Maxwell's Equations Explained: Supplement to the History of Maxwell's Eq.
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- Опубліковано 30 чер 2024
- I start with the basics (vectors, dot & cross product) and then give an overview of where all 4 Maxwell's equations came from, what they mean, and why they are important!
To read the script (with citations) and download the pdf of first 3 chapters of "Lightning Tamers" please visit:
www.KathyLovesPhysics.com
order my book:
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A few extra links:
Woman in Faraday Cage (very good BTW): • Prelude To Power: 1931...
Lenz's Law: • Lenz's Law
Clip from "The Simpsons" Season 6, episode 21 (1995) • The Simpsons - Perpetu...
Sand Castle of William Rowan Hamilton and Helen Hamilton carving quaternions on a bridge by Daniel Doyle 2002. Used with permission. Check it out!
peakd.com/hive-156509/@ammoni...
0:00 Introduction
0:38 Vectors & Vector Multiplication
5:02 Del, Divergence and Gauss's Laws
13:25 Maxwell's Equations with Curl
25:35 Maxwell's Equations and Magnets
27:14 Maxwell's Equations and Light
I'm a seasoned electrical engineer, and during my studies I've seen the Maxwell equations in a lot of courses. First in high school, then in the physics course on electromagnetism, then in a dedicated course on antennas and radio wave propagation, then in a course on lasers, then in fundamentals of quantum mechanics and in many other courses. This is the best lecure about he topic that I have ever seen!
Wow, thank you so much. The more I study the history of science, the more strongly I believe that it is the secret to understanding and explaining the science with depth.
@@Kathy_Loves_Physics I agree! By retracing the steps that brought the scientists to a discovery, but at the same time using the modern framework to illuminate the path you can unveil the reasoning process in a way that feels more natural. Plus it is fascinating to learn about the adventurous life of all these extraordinary people!
I'm also an electrical engineer, and mathematician, 40 years. Love all the connections between the scientists and mathematicians to come to these four fundamental equations. Did not get this progression in school which gives me a clearer and deeper understanding of their underlying meaning. Thanks!
@@7sArts I haven't read the original Maxwell equations, but I thought Maxwell had framed them in terms of potentials, rather than the fields themselves. That was Heaviside's work.
@@Kathy_Loves_Physics
Kathy, One of the more memorable experiences I had in college was re-creating the Millikan oil drop experiment. I think it would make a great subject for you to cover. It certainly falls within your area of expertise. I can remember my professor struggling to get that wonderful little drop to become visible under the microscope!
My regret is that at 74 years of age I am now finding these wonderful lectures from an enthusiastic, vibrant and delightful teacher . Would I were 20 again.
tThank you Kathy.
Kathy takes all the tension away from learning physics and math in a way I have never ever seen before. Showing us why and helping understand how Maxwell's equations came to be, all with an honest love of the subject matter sounds impossible but that is what Kathy does. It is an amazing gift. Thank you.
I am a recently retired electrical engineer and I would like to say that this is on of the best explanations on the subjet I have ever seen.
My congratulations from Spain for your excellent work on your UA-cam channel.
By the way, I'm looking forward to reading your book "The Lightning Tamers" which will no doubt be just as fascinating as the videos.
Fabulous presentation of Maxwell's equations. It brought back memories of deriving them from first principles as an electrical engineering major 40 years ago. I particularly liked how you included images from his papers that do not use vector notation.
This is by far the best explanation of this that I have ever seen. You managed to get so much into only 34 minutes, I am truly amazed. Especially how you linked all the discoveries.
That was excellent. I wish this had been available when I was studying this stuff 45 years ago. I knew several bits and pieces, and you've assembled them into a consistent story, which triggers more insights. I can re-watch this several times, and pick up a little more each time. Wouldn't it be fun to bring these fellows back in a time machine (especially Hertz, who foresaw no practical uses) and show them what we do with their discoveries today.
Your vision of showing the history of each one of the ideas that finally were mature enough to bring Maxwell to write them with math equations is really brilliant and very powerful to give us a better understanding of this subject. Congratulations for your work. All of these famous developers would be proud of you. From Brazil, a great hug!! Miguez
Wonderfully explained! Studied these equation in first year physics during under graduation and always had these unsettling feeling of not grasping them fully. Over a period of time, have tried to understand from different literatures and physics videos like this in leisure, but so far have found it just mathematical jugglery. This is the first video that I could relate to and ties everything that I know about magnetism and electricity seamlessly. Thank you very much!
Your enthusiasm is infectious! While I enjoy math, I am not very gifted or fast at it. Most of the math in my electrical apprenticeship was above my full understanding, but I found that treating it like a language really helped. I have forgotten most of it, but so much enjoy your putting together the historical development of the people and the math that have given us so much in our ability to understand our world.
My son is starting his freshman year in college as a physics major (as his dad did) and I have pointed him at your channel. You are truly a gifted teacher!
Also point your son to the online MIT videos by Dr. Walter Lewin.
@@centexrails Thank you so much! I will check it out. He is particularly interested in nuclear physics so I have also pointed him at Dr. Ruzic at the University of Illinois.
@@centexrails I’m a little disturbed to see that Dr. Walter Lewin has had sexual harassment allegations against him. Sad!
I agree 100%. Kathy empowers us with her talent and enthsiasm. What a joy.
What is covered in this video is too advanced for a beginning freshman.
I must say, you really know how to keep a viewer’s interest in a subject, Miss Kathy.
Great job on your latest effort. Thanks so much.
Thank you! 😊
I know you’ve done at least one video on the class and background differences between Faraday and Maxwell, and the general resistance of the aristocratic scientists of the day to properly credit Faraday. So it was nice to see Hertz fully crediting “Faraday-Maxwell” in that paper. Yeah, the science is always cool. But sometimes the personal stories behind the science are equally interesting.
I believe the Faraday-Maxwell equations should be renamed to Faraday-Maxwell-Heaviside (FMH) equations because Oliver Heaviside was the one who created the modern form of the Faraday-Maxwell equations.
@@chiensyang so stupid. Heaviside just put them into a new notation; Maxwell did the lion's share of the work. Faraday came up with the general ideas, so, they should be called "Maxwell's equations," because that is exactly what they are
I studied physics 40 years ago at King's College London, Maxwell's Alma mater. I remember having such a hard time of it with vector analysis. That green Schaum textbook haunts my memories. I never really understood it - I just memorised it all. No such thing as UA-cam and teachers like Kathy and 3blue1brown. I'm really enjoying this series.
Vectors are the easiest thing ever..... you can learn them in 5th grade
@@pyropulseIXXI Differential equations involving the del operator in 5th grade? I'm glad I didn't go to your school ;) Div, grad and curl ... Still make me shudder.
@@CheeseAlarm Yeah. Calculus is easier than algebra, and I learned algebra in 5th grade. I read books on my own, the school didn't teach me about calculus
I suppose vector calculus gets a little intense when you start integrating over parametric surfaces, over volumes, divergence, and curl, and the Jacobian
But the concepts are simple; the calculations sometimes take some work
This thing remained sort of a mystery to me for about three decades. Now I believe I've seen the light.
Thank.
You.
Maxwell, the most underrated physicist of all time.
United two separate forces into one mathematically on his own and proved that light was an electromagnetic wave.
Even Einstein saw him as a role model.
Your videos always make me want to learn more, even when I think I know the subject well.
Your balance of the history behind and not, “dumbing down,” the maths is beautifully unique! Thank you.
So glad you liked it. Love your name
@@Kathy_Loves_Physics Danke! Sometimes before creation, there must be destruction. ^.^
I have bought your book by preorder and a few minutes ago it is delivered to me! Live long and prosper, ma’am!
Kathy, where were you when I was an undergrad? If these issues had been explain with this clarity, my life would have been MUCH happier. Nice job!
BTW, I have one of those t-shirts. I've worn it out, so time for a new one!
I've seen many videos on vector multiplication and the divergence and curl operations, but yours finally made it "click." Thanks for being such a good teacher, Kathy.
Glad it was helpful!
One of my favorite moments in college physics was when we combined Maxwell's equations and it turns into a wave equation with the constants in the place of velocity, adding more evidence light is composed of electric and magnetic waves.
Thank you for the wonderful content. I’m 57 and still enjoy learning. Your videos are accurate and not dumbed down. You keep me thinking. Thanks again.
Very conceptual explanation! The historical background and narratives help me understand more the concept. As a graduate of Electrical Engineering, I salute you for tackling this difficult subject. Job well done, Kathy.
At about 14:52, "french polymorph" probably wanted to be polymath, but I'm enjoying the idea of ampere as a shapeshifter.
You’re one of my favourite professors, Kathy. Thank you for this incredible work.
The subject of this one video would make a great book all by itself. I love how you reference and highlight key sections of original sources in your videos. Great summary of great scientists building on their predecessors work. Thanks so much.
I'm guessing she will make it a book.
Kathy amazing lecture, and the illustrations made it powerful and easier to understand. Thanks
Another electrical engineer. That was an excellent lecture. It has greatly helped my understanding of things I always struggled with. And all done with such infectious enthusiasm. So interesting to hear it through the histroy of how different peoples contributions combined to help advance knowledge.
Well Kathy, I'm just a software engineer that learned Maxwell equations at university, but I must say your video was so enjoyable to me like listening a classical symphony in four parts, with an apotheotic conclusion!
Congratulations and all the best!
Thank you Kathy! For weird reasons, I skipped physics in college. This would likely have been day 1, and I have been mystified about Maxwell's Equations for 40 years. Until now.
So I am going to buy your book, and maybe more of physics will open up to me.
The best explanation of Maxwell's equation. Loved the part which states "what the equation (statement) means".
Also, the historical contexts were quite interesting.
Thanks for posting such detailed informative session.
Beautifully explained and contextualized with historical references to what birthed these equations. Bravo!
This is really amazing,
I usually read history before studying any theory, to learn which events made discoveres to assert something about the theory,
But unfortunately, I don't know where to look for proper information and it takes so much time,
But here, In one video, I learn your research, which is tedious in real, for the history of the events and also detailed implications ,, This is just amazing ma'am,
Thank you for these videos 😊
Best explanation anywhere! Kathy is both brilliant and sublime. 🎉😊
The curl of your articulate speech and divergence of knowledge together create dazzling light.. I loved this video.. it is one of those which u will see over and over again.
Hi Kathy, one of your patrons, will have this as required watching for my students
Thank You captain 🫡
I love your videos. I am a biochemist, but I find physics fascinating. If I was more intelligent I would have went into physics. At least I can watch these videos from you and gain a better understanding, you make it so accessible!
I’m so glad you liked it but I wish you wouldn’t insult yourself. I have no idea how to do bio chemistry but I wouldn’t call myself stupid. ❤️
@@Kathy_Loves_Physics - Ah, thank you! I had to keep a 3.5 GPA or above, so the senior physics courses intimidated me. I would love to go back and take some now with no pressure though!
@@ElectronFieldPulse I did a physics major and got a 3.98 gpa at Cal. I didn't know that an A- would lower my GPA..... and my first A- was in quantum mechanics I, cuz I thought getting an A- counted as a 4.0..... pisssed me off
I wanted to do Biophysics..... there is evidence that plants can transmute elements, and there is evidence that the transmutations occur via 'particle accelerators.' So efficient that it takes an array of 5 of the same molecule, that binds in such a way that the center is charged, but 'hollow,' and accelerates an atom through and can transmute it to the next highest element by smashing it into a proton. (hydrogen ion)..
Great stuff Kathy. Love your authenticity.
Thanks Kathy for bringing to life all those equations that innervated my engineering schooling and life!
Fantastic! You are an extremely gifted educator!
Thanks for making these videos you are making , a little history while learning concepts makes it more interesting. Actually it gives some intuition how those people used to think, what was the reason to do a certain experiment and much more. Thank You !!
what an impressive and important video!!!!! Amazing work and a gift to us Luddites! Am now retired but have worked with so many that didn't know 'the story' about why these are so important today in the way they are. Electromagnetism and algebra Cadabra. Brilliant. The universerse is Gaussian.
Amazig lecture. The best I have ever seen. The explanation leaves Feyman in the dust. Thank you.
My goodness, what a great complement! Thank you thank you thank you and I’m glad you liked it.
Whew - Might need to watch this several times to actually understand such a fundamental mathematical view of physics. Great Video
Notifications: ON
I'm excited about your next video on Quaternions. I was playing with Quaternions while watching your video. What a coincidence. Was trying to make a knot function.
I enjoy your videos. They are on point, deep, dense and compressed. You tell things in an inspiring way. Thanks for your hard work. I appreciate it as much as everyone else. A lot.
Jeff, that is funny! Well, I hope you end up feeling like I do quaternions justice.
tied the knot using quaternions ua-cam.com/video/IyUK_LWZ9_Q/v-deo.html
beautiful recap and explanations, thank you. i always recommend your channel to my students.
Thank you! I am a fanatic of Maxwell's equations and you just gave me the historic perspective and how everything came nicely together. I never had the time to research the history of the equations, and you just served them to me on a silver plate. Keep going the good work.
I’m so glad you liked it and that I saved you some effort.
I love these videos so much. Thank you for making them.
Kathy, that was a great explanation! And by the way, I love your enthusiasm! Thank you very much
really accessible, well-explained and has a the right thread to make it easier to follow
Beautiful presentation!
This is, again, a _very good video!_
Geometric Algebra, also called Clifford Algebra, makes the Maxwell's equations a lot simpler. Especially because with Geometric Algebra you can represent rotations directly, without needing a vector perpendicular to the distance and force vector.
A torque is then a rotation given by what is called an outer product: *_Rot_* = *_F_* ^ *_d_* _in the plane of the vectors_ *_F_* and *_d_* . Both the dot product and the outer product, which replaces the cross product are united in a single product, which is called the Geometric Product, from which the magnitudes and the rotation direction of the outer product, and the inner product can be obtained in the following way:
Given two vectors *_u_* and *_v_* in space, you can form the geometric product simply by *_uv_* which is not a commutative product. *_uv_* is not equal to *_vu_* .If the vectors *_u_* and *_v_* are parallel, then the Geometric Product is commutative: *_uv_* = *_vu_* . If the vectors *_u_* and *_v_* are perpendicular, then their Geometric Product is anti-commutative: *_uv_* = - *_vu_* . In all other cases, the Geometric Product is neither commutative nor anti-commutative.
The dot product is then given by *_u . v_* = ( *_uv_* + *_vu_* )/2, which is a scalar, and the outer product, which represents a rotation from *_u_* onto *_v_* is then given by *_u ^ v_* = ( *_uv_* - *_vu_* )/2 which is a rotation of *_u_* onto *_v_* .
The Geometric Product is then given by *_uv_* = *_u . v_* + *_u ^ v_* In other words, a geometric product is a sum of a scalar and somehting that rotates.
A nice thing about the Geometric Product, is that you can both multiply and divide vectors and any formula consisting of sums and geometric products. In other words, Geometric Algebra is a complete algebra, while using scalars, inner products and vector products this cannot be done, because they are all different algebraic entities.
Rotations can only be represented by vectors in 3D space. In 1D space you have just a fixed direction, and therefore zero rotation. In the plane you can have just one rotation. And in 4D space you have 6 rotations, not 4!
It is just an accident of mathematics, that in 3D space you have as many translations as rotations. This is because any rotation happens in a plane defined by two axes. In 1D space you have only one axis, therefore it is impossible to rotate something in a 1D space. In a 2D space you have two axes, and therefore exactly one kind of rotation. In 3D space there are exactly 3 ways to select 3 axes defining a plane of rotation. That is why you can represent rotations by vectors. But if you do that, you obfuscate, even eliminate _the fundamental difference_ that exists between translations and rotations.
In 4D space there are 6 ways to select 2 axes. That is why in 4D space you have 6 rotations. And in 5D space you have 10 rotations, because there are 10 ways to select two axes out of 5 that form a plane within which you can rotate.
Happy to know that there are humans who can understand and make sense of such things. :-)
I have a Ph.D in EE. My specialty is Telecommunication. I am retired now. Of course, I took grduate courses in EM and antenna theories in one of the good EE Deparment in the US (Polytechnic/NYU). I did not have a textbook that went into the background of these 4 equations, and the professors who lectured in this subject to teach as simplified as Kathy did. It was always bogged down with heavy math without touching the Physical aspects of the phenomena. Students are forced to memorize them even in fine instutions without capturing the essence of them. I do not know your background, Kathy. I wish I had someone like you then to simplify that genius work.
Kathy, what a great channel, I absolutely love it!!!
Very good videos. I check my phone for your postings. And wait in anticipation of there releases. Great job.
What a wonderful teacher Kathy is. Wished I had a teacher like her during my undergraduate days.
Thank you very much for the wonderful lecture, we were never taught like this.
Congratulations, Kathy!
Woohoo! A recent subscriber here. I really enjoy getting notifications from your channel. Looking forward to getting your book too. Not long to go now!
So exciting!! Thanks
this is amazing work, thanks for posting.
What a beautiful summary!! You are an amazing communicator.
Thanks for another great video. Waiting for the book to arrive.
When I took differential forms my professor Ted Frankel derived some of Maxwell's equations from using the static equations and the Lorentz Transformations as axioms. It was very exciting. I am 75 now and have forgotten some of the details, but it was pretty fun.
I cannot wait to watch the Sir William Rowan Hamilton video! Last month I purchased an original Introduction To Quaternions by Kelland and Tait 1873. Honestly, I have been on one Gnarly ride learning from and reading about the many great minds that once walked the halls at the University of Edinborough. Sooo, Freaking Many! Geez...
Last week, I felt alone on this journey. This week, here you go again bringing the heat!
Gracias
Side note: Oliver Heaviside -NATURE -1893 - Vectors Versus Quaternions
I found a hysterical article from Nature by Oliver Heaviside in 1893 with regard to Maxwell's use of quaternions. It's not at all the way I remembered learning about the Heaviside story nor how history has framed him as the first-rate oddity" and Maxwellian Fanboy #1. That was definitely not the case. He's an insult guru of the highest degree..lol I mention it because it relates to your content and is one amusing read.
Cheers
Techinically del is an operator as it transforms its argument (a multivariable differentiable function) into another object (in this case a vector) according to a well defined law (taking the partial derivatives)
The dot product between del and a vector field F is called divergence.
Its result is the sum of the partial derivatives of the vector field's components.
It represents the tendency of F to clump (negative divergence) orr spread out (positive divergence) in an infititesimal volume dV.
If the divergence is 0 then F is said to be solenoidal.
The curl on the other hand is the cross product betwwen del and F.
It represents the rotantion of F in an infinitesimale volume dV.
By convention the positive ditection of Curl(F) corresponds to a couterclockwise rotation of del around F.
If the curl is null then F is irrotationak.
I so wish I'd had your videos available when I was taking physics.
Good work.
Dr Kathy is a gifted teacher.
Your Videos are amazing Kathy
After 50 years of mystery Maxwells equations are making sense! And thanks for explaining the T shirt!
Curiosity got the better of me. I dug out my Michigan Tech transcripts. Fall 1971 EE325 Electro Magnetic Theory. I got an A so I must have understood it well enough 51 years ago.
exceptionally thorough presentation and flow. appreciate explaining the difference between mathematical scalar vs vector. so many brilliant minds leading to this understanding. MAXWELL RULES.
thank you kindly madame
Saving this one to my playlist! Great explanations and refreshers
This is great! I particularly like the historical approach, showing the original published work of the scientists involved. The names are familar to me, but your video helps us learn the sequence of events, and who influenced whom.
Now try doing it again, in spherical coordinates.
Best video I have ever seen about Maxwell’s equations! Respect!
Kathy - excellent Job!
Always love your videos.
I have a flue and enjoying your video yet again over a cup if tea.. thank you maam!!
Just to add to the confusion, James Clerk Maxwell also has another set of relations expressed in terms of partial derivatives in thermodynamics that are named after him, i.e., the 'Maxwell Relations'.
Thank you so much for this video. I can't quite fiollow the math, but I do follow the conclusions and how they relate. I find it all fascinating!
Love your videos! The history of physics discovers and inventors is amazing! Great job!
Very Well done ! Hystory citing primary bibliographic sources and mathematical rigor side by side. Fantastic!
Saw and very useful I hope you make this in a playlist
Thank you Kathy, that was a great lesson very well explained. You're brilliant!
Excellent exposition! I would add for those interested, a reference to Oliver Heaviside who according to Wikipedia: " In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations."
Fantastic presentation! Thank you
I'm very glad for the scripts. Watching the video's is great, but to really understand these things I have to sit with pencil and paper and write them out. Sometimes I wish I could go back to school.
I am so glad you found it helpful. Putting all those equations on a website was surprisingly difficult but I felt it was necessary for many people.
It's counter-wise. The changes in the magnetic field changes the distribution of charges, therefore the electric field around them. Great videos, all of them!
You is the best channel on the YT, I love your videos! Big Thank, you!!!
Thank you
This is a phenomenal video; I love your content!😁
Lovely explanation, much appreciated. Inwork with ship power generation and distribution plants, and this video makes me feel that the earth itself is a gwnerator, albeit, without an obvious prime mover.
Kathy, I've just bought your book, and will read it whilst heading overseas for work. Thank you for the great video series.
Kathy, I think you are explaining the calculus very well of the maxwell equations. Perhaps you can also show examples of what the equations mean in real life eg by showing a magnetic field or induction. Like Maxwell said: what is the go of it. Intuitively understanding what is happening that is more important than the equations themselves.
Cool to see a more math-oriented video. I feel there's always more to learn about the math of Maxwell's Equations. I'm curious if you've ever seen the "Geometric Algebra" (aka Clifford Algebra) approach to Maxwell's Equations? It lets you combine all 4 into a single equation. It's a bit abstract and probably not very useful, practically speaking, but it's neat.
Interesting. There's also the antisymmetric 4x4 EM tensor which contains components of E and B fields. You get two of the Maxwell equations popping out of that by applying the Bianchi identity and you get the other two by applying a second kind of (first) derivative operation. I don't know how useful that really is either but it gets you a fun derivation of all four Maxwell equations.
p.s. I love your name. It's a nice way of implying you belong to yourself while also being a property of The Matrix.
@@muttleycrew Yeah, the anti-symmetric EM tensor is used in the geometric algebra formulation as well. en.wikipedia.org/wiki/Geometric_algebra#Spacetime_model
Lately I've been reading that the formulation you mentioned (with the Bianchi identity) has some commonalities with the Bianchi identity in general relativity. (GR uses the Riemann Curvature Tensor, and the equivalent curvature tensor in E&M is the EM tensor).
The most elegant form, and finally the derivation of i
I have not seen that, sounds interesting. In 1862 and 1864 Maxwell actually derived both of his "Gauss's Law" equations from his curling equations, making 4 equations into 2 (although the electric Gauss's law does need the relationship between the current density, j, and the charge density, rho, which is arguably another equation, so he made 4 equations into 3?).
I've been wondering how this "next step" of tensor notation evolved from Maxwell's equation (A∙B=a -> a=AᵢBᵢ ; A×B=C -> Cᵢⱼ=εᵢⱼₖAⱼBₖ ; Fᵢⱼ=∂ᵢAⱼ-∂ⱼAᵢ), why is it (sometimes?) associated with Einstein, and why so many people involved seem to suffer from this Gell-Civitia syndrome. I'm worrying it's just a really boring story, since there is no shocking people involved.
What a great explanation to vectors used in engineering. Thank you!!!!
It is not easy to be an educator. You don’t just tell people stuff. I, like most, suffered through high school and college believing that I just couldn’t understand some things. I have only lately become aware that poor curriculum development and instruction can doom all but the Einsteins and Faradays of the world. We don’t have to be one of these people to understand physics. We just need outstanding teachers like Kathy.
I wrote more but deleted it. Being a good teacher is incredibly difficult.
Can't wait for my book!
I like the way you casually refer to Gauss as “a German mathematician.”
Kathy should write a book by putting all the work to gether. I have EE but never made connection on how science evolved and contribution made by others unkowns. As newton once said he was able to see further by standing on shoulder of giants. Maxwell stood on shoulder of faraday, ampere, gauss and columb and may other unkown.
Maxwell lived short life but made signeficant contribution and influenced our daily life. Imagine your life without modern communication.
Your insights are so inspiring. Sharing you with many. My 12 y.o. watches again and again (me, too!).
This video content is awesome! your presentation is simply excellent!!!!
You are a great teacher.
Beautifully done.
Thank you.