 1.7 secs. t² is 2.89. Multiplying them makes 28.35, which, halved, is 14.175. metres. At least I got the calculation right, but as to the why….
References
https://sites.google.com/a/vistausd.org/physicsgraphicalanalysis/displacement-position-vs-time-graph
advancing solar, the photovoltaic effect, p-type semiconductors and the fiendishness of human manipulation
how to enslave electrons – human, all too human – stolen from E4U
Canto: Back to practical stuff for now (not that integral calculus isn’t practical), and the efficiencies in solar panels among other green technologies. Listening to podcasts such as those from SGU and New Scientist while walking the dog isn’t the best idea, what with doggy distractions and noise pollution from ICEs, so we’re going to take some of the following from another blog, Neurologica, which was also summarised on a recent SGU podcast.
Jacinta: Yes it’s all about improvements in solar panels, and the materials used in them, over the past couple of decades. We’re talking about improvements in lifespan and overall efficiency, not to mention cost to the consumer. Your standard silicon solar panels have improved efficiency since the mid 2000s from around 11% to around 28% – something like a 180% improvement. Is that good maths? Anyway, it’s the cheapest form of new energy and will become cheaper. And there’s also perovskite for different solar applications, and the possibility of quantum hi-tech approaches, using advanced AI technology to sort out the most promising. So the future is virtually impossible for we mere humans to predict.
Canto: Steven Novella, high priest of the SGU and author of the Neurologica post, suggests that with all the technological focus in this field today, who knows what may turn up – ‘researchers are doing amazing things with metamaterials’. He takes a close look at organic solar cells in particular, but these could possibly be combined with silicon and perovskite in the future. Organic solar cells are made from carbon-based polymers, essentially forms of plastic, which can be printed on various substrates. They’re potentially very cheap, though their life-span is not up to the silicon crystal level. However, their flexibility will suit applications other than rooftop solar – car roofs for example. They’re also more recyclable than silicon, which kind of solves the life-span problem. Their efficiency isn’t at the silicon level either, but that of course may change with further research. Scaling up production of these flexible organic solar materials has already begun.
Jacinta: So, I’ve mentioned perovskite, and I barely know what I’m talking about. So… some basic research tells me it’s a calcium titanium oxide mineral composed of calcium titanate (chemical formula CaTiO3), though any material with the ‘perovskite structure’ can be so called. It’s found in the earth’s mantle, in some chondritic meteorites, ejected limestone deposits and in various isolated locations such as the Urals, the Kola Peninsula in Russia, and such other far-flung places as Sweden and Arkansas. But I think the key is in the crystalline structure, which can be found in a variety of compounds.
Canto: Yes, worth watching perovskite developments in the future. I’m currently watching a video from Real Engineering called ‘the mystery flaw of solar panels’, which argues that this flaw has been analysed and solutions are being found. So, it starts with describing the problem – light-induced degradation, and explaining the photovoltaic effect:
The photovoltaic effect is the generation of voltage and electric current in a material upon exposure to light. It is a physical and chemical phenomenon.
Jacinta: Okay can we get clear again about the difference between voltage and current? I know that one is measured in volts and the other in amps but that explains nout.
Canto: Well, here’s one explanation – voltage, or emf, is the difference in electric potential between one point and another. Current is the rate of flow of an electric charge at any particular point. Check the references for more detail on that. Anyway we really are in the middle of a solar revolution, but the flaw in current solar panels is that newly manufactured solar cells are being tested at a little over 20% efficiency, that’s to say, 20% of the energy input from the sun is being converted into electric current. But within hours of operation the efficiency drops to 18% or so. That’s a 10% drop in generation, which becomes quite substantial on a large scale, with solar farms and such. So this is the problem of light-induced degradation, as mentioned. So, to quote the engineering video, ‘[the photovoltaic effect] is where photons of a particular threshold frequency, striking a material, can cause electrons to gain enough energy to free them from their atomic orbits and move freely in the material’. Semiconductors, which are sort of halfway between conductors and insulators, are the best materials for making this happen.
Jacinta: That’s strange, or counter-intuitive. Wouldn’t conductors be the best for getting electrons moving? Isn’t that why we use copper in electric wiring?
Canto: That’s a good question, which we might come back to. The first semiconducting material used, back in the 1880s, was (very expensive) selenium, which managed to create a continuous current with up to 1% efficiency. And so, silicon.
Jacinta: Which is essentially what we use, in inedible chip form, in all our electronic devices. Pretty versatile stuff. Will we always have enough of it?
Canto: Later. So when light hits this silicon crystal material, it can either be reflected, absorbed or neither – it may pass through without effect. Only absorption creates the photovoltaic effect. So, to improve efficiency we need to enhance absorption. Currently 30% of light is reflected from untreated silicon panels. If this wasn’t improved, maximum efficiency could only reach 70%. So we treat the panels with a layer of silicon monoxide reducing reflection to 10%. Add to that a layer of titanium dioxide, taking reflection to as low as 3%. A textured surface further enhances light absorption – for example light might be reflected sideways and hit another bump, where it’s absorbed. Very clever. But even absorbed light only has the potential to bring about the photovoltaic effect.
Jacinta: Yes, in order to create the effect, that is, to get electrons shifted, the photon has to be above a certain energy level, which is interesting, as photons aren’t considered to have mass, at least not when they’re at rest, but I’m not sure if photons ever rest… As the video says, ‘a photon’s energy is defined by multiplying Planck’s constant by its frequency’. That’s E = h.f, where h is Planck’s constant, which has been worked out by illustrious predecessors as 6.62607015 × 10−34 joule-seconds, according to the International System of Units (SI). And with silicon, the photons need an electromotive force of 1.1 electron volts to produce the photovoltaic effect, which can be converted, apparently, to a wavelength of 1,110 nanometres. That’s in the infrared, on the electromagnetic spectrum, near visible light. Any lower, in terms of energy (the lower the energy, the lower the frequency, the longer the wavelength, I believe), will just create heat and little light, a bit like my brain.
Canto: I couldn’t possibly comment on that, but the video goes on to explain that the solar energy we get from the sun, shown on a graph, is partially absorbed by our atmosphere before it reaches our panels. About 4% of the energy reaching us is in the ultraviolet, 44% is in the visible spectrum and 52% is in the infrared, surprisingly enough. Infrared red light has lower energy than visible light but it has a wider spectrum so the total energy emitted is greater. Now, silicon cannot use light above 1,110 nms in wavelength, meaning that some 19% of the sun’s energy can’t be used by our panels.
Jacinta: Yes, and another thing we’re supposed to note is that higher energy light doesn’t release more electrons, just higher energy electrons…
Canto: And presumably they’re talking about the electrons in the silicon structure?
Jacinta: Uhh, must be? So blue light – that’s at the short-wavelength end of the visible spectrum – blue light has about twice the energy of red light, ‘but the electrons that blue light releases simply lose their extra energy in the form of heat, producing no extra electricity. This energy loss results in about 33% of sunlight’s energy being lost.’ So add that 33% to the 19% lost at the long-wavelength end, that’s 52% of potential energy being lost. These are described as ‘spectrum losses’.
Canto: Which all sounds bad, but silicon, or its reaction with photons, has a threshold frequency that ‘balances these two frequency losses’. So, it captures enough of the low-energy wavelengths (the long wavelengths beyond the infra-red), while not losing too much efficiency due to heat. The heat problem can be serious, though, requiring active cooling in some climates, thus reducing efficiency in a vicious circle of sorts. Still, silicon is the best of threshold materials we have, presumably.
Jacinta: So, onto the next piece of physics, which is that there’s more to creating an electric current than knocking an electron free from its place in ye olde lattice, or whatever. For starters, ye olde electron just floats about like a lost lamb.
Canto: No use to anyone.
Jacinta: Yeah, it needs to be forced into doing work for us.
Canto: Because humans are arseholes who make slaves of everything that moves. Free the electrons!
Jacinta: You’ve got it. They need to be forced to work an electric circuit. And interestingly, the hole left when we’ve knocked an electron out of its happy home, that hole is also let loose to roam about like a lost thing. Free electrons, free holes, when they meet, they’re happy but the circuit is dead before it starts.
Canto: This sounds like a tragicomedy.
Jacinta: So we have to reduce the opportunities for electrons and holes to meet. Such is the cruelty of progress. For of course, we must needs use force, taking advantage of silicon’s unique properties. The most excellent crystal structure of the element is due to its having 4 electrons in its outer shell. So it bonds covalently with 4 other silicon atoms. And each of those bonds with 3 others and so on. A very stable balance. So the trick that we manipulative humans use to mess up this divine balance is to introduce impurities called dopants into the mix. If we add boron, which has 3 outer electrons, into the crystal lattice, this creates 3 covalent bonds with silicon, leaving – a hole!
Canto: How fiendishly clever!
Jacinta: It’s called a p-type trick, as it has this ‘positive’ hole just waiting for an electron to fill it. Sounds kind of sexy really.
Canto: Manipulation can be sexy in a perverse way. Stockholm syndrome for electrons?
Jacinta: Okay, there’s a lot more to this, but we’ve gone on long enough. I’ve had complaints that our blog posts are too long. Well, one complaint, because only one or two people read our stuff…
Canto: No matter – at least we’ve learned something. Let’s continue to rise above ourselves and grasp the world!
Jacinta: Okay, to be continued….
References
https://www.theskepticsguide.org/podcasts
https://news.mit.edu/2022/perovskites-solar-cells-explained-0715
The Mystery Flaw of Solar Panels (Real Engineering video)
https://byjus.com/physics/difference-between-voltage-and-current/
an interminable conversation 8: eddy currents, Ampere’s Law and other physics struggles
easy peasy
Canto: So we were talking about eddy currents, but before we get there, I’d like to note that, according to one of the various videos I’ve viewed recently, this connection between electricity and magnetism, first observed by Faraday and Henry, and brilliantly mathematised by James Clerk Maxwell, has transformed our human world perhaps more than any other discovery in our history. I think this is why I’m really keen to comprehend it more thoroughly before I die.
Jacinta: Yeah very touching. So what about eddy currents?
Canto: Okay, back to Wikipedia:
Eddy currents (also called Foucault’s currents) are loops of electrical current induced within conductors by a changing magnetic field in the conductor according to Faraday’s law of induction or by the relative motion of a conductor in a magnetic field. Eddy currents flow in closed loops within conductors, in planes perpendicular to the magnetic field. They can be induced within nearby stationary conductors by a time-varying magnetic field created by an AC electromagnet or transformer, for example, or by relative motion between a magnet and a nearby conductor.
Jacinta: Right. All is clear. End of post?
Canto: Well, this ‘perpendicular’ thing has been often referred to. I’ll steal this Wikipedia diagram, and try to explain it in my own words.
So, the eddy currents are drawn in red. They’re induced in a metal plate (C)…
Jacinta: What does induced actually mean?
Canto: That’s actually quite a difficult one. Most of the definitions of electrical induction I’ve encountered appear to be vague if not circular. Basically, it just means ‘created’ or ‘produced’.
Jacinta: Right. So, magic?
Canto: The fact that an electric current can be produced (say in a conductive wire like copper) by the movement of a magnet suggests strongly that magnetism and electricity are counterparts. That’s the central point. That’s why we refer to electromagnetism, and electromagnetic theory, because the connections – between the conductivity and resistance of the wire and the strength and movement of the magnet (for example it can be made to spin) will determine the strength of the electric field, or the emf, and all this can be calculated precisely via an equation or set of equations, which helps us to use the emf to create useful energy.
Jacinta: Okay, so this metal plate is moving, and I’m guessing V stands for velocity. The plate is a conductor, and the nearby magnet (N – that’s the magnet’s north pole) produces, or induces, a magnetic field (B) – or it just has a magnetic field, being a magnet, and this creates a current in the plate.
Canto: Which is perpendicular to the magnetic field, because what causes the current in the plate is the movement of electrons, which can’t jump out of the plate after all, but move within the plane of the plate. And the same would go for a wire. There’s also the matter of the direction, within the plane, of the current – clockwise or anticlockwise? And many other things beyond my understanding.
Jacinta: Would it help to try for a historical account, going back to the 18th century – Franklin, Cavendish, even Newton? The beginning of the proper mathematisation of physical forces? I mean, all I wanted to know was how an induction stovetop worked.
Canto: That’s life – you wonder why x does y and you end up reflecting on the origin of the universe. I’ve looked at a couple of videos, and they explain well enough what happens when a magnet goes inside an electrified coil, but never really explain why. But let’s just start with Faraday. He was a great experimenter, as they all tell me, but not too much of a mathematician. Faraday wasn’t the first to connect electricity with magnetism, though. H C Ørsted was the first, I think, to announce, and presumably to discover, that an electric current flowing through a wire produced a magnetic field around it. That was around 1820, which dates the first recognised connection between electricity and magnetism. The discovery was drawn to the attention of Andre-Marie Ampère, who began experimenting with, and mathematising, the relationship. Here’s a quote from Britannica online:
Extending Ørsted’s experimental work, Ampère showed that two parallel wires carrying electric currents repel or attract each other, depending on whether the currents flow in the same or opposite directions, respectively. He also applied mathematics in generalizing physical laws from these experimental results. Most important was the principle that came to be called Ampère’s law, which states that the mutual action of two lengths of current-carrying wire is proportional to their lengths and to the intensities of their currents.
Jacinta: That’s interesting – what does the mutual action mean? So we have two lengths of wire, which could be flowing in the same direction, in which case – what? Do they attract or repel? Presumably they repel, as like charges repel. But that’s magnetism, not electricity. But it’s both, as they were starting to discover. But how, proportional to the lengths of the wire? I can imagine that the intensity of the currents would be proportional to the degree of attraction or repulsion – but the length of the wires?
Canto: You want more bamboozlement? Here’s another version of Ampère’s law:
The integral around a closed path of the component of the magnetic field tangent to the direction of the path equals μ0 times the current intercepted by the area within the path.
The magnetic field created by an electric current is proportional to the size of that electric current with a constant of proportionality equal to the permeability of free space.
Canto: The symbol in in the equation above, (μ0), is a physical constant used in electromagnetism. It refers to the permeability of free space. My guess is that it wasn’t defined that way by Ampère.
Jacinta: I understand precisely nothing about that equation. Please tell me what an integral is, as if that might provide enlightenment.
Canto: It’s about quantifying areas defined by or under curves. And a tangent – but let’s not get into the maths.
Jacinta: But we have to!
Canto: Well, briefly for now, a tangent in maths can sort of mean more than one thing, I think. If you picture a circle, a tangent is a straight line that touches once the circumference of the circle. So that straight line could be horizontal, vertical or anything in between.
Jacinta: Right. And how does that relate to electromagnetism?
Canto: Okay, let’s return to Ampère’s experiment. Two parallel wires attracted each other when their currents were running in the same direction, and repelled each other when they were running in the opposite direction. It’s also the case – and I don’t know if this was discovered by Ampère, but never mind – that if you coil up a wire (carrying a current), the inside of the coil acts like a magnet, with a north and south pole. Essentially, what is happening is that the current in a wire creates a magnetic field around it, circling in a particular direction – either clockwise or anti-clockwise. The magnetic field is ‘stronger’ the closer it is to the wire. So there’s clearly a relationship between distance from the wire and field strength. And there’s also a relationship between field strength and the strength of the current in the wire. It’s those relations, which obviously can be mathematised, that are the basis of Ampère’s Law. So here’s another definition – hopefully one easier to follow:
The equation for Ampère’s Law applies to any kind of loop, not just a circle, surrounding a current, no matter how many wires there are, or how they’re arranged or shaped. The law is valid as long as the current is constant.
That’s the easy part, and then there’s the equation, which I’ll repeat here, and try to explain:
reading matters 1
The universe within by Neil Turok (theoretical physicist extraordinaire)
Content hints
– Massey Lectures, magic that works, the ancient Greeks, David Hume and the Scottish Enlightenment, James Clerk Maxwell, quantum mechanics, entanglement, expanding and contracting universes, the square root of minus one, mathematical science in Africa, Paul Dirac, beauty and knowledge, the vitality of uncertainty, Mary Shelley, quantum computing, digital and analogue, Richard Feynman, science and humanity, humility, education, love, collaboration, creativity and thrill-seeking.
towards James Clerk Maxwell 6: Newton’s universal law of gravitation and G
Newton’s law of gravity goes like this:
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.
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
towards James Clerk Maxwell 4: a detour into dimensional analysis and Newton’s laws
Canto: Getting back to J C Maxwell, I’m trying to learn some basic physics, which may or may not be relevant to electromagnetism, but which may help me to get in the zone, so to speak.
Jacinta: Yes, we’re both trying to brush up on physics terms and calculations. For example, acceleration is change in velocity over time, which is hard to put in notation form in a blog post, but I can steal it from elsewhere
in which the triangle represents ‘change in’. Now velocity is a vector quantity, therefore so is acceleration – it’s a particular magnitude in a particular direction. So imagine a car that goes from stationary to, say 50 kms/hour in 5 seconds, what’s the acceleration? According to the formula, it’s 50 – 0 kph/5 seconds, or 10kph/sec, which we can write out as a change of velocity of ten kilometres per hour per second.
Canto: So every second, the velocity of the car is increasing by 10 kilometres per hour. I’m trying to picture that. It’s quite hard.
Jacinta: Okay while you’re doing that, let’s introduce dimensional analysis, so that we reduce everything to the same dimension, sort of. I mean, we have hours and seconds here, so let’s take it all to seconds. I won’t be able to do this properly without an equation-writing plug-in, which I can’t work out how to get without paying. Anyhow..
10 kms/hour.second.1/3600 hour/second. Cancelling out the hours, you get 10 kms/3600 seconds squared, or 1/360 km/s2
Canto: I wonder if there’s a way of hand-writing equations in the blog, that’d be more fun and easy. So can you briefly explain dimensional analysis?
Jacinta: Well physical quantities are often measured in different units – for example, quantities of time – time is called the base quantity – are measured in seconds, hours, days etc. So, it’s just a matter of getting such measurements to be commensurate, so that an equation can be simplified – all in seconds, or all in metres, when they can be. Though actually it’s more complicated than that, and I’ve probably got it wrong.
Canto: So talking of brushing up on stuff, or actually knowing about stuff for the first time, I thought it might be good to go back to Newton, his three laws of motion, in written and mathematical form.
Jacinta: Go ahead.
Canto: Well, the first law, which really comes from Galileo, is often called the law of inertia. Newton formulated it this way, in the Principia (translated from Latin):
Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.
And as Sal Khan and others point out, Newton is talking about an unbalanced force, one that isn’t matched by an equal and opposite force (which would be a balanced force – see Newton’s third law). This law doesn’t come with a mathematical formula.
The second law, which I filched from The Physics Classroom, can be stated thus:
The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
It’s famous formula is this:
Fnet = m • a
It can be written different ways, for example simply F =m.a, or with the vector sign (an arrow) above force (F) and acceleration (a), showing the same direction, but it’s certainly important to explain net force here. It’s essentially the sum of all the forces acting on the mass, in vector or directional terms. It’s this net force that produces the acceleration.
So to the third law, and this is how Newton presented it, again translated from Latin:
To every action there is always an equal and opposite reaction: or the forces of two bodies are always equal and are directed in opposite directions.
It’s often stated in this ‘wise proverb’ sort of way: ‘for every action there’s an equal and opposite reaction’.
Jacinta: What goes around comes around.
Canto: That’s more of a wise-guy thing. Anyway, the best formula for the third law is:
FA = −FB
where force A is the action and force B the reaction. This law is sort of counter-intuitive and also sort of obvious at the same time! I think it’s the most brilliant law. Sal Khan gives a nice extra-terrestrial example of how you might utilise it. Imagine you’re in outer space and you’ve been cut off from your spaceship and are accelerating away from it. To save yourself, take something massive, if you can, something on your suit or a tool you’re carrying, and push it hard away from you in the opposite direction to the ship, and this should send you accelerating back to the ship. But make sure your aim is true!
Jacinta: Okay, so this seems to have taken us absolutely no closer to Maxwell’s equations.
Canto: Well, yes and no. It makes us think of forces and energy, albeit of a different kind, and it makes us think in a logical, semi-mathematical way. but we’ve certainly got a long way to go…
References
https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion
https://www.physicsclassroom.com/class/newtlaws/Lesson-1/Newton-s-First-Law
https://www.livescience.com/46558-laws-of-motion.html
https://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
On electrickery, part 1 – the discovery of electrons
Canto: This could be the first of a thousand-odd parts, because speaking for myself it will take me several lifetimes to get my head around this stuff, which is as basic as can be. Matter and charge and why is it so and all that.
Jacinta: so let’s start at random and go in any direction we like.
Canto: Great plan. Do you know what a cathode ray is?
Jacinta: No. I’ve heard of cathodes and anodes, which are positive and negative terminals of batteries and such, but I can’t recall which is which.
Canto: Don’t panic, Positive is Anode, Negative Is Cathode. Though I’ve read somewhere that the reverse can sometimes be true. The essential thing is they’re polar opposites.
Jacinta: Good, so a cathode ray is some kind of negative ray? Of electrons?
Canto: A cathode ray is defined as a beam of electrons emitted from the cathode of a high-vacuum tube.
Jacinta: That’s a pretty shitty definition, why would a tube, vacuum or otherwise, have a cathode in it? And what kind of tube? Rubber, plastic, cardboard?
Canto: Well let’s not get too picky. I’m talking about a cathode ray tube. It’s a sealed tube, obviously, made of glass, and evacuated as far as possible. Sciencey types have been playing around with vacuums since the mid seventeenth century – basically since the vacuum pump was invented in 1654, and electrical experiments in the nineteenth century, with vacuum tubes fitted with cathodes and anodes, led to the discovery of the electron by J J Thomson in 1897.
Jacinta: So what do you mean by a beam of electrons and how is it emitted, and can you give more detail on the cathode, and is there an anode involved? Are there such things as anode rays?
Canto: I’ll get there. Early experiments found that electrostatic sparks travelled further through a near vacuum than through normal air, which raised the question of whether you could get a ‘charge’, or a current, to travel between two relatively distant points in an airless tube. That’s to say, between a cathode and an anode, or two electrodes of opposite polarity. The cathode is of a conducting material such as copper, and yes there’s an anode at the other end – I’m talking about the early forms, because in modern times it starts to get very complicated. Faraday in the 1830s noted a light arc could be created between the two electrodes, and later Heinrich Geissler, who invented a better vacuum, was able to get the whole tube to glow – an early form of ‘neon light’. They used an induction coil, an early form of transformer, to create high voltages. They’re still used in ignition systems today, as part of the infernal combustion engine
Jacinta: So do you want to explain what a transformer is in more detail? I’ve certainly heard of them. They ‘create high voltages’ you say. Qu’est-ce que ça veux dire?
Canto: Do you want me to explain an induction coil, a transformer, or both?
Jacinta: Well, since we’re talking about the 19th century, explain an induction coil.
Canto: Search for it on google images. It consists of a magnetic iron core, round which are wound two coils of insulated copper, a primary and secondary winding. The primary is of coarse wire, wound round a few times. The secondary is of much finer wire, wound many many more times. Now as I’ve said, it’s basically a transformer, and I don’t know what a transformer is, but I’m hoping to find out soon. Its purpose is to ‘produce high-voltage pulses from a low-voltage direct current (DC) supply’, according to Wikipedia.
Jacinta: All of this’ll come clear in the end, right?
Canto: I’m hoping so. When a current – presumably from that low-volage DC supply – is passed through the primary, a magnetic field is created.
Jacinta: Ahh, electromagnetism…
Canto: And since the secondary shares the core, the magnetic field is also shared. Here’s how Wikipedia describes it, and I think we’ll need to do further reading or video-watching to get it clear in our heads:
The primary behaves as an inductor, storing energy in the associated magnetic field. When the primary current is suddenly interrupted, the magnetic field rapidly collapses. This causes a high voltage pulse to be developed across the secondary terminals through electromagnetic induction. Because of the large number of turns in the secondary coil, the secondary voltage pulse is typically many thousands of volts. This voltage is often sufficient to cause an electric spark, to jump across an air gap (G) separating the secondary’s output terminals. For this reason, induction coils were called spark coils.
Jacinta: Okay, so much for an induction coil, to which we shall no doubt return, as well as to inductors and electromagnetic radiation. Let’s go back to the cathode ray tube and the discovery of the electron.
Canto: No, I need to continue this, as I’m hoping it’ll help us when we come to explaining transformers. Maybe. A key component of the induction coil was/is the interruptor. To have the coil functioning continuously, you have to repeatedly connect and disconnect the DC current. So a magnetically activated device called an interruptor or a break is mounted beside the iron core. It has an armature mechanism which is attracted by the increasing magnetic field created by the DC current. It moves towards the core, disconnecting the current, the magnetic field collapses, creating a spark, and the armature springs back to its original position. The current is reconnected and the process is repeated, cycling through many times per second.
A Crookes tube showing green fluorescence. The shadow of the metal cross on the glass showed that electrons travelled in straight lines
Jacinta: Right so now I’ll take us back to the cathode ray tube, starting with the Crookes tube, developed around 1870. When we’re talking about cathode rays, they’re just electron beams. But they certainly didn’t know that in the 1870s. The Crookes tube, simply a partially evacuated glass tube with cathode and anode at either end, was what Rontgen used to discover X-rays.
Canto: What are X-rays?
Jacinta: Electromagnetic radiation within a specific range of wavelengths. So the Crookes tube was an instrument for exploring the properties of these cathode rays. They applied a high DC voltage to the tube, via an induction coil, which ionised the small amount of air left in the tube – that’s to say it accelerated the motions of the small number of ions and free electrons, creating greater ionisation.
x-rays and the electromagnetic spectrum, taken from an article on the Chandra X-ray observatory
Canto: A rapid multiplication effect called a Townsend discharge.
Jacinta: An effect which can be analysed mathematically. The first ionisation event produces an ion pair, accelerating the positive ion towards the cathode and the freed electron toward the anode. Given a sufficiently strong electric field, the electron will have enough energy to free another electron in the next collision. The two freed electrons will in turn free electrons, and so on, with the collisions and freed electrons growing exponentially, though the growth has a limit, called the Raether limit. But all of that was worked out much later. In the days of Crookes tubes, atoms were the smallest particles known, though they really only hypothesised, particularly through the work of the chemist John Dalton in the early nineteenth century. And of course they were thought to be indivisible, as the name implies.
Canto: We had no way of ‘seeing’ atoms in those days, and cathode rays themselves were invisible. What experimenters saw was a fluorescence, because many of the highly energised electrons, though aiming for the anode, would fly past, strike the back of the glass tube, where they excited orbital electrons to glow at higher energies. Experimenters were able to enhance this fluorescence through, for example, painting the inside walls of the tube with zinc sulphide.
Jacinta: So the point is, though electrical experiments had been carried out since the days of Benjamin Franklin in the mid-eighteenth century, and before, nobody knew how an electric current was transmitted. Without going into much detail, some thought they were carried by particles (like radiant atoms), others thought they were waves. J J Thomson, an outstanding theoretical and mathematical physicist, who had already done significant work on the particulate nature of matter, turned his attention to cathode rays and found that their velocity indicated a much lighter ‘element’ than the lightest element known, hydrogen. He also found that their velocity was uniform with respect to the current applied to them, regardless of the (atomic) nature of the gas being ionised. His experiments suggested that these ‘corpuscles’, as they were initially called, were 1000 times lighter than hydrogen atoms. His work was clearly very important in the development of atomic theory – which in large measure he initiated – and he developed his own ‘plum pudding’ theory of atomic structure.
Canto: So that was all very interesting – next time we’ll have a look at electricity from another angle, shall we?
So why exactly is the sky blue? SfD tries to investigate
Canto: Well, Karl Kruszelnicki is one of our best science popularisers as you know, and therefore a hero of ours, but I have to say his explanation of the blueness of our daily sky in his book 50 Shades of Grey left me scratching my head…
Jacinta: Not dumbed-down enough for you? Do you think we could form a Science for Dummies collaboration to do a better job?
Canto: Well that would really be the blind leading the blind, but at least we’d inch closer to understanding if we put everything in our own words… and that’s what I’m always telling my students to do.
Jacinta: So let’s get down to it. The day-sky is blue (or appears blue to we humans?) because…?
Canto: Well the very brief explanation given by Dr Karl is that it’s about Rayleigh scattering. Named for a J W Strutt, aka Lord Rayleigh, who first worked it out.
Jacinta: So let’s just call it scattering. What’s scattering?
Canto: Or we might call it light scattering. Our atmosphere is full of particles, which interfere with the light coming to us from the sun. Now while these particles are all more or less invisible to the naked eye, they vary greatly in size, and they’re also set at quite large distances from each other, relative to their size. The idea, broadly, is that light hits us from the sun, and that’s white light, which as we know from prisms and rainbows is made up of different wavelengths of light, which we see, in the spectrum that’s visible to us, as Roy G Biv, red orange yellow green blue indigo violet, though there’s more of some wavelengths or colours than others. Red light, because it has a longer wavelength than blue towards the other end of the spectrum, tends to come straight through from the sun without hitting too many of those atmospheric particles, whereas blue light hits a lot more particles and bounces off, often at right angles, and kind of spreads throughout the sky, and that’s what we mean by scattering. The blue light, or photons, bounce around the sky from particle to particle before hitting us in the eye so to speak, and so we see blue light everywhere up there. Now, do you find that a convincing explanation?
Jacinta: Well, partly, though it raises a lot of questions.
Canto: Excellent. That’s science for you.
Jacinta: You say there are lots of particles in the sky. Does the size of the particle matter? I mean, I would assume that the light, or the photons, would be more likely to hit large particles than small ones, but that would depend on just how many large particles there are compared to small ones. Surely our atmosphere is full of molecular nitrogen and oxygen, mostly, and they’d be vastly more numerous than large dust particles. Does size matter? And you say that blue light, or blue photons, tend to hit these particles because of their shorter wavelengths. I don’t quite get that. Why would something with a longer wavelength be more likely to miss? I think of, say, long arrows and short arrows. I see no reason why a longer arrow would tend to miss the target particles – not that they’re aiming for them – while shorter arrows hit and bounce off. And what makes them bounce off anyway?
Canto: OMG what a smart kid you are. And I think I can add more to those questions, such as why do we see different wavelengths of light as colours anyway, and why do we talk sometimes of waves and sometimes of particles called photons? But let’s start with the question of whether size matters. All I can say here is that it certainly does, but a fuller explanation would be beyond my capabilities. For a start, the particles hit by light are not only variable by size but by shape, and so potentially infinite in variability. Selected geometries of particles – for example spherical ones – can yield solutions as to light scattering based on the equations of Maxwell, but that doesn’t help much with random dust and ice particles. Rayleigh scattering deals with particles much smaller than the light’s wavelength but many particles are larger than the wavelength, and don’t forget light is a bunch of different wavelengths, striking a bunch of different sized and shaped particles.
Jacinta: Sounds horribly complex, and yet we get this clear blue sky. Are you ready to give up now?
Canto: Just about, but let me tackle this bouncing off thing. Of course this happens all the time, it’s called reflection. You see your reflection in the mirror because mirrors are designed as highly reflective surfaces.
Jacinta: Highly bounced-off. So what would a highly unreflective surface look like?
Canto: Well that would be something that lets all the light through without reflection or distortion, like the best pane of glass or pair of specs. You see the sky as blue because all these particles are absorbing and reflecting light at particular wavelengths. That’s how you see all colours. As to why things happen this way, OMG I’m getting a headache. The psychologist Thalma Lobel highlights the complexity of it all this way:
A physicist would tell you that colour has to do with the wavelength and frequency of the beams of light reflecting and scattering off a surface. An ophthalmologist would tell you that colour has to do with the anatomy of the perceiving eye and brain, that colour does not exist without a cornea for light to enter and colour-sensitive retinal cones for the light-waves to stimulate. A neurologist might tell you that colour is the electro-chemical result of nervous impulses processed in the occipital lobe in the rear of the brain and translated into optical information…
Jacinta: And none of these perspectives would contradict the others, it would all fit into the coherence theory of truth…
Canto: Not truth so much as explanation, which approaches truth maybe but never gets there, but the above quote gives a glimpse of how complex this matter of light and colour really is…
Jacinta: And just on the physics, I’ve looked at a few explanations online, and they don’t satisfy me.
Canto: Okay, I’m going to end with another quote, which I’m hoping may give you a little more satisfaction. This is from Live Science.
The blueness of the sky is the result of a particular type of scattering called Rayleigh scattering, which refers to the selective scattering of light off of particles that are no bigger than one-tenth the wavelength of the light.
Importantly, Rayleigh scattering is heavily dependent on the wavelength of light, with lower wavelength light being scattered most. In the lower atmosphere, tiny oxygen and nitrogen molecules scatter short-wavelength light, such as blue and violet light, to a far greater degree than long-wavelength light, such as red and yellow. In fact, the scattering of 400-nanometer light (violet) is 9.4 times greater than the scattering of 700-nm light (red).
Though the atmospheric particles scatter violet more than blue (450-nm light), the sky appears blue, because our eyes are more sensitive to blue light and because some of the violet light is absorbed in the upper atmosphere.
Jacinta: Yeah so that partially answers some of my questions… ‘selective scattering’, there’s something that needs unpacking for a start…
Canto: Well, keep asking questions, smart ones as well as dumb ones…
Jacinta: Hey, there are no dumb questions. Especially from me. Remember this is supposed to be science for dummies, not science by dummies.
Canto: Okay then. So maybe we should quit now, before we’re found out…
References:
‘Why is the sky blue?’, from 50 shades of grey matter, Karl Kruszelnicki, pp15-19
‘Blue skies smiling at me: why the sky is blue’, from Bad astronomy, Philip Plait, pp39-47
http://www.livescience.com/32511-why-is-the-sky-blue.html
more on Einstein, black holes and other cosmic stuff
Einstein at Mount Wilson in 1931, staring at a wee bit of the universe, with Edwin Hubble
Jacinta: Well Canto I’d like to get back to Einstein and space and time and the cosmos, just because it’s such a fascinating field to inhabit and explore.
Canto: Rather a big one.
Jacinta: I’ve read, or heard, that Einstein’s theory, or one of them, predicted black holes, though he didn’t necessarily think that such entities really existed, but now black holes are at the centre of everything, it seems.
Canto: Including our own galaxy, and most others.
Jacinta: Yes, and there appears to be a correlation between the mass of these supermassive black holes at the centres of galaxies and the mass of the galaxies themselves, indicating that they appear to be the generators of galaxies. Can you expand on that?
Canto: Well the universe seems to be able to expand on that better than I can, but I’ll try. Black holes were first so named in the 1960s, but Einstein’s theory of general relativity recast gravity as a distortion of space and time rather than as a Newtonian force, with the distortion being caused by massive objects. The greater the mass, the greater the distortion, or the ‘geodetic effect’, as it’s called. The more massive a particular object, given a fixed radius, the greater is the velocity required for an orbiting object to escape its orbit, what we call its escape velocity. That escape velocity will of course, approacher closer and closer to the speed of light, as the object being orbited becomes more massive. So what happens when it reaches the speed of light? Then there’s no escape, and that’s where we enter black hole territory.
Jacinta: So, let me get this. Einstein’s theory is about distortions of space-time (and I’m not going to pretend that I understand this), or geodetic effects, and so it has to account for extreme geodetic effects, where the distortion is so great that nothing, not even light, can escape, and everything kind of gets sucked in… But how do these massive, or super-massive objects come into being, and won’t they eventually swallow all matter, so that all is just one ginormous black hole?
Canto: Okay I don’t really get this either but shortly after Einstein published his theory it was worked out by an ingenious astrophysicist, Karl Schwarzschild – as a result of sorting out Einstein’s complex field equations – that if matter is severely compressed it will have weird effects on gravity and energy. I was talking a minute ago about increasing the mass, but think instead of decreasing the radius while maintaining the mass as a constant…
Jacinta: The same effect?
Karl Schwarzschild
Canto: Well, maybe, but you’ll again reach a point where there’s zero escape, so to speak. In fact, what you have is a singularity. Nothing can escape from the object’s surface, whether matter or radiation, but also you’ll have a kind of internal collapse, in which the forces that keep atoms and sub-atomic particles apart are overcome. It collapses into an infinitesimal point – a singularity. It was Schwarzschild too who described the ‘event horizon’, and provided a formula for it.
Jacinta: That’s a kind of boundary layer, isn’t it? A point of no return?
Canto: Yes, a spherical boundary that sort of defines the black hole.
Jacinta: So why haven’t I heard of this Schwarzschild guy?
Canto: He died in 1916, shortly after writing a paper which solved Einstein’s equations and considered the idea of ‘point mass’ – what we today would call a singularity. But both he and Einstein, together with anyone else in the know, would’ve considered this stuff entirely theoretical. It has only become significant, and very significant, in the last few decades.
Jacinta: And doesnt this pose a problem for Einstein’s theory? I recall reading that this issue of ‘point mass’, or a situation where gravity is kind of absolute, like with black holes and the big bang, or the ‘pre-big bang’ if that makes sense, is where everything breaks down, because it seems to bring in the mathematical impossibility of infinity, something that just can’t be dealt with mathematically. And Einstein wasn’t worried about it in his time because black holes were purely theoretical, and the universe was thought to be constant, not expanding or contracting, just there.
Canto: Well I’ve read – and I dont know if it’s true – that Einstein believed, at least for a time, that black holes couldn’t actually exist because of an upper limit imposed on the gravitational energy any mass can produce – preventing any kind of ‘infinity’ or singularity.
Jacinta: Well if that’s true he was surely wrong, as the existence of black holes has been thoroughly confirmed, as has the big bang, right?
Canto: Well of course knowledge was building about that in Einstein’s lifetime, as Edwin Hubble and others provided conclusive evidence that the universe was expanding in 1929, so if this expansion was uniform and extended back in time, it points to an early much-contracted universe, and ultimately a singularity. And in fact Einstein’s general relativity equations were telling him that the universe wasn’t static, but he chose to ignore them, apparently being influenced by the overwhelming thinking of the time – this was 1917 – and he introduced his infamous or famous cosmological constant, aka lambda.
Jacinta: And of course 1917 was an early day in the history of modern astronomy, we hardly knew anything beyond our own galaxy.
Canto: Or within it. One of the great astrophysicists of the era, Sir Arthur Eddington, believed at the time that the sun was at the centre of the universe, while admitting his calculations were ‘subject to large uncertainties’.
Jacinta: Reminds me of Lord Kelvin on the age of the Earth only a few generations before.
Canto: Yes, how quickly our best speculations can become obsolete, but that’s one of the thrills of science. And it’s worth noting that the work of Hubble and others on the expansion of the universe depended entirely on improved technology, namely the 100-inch Hooker telescope at Mount Wilson, California.
Jacinta: Just as the age of the Earth problem was solved through radiometric dating, which depended on all the early twentieth century work on molecular structure and isotopes and such.
Canto: Right, but now this lambda (λ) – which Einstein saw as a description of some binding force in gravity to counteract the expansion predicted by his equations – is very much back in the astrophysical frame. The surprising discovery made in 1998 that the universe’s expansion is accelerating rather than slowing has, for reasons I can’t possibly explain, brought Einstein’s lambda in from the cold as an explanatory factor in that discovery, which is also somehow linked to dark energy.
Jacinta: So his concept, which he simply invented as a ‘fix-it’ sort of thing, might’ve had more utility than he knew?
Canto: Well the argument goes, among some, that Einstein was a scientist of such uncanny insight that even his mistakes have proved more fruitful than others’ discoveries. Maybe that’s hero worship, maybe not.
Jacinta: So how does lambda relate to dark energy, and how does dark energy relate to dark matter, if you please?
Canto: Well the standard model of cosmology (which is currently under great pressure, but let’s leave that aside) has been unsuccessful in trying to iron out inconsistent observations and finding with regard to the energy density of the universe, and so dark energy and what they call cold dark matter (CDM) have been posited as intellectual placeholders, so to speak, to make the observations and equations come out right.
Jacinta: Sounds a bit dodgy.
Canto: Well, time will tell how dodgy it is but I don’t think anyone’s trying to be dodgy, there’s a great deal of intense calculation and measurement involved, with so many astrophysicists looking at the issue from many angles and with different methods. Anyway, to quickly summarise CDM and dark energy, they together make up some 96% of the mass-energy density of our universe according to the most currently accepted calculations, with dark energy accounting for some 69% and CDM accounting for about 27%.
Jacinta: Duhh, that does sound like a headachey problem for the standard model. I mean, I know I’m only a dilettanty lay-person, but a model of universal mass-energy that only accounts for about 4% of the stuff, that doesn’t sound like much of a model.
Canto: Well I can assure they’re working on it…
Jacinta: Or working to replace it.
Canto: That too, but let me try to explain the difference between CDM and dark energy. Dark energy is associated with lambda, because it’s the ‘missing energy’ that accounts for the expansion of the universe, against the binding effects of gravity. As it happens, Einstein’s cosmological constant pretty well perfectly counters this expansive energy, so that if he hadn’t added it to his equations he would’ve been found to have predicted an expanding universe decades before this was confirmed by observation. That’s why it was only in the thirties that he came to regret what he called the greatest mistake of his career. Cold dark matter, on the other hand, has been introduced to account for a range of gravitational effects which require lots more matter than we find in the observed (rather than observable) universe. These effects include the flat shapes of galaxies, gravitational lensing and the tight clustering of galaxies. It’s described as cold because its velocity is considerably less than light-speed.
the lambda- cold dark matter model
Jacinta: Okay, so far so bad, but let’s get back to black holes. Why are they so central?
Canto: Well, that’s perhaps the story of supermassive black holes in particular, but I suppose I should try to tell the story of how astronomers found black holes to be real. As I’ve said, the term was first used in the sixties, 1967 to be precise, by John Wheeler, at a time when their actual existence was being considered increasingly likely, and the first more or less confirmed discovery was made in 1971 with Cygnus x-1. You can read all about it here. It’s very much a story of developing technology leading to increasingly precise observational data, largely in the detecting and measuring of X-ray emissions, stuff that was undetectable to us with just optical instruments.
Jacinta: Okay, go no further, I accept that there’s been a lot of data from a variety of sources that have pretty well thoroughly confirmed their existence, but what about these supermassive black holes? Could they actually be the creators of matter in the galaxies they’re central to? That’s what I’ve heard, but my reception was likely faulty.
Canto: Well astrophysicists have been struggling with the question of this relationship – there clearly is a relationship between supermassive black holes and their galaxies, but which came first? Now supermassive black holes can vary a lot – our own ‘local’ one is about 4 million solar masses, but we’ve discovered some with billions of solar masses. But it was found almost a decade ago that there is correlation between the mass of these beasties and the mass of the inner part of the galaxies that host them – what they call the galactic bulge. The ratio is always about 1 to 700. Obviously this is highly suggestive, but it requires more research. There are some very interesting examples of active super-feeding black holes emitting vast amounts of energy and radiation, which is both destructive and productive in a sense, creating an active galaxy. Our own Milky Way, or the black hole at its centre, is currently quiescent, which is just as well.
Jacinta: You mean if it starts suddenly feeding, we’re all gonna die?
Canto: No probably not, the hole’s effects are quite localised, relatively speaking, and we’re a long way from the centre.
Jacinta: Okay thanks for that, that’s about as much about black holes as I can stand for now.
Canto: Well I’m hoping that in some future posts we can focus on the technology, the ground-based and space-based telescopes and instruments like Hubble and Kepler and James Webb and so many others that have been enhancing our knowledge of black holes, other galaxies, exoplanets, all the stuff that makes astrophysics so rewarding these days.
Jacinta: You’re never out of work if you’re an astrophysicist nowadays, so I’ve heard. Halcyon days.
an x-ray burst from a supermassive black hole – artist’s impression
Einstein, science and the natural world: a rabid discourse
Einstein around 1915
Canto: Well, we’re celebrating this month what is surely the greatest achievement by a single person in the history of science, the general theory of relativity. I thought it might be a good time to reflect on that achievement, on science generally, and on the impetus that drives us to explore and understand as fully as possible the world around us.
Jacinta: The world that made us.
Canto: Précisément.
Jacinta: Well, first can I speak of Einstein as a political animal, because that has influenced me, or rather, his political views seem to chime with mine. He’s been described as a supra-nationalist, which to me is a kind of political humanism. We’re moving very gradually towards this supra-nationalism, with the European Union, the African Union, and various intergovernmental and international organisations whose goals are largely political. Einstein also saw the intellectual venture that is science as an international community venture, science as an international language, and an international community undertaking. And with the development of nuclear weapons, which clearly troubled him very deeply, he recognised more forcefully than ever the need for us to take international responsibility for our rapidly developing and potentially world-threatening technology. In his day it was nuclear weapons. Today, they’re still a threat – you’ll never get that genie back in the bottle – but there are so many other threats posed by a whole range of technologies, and we need to recognise them, inform ourselves about them, and co-operate to reduce the harm, and where possible find less destructive but still effective alternatives.
Canto: A great little speech Jas, suitable for the UN general assembly…
Jacinta: That great sinkhole of fine and fruitless speeches. So let’s get back to general relativity, what marks it off from special relativity?
Canto: Well I’m not a physicist, and I’m certainly no mathematician, but broadly speaking, general relativity is a theory of gravity. Basically, after developing special relativity, which dealt with the concepts of space and time, in 1905, he felt that he needed a more comprehensive relativistic theory incorporating gravity.
Jacinta: But hang on, was there really anything wrong with space and time before he got his hands on them? Why couldn’t he leave them alone?
Canto: OMG, you’re taking me right back to basics, aren’t you? If I had world enough, and time…
Jacinta: Actually the special theory was essentially an attempt – monumentally successful – to square Maxwell’s electromagnetism equations with the laws of Newton, a squaring up which involved enormous consequences for our understanding of space and time, which have ever since been connected in the concept – well, more than a concept, since it has been verified to the utmost – of the fourth, spacetime, dimension.
Canto: Well done, and there were other vital implications too, as expressed in E = mc², equivalating mass and energy.
Jacinta: Is that a word?
Canto: It is now.
Jacinta: So when can we stop pretending that we understand any of this shite?
Canto: Not for a while yet. The relevance of relativity goes back to Galileo and Newton. It all has to do with frames of reference. At the turn of the century, when Einstein was starting to really focus on this stuff, there was a lot of controversy about whether ‘ether’ existed – a postulated quasi-magical invisible medium through which electromagnetic and light waves propagated. This ether was supposed to provide an absolute frame of reference, but it had some contradictory properties, and seemed designed to explain away some intractable problems of physics. In any case, some important experimental work effectively quashed the ether hypothesis, and Einstein sought to reconcile the problems by deriving special relativity from two essential postulates, constant light speed and a ‘principle of relativity’, under which physical laws are the same regardless of the inertial frame of reference.
the general theory – get it?
Jacinta: What do you mean, ‘the initial frame of reference’?
Canto: No, I said ‘the inertial frame of reference’. That’s one that describes all parameters homogenously, in such a way that any such frame is in a constant motion with respect to other such frames. But I won’t go into the mathematics of it all here.
Jacinta: As if you could.
Canto: Okay. Okay. I won’t go any further in trying to elucidate Einstein’s work, to myself, you or anyone else. At the end of it all I wanted to celebrate the heart of Einstein’s genius, which I think represents the best and most exciting element in our civilisation.
Jacinta: Drumroll. Now, expose this heart to us.
Canto: Well we’ve barely touched on the general theory, but what Einstein’s work on gravity teaches us is that by not leaving things well alone, as you put it, we can make enormous strides. Of course it took insight, hard work, and a full and deep understanding of the issues at stake, and of the work that had already been done to resolve those issues. And I don’t think Einstein was intending to be a revolutionary. He was simply exercised by the problems posed in trying to understand, dare I say, the very nature of reality. And he rose to that challenge and transformed our understanding of reality more than any other person in human history. It’s unlikely that anything so transformative will ever come again – from the mind of a single human being.
Jacinta: Yes it’s an interesting point, and it takes a particular development of culture to allow that kind of transformative thinking. It took Europe centuries to emerge from a sort of hegemony of dogmatism and orthodoxy. During the so-called dark ages, when warfare was an everyday phenomenon, and later too, right through to the Thirty Years War and beyond, one thing that could never be disputed amongst all that disputation was that the Bible was the word of God. Nowadays, few people believe that, and that’s a positive development in the evolution of culture. It frees us to look at morality from a broader, richer, extra-Biblical perspective..
Canto: Yes we no longer have to even pretend that our morality comes from such sources.
Jacinta: Yes and I’m thinking of other parts of the world that are locked in to this submissive way of thinking. A teaching colleague, an otherwise very liberal Moslem, told me the other day that he didn’t believe in gay marriage, because the Qu-ran laid down the law on homosexuals, and the Qu-ran, because written by God, is perfect. Of course I had to call BS on that, which made me quite sad, because I get on very well with him, on a professional and personal basis. It just highlights to me the crushing nature of culture, how it blinds even the best people to the nature of reality.
Canto: Not being capable of questioning, not even being aware of that incapability, that seems to me the most horrible blight, and yet as you say, it wasn’t so long ago that our forebears weren’t capable of questioning the legitimacy of Christianity’s ‘sacred texts’, in spite of interpreting those remarkably fluid texts in myriad ways.
Jacinta: And yet out of that bound-in world, modern science had its birth. Some modern atheists might claim the likes of Galileo and Francis Bacon as one of their own, but none of our scientific pioneers were atheists in the modern sense. Yet the principles they laid down led inevitably to the questioning of sacred texts and the gods described in them.
Canto: Of course, and the phenomenal success of the tightened epistemology that has produced the scientific and technological revolution we’re enjoying now, with exoplanets abounding, and the revelations of Homo floresiensis, Homo naledi and the Denisovan hominin, and our unique microbiome, and recent work on the interoreceptive tract leading to to the anterior insular cortex, and so on and on and on, and the constant shaking up of old certainties and opening up of new pathways, all happening at a giddying accelerating rate, all of this leaves the ‘certainty of faith’ looking embarrassingly silly and feeble.
Jacinta: And you know why ‘I fucking love science’, to steal someone else’s great line? It’s not because of science itself, that’s only a means. It’s what it reveals about our world that’s amazing. It’s the world of stuff – animate and inanimate – that’s amazing. The fact that this solid table we’re sitting at is made of mostly empty space – a solidity consisting entirely of electrochemical bonds, if that’s the right term, between particles we can’t see but whose existence has been proven a zillion times over, and the fact that as we sit here on a still, springtime day, with a slight breeze tickling our faces, we’re completely oblivious of the fact that we hurtling around on the surface of this earth, making a full circle every 24 hours, at a speed of nearly 1700 kms per hour. And at the same time we’re revolving around the sun at a far greater speed, 100,000 kms per hour. And not only that, we’re in a solar system that’s spinning around in the outer regions of our galaxy at around 800,000 kilometres an hour. And not only that… well, we don’t feel an effing thing. It’s the counter-intuitive facts about the natural world that our current methods of investigation reveal – these are just mind-blowing. And if your mind doesn’t get blown by it, then you haven’t a mind worth blowing.
Canto: And we have two metres of DNA packed into each nucleus of the trillions of cells in our body. Who’d’ve thunkit?
whatever