Deriving the speed of light from Maxwell's Equations using A-Level mathematics! 

Thank you. This is a regular feature of my lessons at school, but I have the good fortune of teaching in a school where the majority of students will go on to study Engineering or Physics at university (with a large minority going on to study medicine). When I teach relativity, I go a bit beyond the specification too. I have found that the specification stops just short of the topics being understandable, so going just a bit further has a huge impact on the students' conceptual understanding.

I remember Maxwell's velocity distribution being covered in chemistry, for catalysts.

Thank you, I'm glad you find it helpful. Sometimes low-tech has its uses 😂

That's a very kind thing to say, thank you

Thank you. I've seen the speed of light derived by taking the curl of both sides and comparing to the others wave equation before, but I don't think that's very A-Level accessible.

😂

Thank you!

Thank you, and what wonderful timing! 😂

Thank you very much!

Thank you. I know the mathematics goes on a bit, but I really dislike it when videos skip too many steps

Well done for persevering!

Thanks a lot! See why I wanted to use this method? 😂

A comment from the legend! Thanks Lewis ♥️

Thank you 🙂

Best wishes for your standardized tests 🙏

You're very kind, thank you

Well, much of this is "beyond A-Level" in the sense that it isn't on the specification, but I still teach it to A-Level students and it's definitely A-Level accessible

Thank you 🙂

Thank you 🙂

Yes, I decided that I would take Maxwell's equations as a starting point. They are met very briefly by A-Level Physicists (if they are doing AQA Option D, for example), but not used at all. I think the first three are fairly easy to follow, but the fourth would need a bit more time and I knew the video would be too long as it was. Maybe I'll derive Maxwell's Equations another time

Thank you

They don’t have any geometry. These functions describe the amplitudes of the electric and magnetic fields, they travel together as waves outwards but they don’t look like anything.the electric and magnetic fields being perpendicular is a mathematical thing, but there’s are more advanced theorems called the Divergence Theorem, Stokes and Greens theorem, that end up showing you that E and B must be perpendicular.

I'm modelling the E- and B-fields using cosines that are perpendicular because, for plane polarised light, that's exactly what we have. That was demonstrated experimentally by Heinrich Hertz in 1888 (I think). Each of Maxwell's equations stands alone describing these fields, and can be verified independently from each other. Maxwell's third equation, for example, can help us understand how transformers work.

You are right that the model for the "electromagnetic wave" owes itself to Hetz, and the notes he made when discovering radio waves. However, this is something of a contentious issue, because light is not affected by magnetic fields, implying that it is an electrostatic wave, without a magnetic component. It is possible that Hertz was measuring the magnetic field induced within the media in which the change in electric field was present. That was copper ring, held at varying distances from the source of the electrostatic wave. It is a tricky issue. So many people believe in "electromagnetic" waves today, it it almost heresy to point out that magnetic fields don't influence light, or other fluctuations in electrostatic force. Nevertheless, there is a wealth of evidence to show this to be the case, and indeed Maxwell's thesis essentially demands the distinction between a current, which is moving charge, and which generates magnetic fields, on the one hand, and a fluctuation in electrostatic force, which is not a moving charge, on the other. It was a lovely exhibition of math, in any case.

Thank you

Yes, I could have finished off by substituting the constants, that's a good point. The video was already quite long, so I'm not sure if a quick dimensions check would have been a good idea, but I agree it's always worth doing when you do a derivation 👍

Welcome to the club (I am 66 years old). I am located in the French ALPS. I love watching maths and coding on youtube. Take care

Thank you for the kind comment 🙂 you're quite right that curl and partial derivatives aren't included on A-Level mathematics syllabuses; however, differentiation is included, as is using vectors. The reason I spent so long at the start of the video deriving the expression for curl is precisely because A-Level mathematics students are unlikely to have seen it before. I do think it's A-Level accessible though, which was the point of this video. A-Level mathematics was the starting point and it doesn't really get any more advanced than that. A good A-Level mathematics student will be able to follow all of the steps and be able to do this derivation themselves, and in doing so gain some context as to why we learn what we learn in A-Level mathematics

@@PhysicswithKeithI am surprised to hear that about partial derivatives, can't recall if we did it for C4 or FP1 though.

Heinrich Hertz demonstrated this was true for plane polarised waves by using a split ring to detect magnetic fields and a dipole detector for electric fields

They indicate the strengths of the electric and magnetic fields. The greater epsilon, for example, the weaker the electric field for a given charge at a given distance.

@@PhysicswithKeith the point is the WHY of c=299,792,458 m/s is still a mystery. Or the specific values of e0 and u0 being what they are to make c what it is. Last I checked this was a mystery as one would assume a more powerful theory would make specific values an emergent property. Otherwise we have a multiverse with every possible combination.

​​@@lucasbachmann glad to see people who seem to share my intuitive belief on why the claim in the title is misleading. My uni days are way too far behind me to be able to follow the match however by skipping to the end i gather that this is only an indirect way to measure the speed of light by measuring electric and magnetic fields. Which seems ridiculous as the propagation equation is based on the assumption that the speed of light is a limit. The whole thing is in no way a validation of the speed of light being an immutable constant that can be derived from fundamental electromagnetism but a circonvoluted way to get back a constant which was here by definition in the first place.

@@lucasbachmann best comment here!

@@PhysicswithKeith But c is still fundamental. The relation between c, e0 and u0 does not change this. All three are fundamental, or at least two. Not defined which one.

My caveat is that I don't teach A-Level mathematics so it might be that unit vectors don't need the time I've given them, but sadly unit vectors aren't on the A-Level physics specification. I teach them anyway, because they make working with vectors far easier

They use "Further Maths" A-Levels nowadays to separate the geeks from the normies.

It's the UK education system. A-Level is the qualification 16-18 year olds can choose to study. It's short for AGCE, which means Advanced General Certificate of Education. The most advanced A-Level stuff is broadly equivalent in difficulty to first-year undergraduate degree study.

Well, quite! 😂

Light was observed to diffract, so the prevailing theory at the time was wave theory, which predicted diffraction (corpuscular theory didn't). The simplest wave function is a cosine. It's just modelling; finding a function and trying it out. Hertz identified that radio waves had an electric field and magnetic field component and that they were orthogonal to each other, hence the E and B equations modelled as perpendicular cosine wave functions

@@PhysicswithKeith So are periodic functions the only ones that can describe light?

Electromagnetic radiation is a particular consequence of the way electric and magnetic fields behave together. You can get effects such as what happens in transformers where energy is transferred from the primary to the secondary coil, where you wouldn't call that an electromagnetic wave, but it's essentially the same physics (and even in transformers we have periodic functions). Even a pulse can be described as an infinite sum of sinusoidal functions (Fourier series). So, I reckon the answer to your question is 'yes', unless you can think of an alternative?

Not me, I was probably the kid eating the glue in the corner

😂 Poynting

That's such a kind thing to say, thank you.

They don't, but they do learn differentiation and they do learn scalar and vector products, so I think it's A-Level accessible. That's why I explain div and curl in so much detail; they'll be new to A-Level students.

I'm glad you enjoyed it 🙂

The school bell ringing? Yes, it is too loud.

Namely because it is the partial derivative.

Tekky little step

No idea, I'm not a mathematics teacher, but I was told by someone who is a mathematics teacher that matrices aren't part of A-Level mathematics 🤷

You learn it in A-Level Further Maths

We sure as hell didn't learn matrices or the Dot and Cross products in A level math

Ah that's good to know, thank you. Maybe I should record a video to show how to do the same derivation using matrices so that A-Level further mathematics students have some context when they learn matrices. I'm afraid I've seen far too much mathematics teaching where the context has been stripped away to the point that the maths looks far too abstract. My own A-Levels were like that, and I performed woefully in Mathematics because of it, but as soon as I was taught mathematics by a physicist at university it all clicked into place.

​@@PhysicswithKeith that would be great!

I've a few derivations where I've used a bit more tech, but the advantage of pen and paper here was the ability to easily manipulate the pages so that different written parts could be shown alongside each other. I think the physics here stands up regardless of the delivery medium, and I'm hoping I was able to make it clearer using the media that students would be using in the classroom.

@@PhysicswithKeith I love the explanations on paper idk

@@PhysicswithKeithI would say no need to listen to him. The explanations on the paper looks amazing and i can follow it up easily with my own paper and pen.

@@PhysicswithKeith The students are on RU-vid watching 3Blue1Brown, Numberphile, Mathloger, among others. Then they enter the classroom and are faced with that eternal poverty of media and resources. And that's why they come to RU-vid. My advice was exactly to increase your coverage, your audience, so that you can share your excellent teaching with more people.

I personally think it's the physics that's important, not the medium. That said, the above comment does reflect the general consensus of the community, which is, if you want to reach a large audience, make it visually pleasing.