## Archive for the ‘**Coulomb’s Law**’ Category

## what is electricity? part 10 – it’s some kind of energy

Canto: We’ve done nine posts on electricity and it still seems to me like magic. I mean it’s some kind of energy produced by ionisation, which we’ve been able to harness into a continuous flow, which we call current. And the flow can alternate directionally or not, and there are advantages to each, apparently.

Jacinta: And energy is heat, or heat is energy, and can be used to do work, and a lot of work has been done on energy, and how it works – for example there’s a law of conservation of energy, though I’m not sure how that works.

Canto: Yes maybe if we dwell on that concept, something or other will become clearer. Apparently energy can’t be created or destroyed, only converted from one form to another. And there are many forms of energy – electrical, gravitational, mechanical, chemical, thermal, whatever.

Jacinta: Muscular, intellectual, sexual?

Canto: Nuclear energy, mass energy, kinetic energy, potential energy, dark energy, light energy…

Jacinta: Psychic energy… Anyway, it’s stuff that we use to do work, like proteinaceous foodstuff to provide us with the energy to get ourselves more proteinaceous foodstuff. But let’s not stray too far from electricity. Electricity from the get-go was seen as a force, as was gravity, which Newton famously explained mathematically with his inverse square law.

Canto: ‘Every object or entity attracts every other object or entity with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres’, but he of course didn’t know how much those objects, like ourselves, were made up of a ginormous number of particles or molecules, of all shapes and sizes and centres of mass.

Jacinta: But the inverse square law, in which a force dissipates with distance, captured the mathematical imagination of many scientists and explorers of the world’s forces over the following generations. Take, for example, magnetism. It seemed to reduce with distance. Could that reduction be expressed in an inverse square law? And what about heat? And of course electrical energy, our supposed topic?

Canto: Well, some quick net-research tells me that magnetism does indeed reduce with the square of distance, as does heat, all under the umbrella term that ‘intensity’ of any force, if you can call thermal energy a force, reduces in an inverse square ratio from the point source in any direction. As to why, I’m not sure if that’s a scientific question.

Jacinta: A Khan Academy essay tackles the question scientifically, pointing out that intuition sort of tells us that a force like, say magnetism, reduces with distance, as does the ‘force’ of a bonfire, and that these reductions with distance might all be connected, and therefore quantified in the same way. The key is in the way the force spreads out in straight lines in every direction from the source. That’s how it dissipates. When you’re close to the source it hasn’t had a chance to spread out.

Canto: So when you’re measuring the gravitational force upon you of the earth, you have to remember that attractive force is pulling you to the earth’s centre of mass. That attractive force is radiating out in all directions. So if you’re at a height that’s twice the distance between the earth’s surface and its centre of mass, the force is reduced by a particular mathematical formula which has to do with the surface of a sphere which is much larger than the earth’s sphere (though the earth isn’t quite a sphere), but can be mathematically related to that sphere quite precisely, or to a smaller or larger sphere. The surface of a sphere increases with the square of the radius.

Jacinta: Yes, and this inverse square law works for light intensity too, though it’s not intuitively obvious, perhaps. Or electromagnetic radiation, which I think is the technical term. And the keyword is *radiation –* it radiates out in every direction. Think of spheres again. But we need to focus on electricity. The question here is – how does the distance between two electrically charged objects affect the force of attraction or repulsion between them?

Canto: Well, we know that increasing the distance doesn’t *increase *the force. In fact we know – we observe – that increasing the distance *decreases *the force. And likely in a precise mathematical way.

Jacinta: Well thought. And here we’re talking about electrostatic forces. And evidence has shown, unsurprisingly, that the decreased or increased force is an inverse square relationship. To spell it out, double the distance between two electrostatically charged ‘points’ decreases the force (of attraction or repulsion) by two squared, or four. And so on. So distance really matters.

Canto: Double the distance and you reduce the force to a quarter of what it was. Triple the distance and you reduce it to a ninth.

Jacinta: This is Coulomb’s law for electrostatic force. Force is inversely proportional to the square of the distance – . Where F is the electric force, q are the two charges and r is the distance of separation. K is Coulomb’s constant.

Canto: Which needs explaining.

Jacinta: It’s a proportionality constant. This is where we have to understand something of the mathematics of variables and constants. So, Coulomb’s law was published by the brilliant Charles Augustin de Coulomb, who despite what you might think from his name, was no aristocrat and had to battle to get a decent education, in 1785. And as can be seen in his law, it features a constant similar to Newton’s gravitational constant.

Canto: So how is this constant worked out?

Jacinta: Well, think of the most famous equation in physics, E=mc^{2, }which involves a constant, c, the speed of light in a vacuum. This speed can be measured in various ways. At first it was thought to be infinite, which is crazy but understandable. It would mean that that we were seeing the sun and stars as they actually are right now, which I’m sure is what every kid thinks. Descartes was one intellectual who favoured this view. It was ‘common sense’ after all. But a Danish astronomer, Ole Roemer, became the first person to calculate an actual value, when he recognised that there was a discrepancy between his calculation of the eclipse of Io, Jupiter’s innermost moon, and the actual eclipse as seen from earth. He theorised correctly that the discrepancy was due to the speed of light. Later the figure he arrived at was successively revised, by Christiaan Huygens among others, but Roemer was definitely on the right track…

Canto: Okay, I understand – and I understand that the calculation of the gravitational force exerted at the earth’s surface, about 9.8 metres per sec per sec, helps us to calculate the gravitational constant, I think. Anyway, Henry Cavendish was the first to come up with a pretty good approximation in 1798. But what about Coulomb’s constant?

Jacinta: Well I could state it – that’s to say, quote it from a science website – in SI units (the International System of units), but how that was arrived at precisely, I don’t know. It wasn’t worked out mathematically by Coulomb, I don’t think, but he worked out the inverse proportionality. There are explanations online, which invoke Gauss, Faraday, Lagrange and Maxwell, but the maths is way beyond me. Constants are tricky to state clearly because they invoke methods of measurements, and those measures are only human. For example the speed of light is measured in metres per second, but metres and seconds are actually human constructions for measuring stuff. What’s the measure of those measures? We have to use conventions.

Canto: Yes, this has gone on too long, and I feel my electric light is fading. I think we both need to do some mathematical training, or is it too late for us?

Jacinta: Well, I’m sure it’s all available online – the training. Brilliant.org might be a good start, or you could spend the rest of your life playing canasta – chess has been ruined by AI.

Canto: So many choices…

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

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:

Newton saw gravity as a force (*F*), and formulated the theory that this force acted between any two objects (*m _{1}* and

*m*– indicating their masses) in a direct line between their respective centres of mass (

_{2}*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’:

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, *k*_{e}, 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