an autodidact meets a dilettante…

‘Rise above yourself and grasp the world’ Archimedes – attribution

Posts Tagged ‘force

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

leave a comment »

je ne sais pas

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 –     F = k \frac{q_1q_2}{r^2}. 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=mc2, 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…

 

Written by stewart henderson

February 20, 2022 at 2:34 pm

towards James Clerk Maxwell 4: a detour into dimensional analysis and Newton’s laws

leave a comment »

nutshell Newton

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

{\displaystyle {\bar {\mathbf {a} }}={\frac {\Delta \mathbf {v} }{\Delta t}}.}

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

Khan Academy physics

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

Written by stewart henderson

June 17, 2019 at 9:52 pm