The Electromagnetic Field Strength Tensor
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- Опубліковано 6 вер 2024
- Today I talk about the field strength tensor, and go back to basic E&M with maxwells equations and defining the vector potential. I then calculate what the components of the tensor are.
Cross products using levi civita symbol:
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Dot products of four-vectors:
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Video on the metric tensor:
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Im a level -1 tensor boi. I accidentally found this channel and have no idea whats going on but i watch anyways.
+1 for the enthusiasm
You are the only light that can guide me through the dark alley of tensors and four vectors, will never be able to thank you enough! :')
Thanks so much, glad you found it helpful!
Thank you so incredibly much. The fact that the Electromagnetic tensor is simply defined to be the partials of A and the origins of A was extremely confusing to try and self learn. I couldn't be more grateful for your service. I have been looking online for a while and was so confused but you cleared up everything. You're a legend. Thankyou!!! All your videos too not just this one. I always find myself landing on your videos and they make everything seem so clear and straightforward.
Thanks a lot for the kind words! Really glad you found the video helpful!
I was in a bit of a rut with my interest with Physics. Well, i've been binge watching your Tensor videos, so, THANK YOU SO MUCH. I needed a gateway back into enjoying this stuff, and you're the perfect level of difficulty/relevence for me. So, thanks :)
happy to help!
It's true, he's reignited my excitement for learning- to the point where I dreamt I was excited, bouncing up and down with energy, to go into a math class- only to then learn it was the day of the test and consequently wake up haha. xD
First time I've ever clicked on a video within 10 minutes of it's release
first bed wood
Proud of you man.
I dont know what a tensor is, and at this point im too afraid to ask
What is an equation?
But you want to know it, dont you?
Literally just a multi-linear map.
docxy0 If a quantity transforms from one set of coordinates to another via a Jacobian, then it is a tensor. If the transformation is not a Jacobian, or is a Jacobian multiplied by some quantity not equal to the identity, then it is not a tensor.
The simplest way tp think of it is that a tensor is a mathematical object that requires multiple pieces of information to describe, and transforms according to certain tensor transformation properties listed out already by angel mendes-rivera.
A Hodge star video would be great, especially given this video was just made, it would make a nice compact summary of the different EM formulations i.e the dF = 0, d*F = J forms
Clifford algebra my dude
When is the Dotson Product going to be a thing? Let's make it happen.
Andrew Eagon More like the Dot Product, son!
Lol I love this. We had a kid in my stat mech class who referred to theta dot as tater-tot and from that point on everyone called it that ;)
KHaN acadeMy didnt prepare me for this!!!
Yes it did! Sal should have convinced you that with clear direction you CAN learn and understand math that seemed incomprehensible at first glance. Always grateful for his incredible contributions to those of us who are not geniuses!
You should introduce differential forms at some point. Then the definition of the field strength tensor makes sense
Differential forms make the EM field tensor and Maxwell's equations so much more beautiful
Agreed. Also Maxwells eqs written in Clifford Algebra
Just been revising electrodynamics for my third year exams (which are all online now) here in the UK and I just have to say that the expression for the Faraday tensor in terms of the 4-potential and 4-gradient is just one of my favourite equations I've come across so far in my physics career, it's just so satisfying!
Wow, I watched the entire video -- he is a decent teacher.
But mostly, very handsome.
You need an upvote even though I didn't understand a word you said. Also I miss the Physics Phlogs
I wish I took physics... I ended up taking chemistry and bio to fill my science requirements. Maybe i'll use some of your videos to teach myself.
what???
Just went over this in final undergrad E&M semester...great timing!
Thank you explained tensors in a way that wasnt stupid like trying to use twnsors to define cube or problems that arent optimized to be solved with tensors like idk spitting aphere into a mercarder projections because honestly tensors are for turning well sefined easy to understand boundary conditions such as idk spacetime, heat flow, etc. Where there are clear dead stops and complete reflections and so many hardstops that are suited to be solved with continuous functions.
I feel so comfortable watching your videos - they are tough and ambitious but I really enjoy how you explain the things in your point of view.
I am just an ongoing physics teacher in second semester, but I feel like I should actually do all that epic theoretical deep stuff that the "real" physics students also have to do. I just can't stop asking for deeper reasons of why things are the way they are 😄
Sorry I didn’t see your comment sooner, thank you for the kind words!
Good explanation.. thanks, Andrew!
Just one suggestion: keep the use of spatial indices 1,2,3 versus x,y,z consistent.. for instance, in your final tensor, there's both B^3 and B^x entries. (Actually, this is a big problem in a lot of physics material.)
..brillant , as always..hell job ,doctor Dobson!!!
12:52 well that popped out elegant af👌👌
Thanks a lot brother. This was really really helpful. Again your tensor calc playlist is the best source for its introduction available as of now on youtube. Keep up the great job!
I can't wait till i understand these equations...
I’d love a video on how you felt about chemistry - what classes you took, what topics you liked and didn’t like. I am a chemistry major and intimidated by the higher level math of physics but got into physics concepts because it and chemistry go hand in hand in a lot ways! Physical chemistry is scaring me to death :-)
Excellent vids! Can't wait for dual tensor derivation!
Why do I watch these I don't understanding anything lol.
I dont know if this is a better question for a mathematician, but so just finished reading Flatland and it got me thinking. Can you do a video on the 4th dimension and how it can be interpreted by us? I that there are some things which have a sort of 4th dimensional connection like using a triple integral to find mass and total electric charge. Are there any other examples or is the concept of a 4th dimensional totally out of our use?
Thank you, I finally have a basic understanding of the Faraday tensor that I can build on. You the man! 😊🙌🏽🎊
I think it's called scalar potential only because it is the gradient of a function of R^n->R therefore called a scalar field.
Thanks a lot, it helped a lot
sweeeeeet, I saw this in Griffiths and have been looking for a good walk through.
Hey Andrew, could you do a video "deriving" the existence of the magnetic field from Coloumb's law and special relativity. I've done a simplified version of the derivation using the trick of a current flowing down a wire and then applying the Lorentz transformation to a moving charge and you can work out that the force is precisely qv(mu_0 I/2pi r), but I couldn't work it out more generally. Say you have an arbitrary E field, applying the Lorentz transformation in a moving reference frame, charged particles should be deflected in unexpected ways, essentially constituting a "derivation" of the Lorentz force law, F_B = qv x B, where B is some auxiliary field that would satisfy Maxwell's equations, and this would constitute a theoretical "discovery" of magnetism.
Love this video btw!
Amazing video, as always. Please do make a video on the Hodge dual of the electromagnetic field tensor.
Also, question: why do we introduce the vector potential? Does it have any uses other than defining this tensor? If not, wouldn't it be simpler just to define the field strength tensor component-wise?
The vector potential is a very important part of electromagnetism. It's a much simpler value to find than the magnetic field from which you can get directly from it, so it has use in problem solving. Or do you mean the four-vector potential?
I recommend looking more into electromagnetism before any other physics, as it is one of the most complete and all encompassing fields of physics.
The purpose of the 4-potential is to derive Maxwell’s equations via the Euler-Lagrange equation. In order to do that, you have to work in terms of the potential, rather than the field tensor
Thank you so much for your videos! Some questions to this one: 1. If you say A^\mu = (..,...) you're actually saying that a component equals a vector. Is this just sloppy physicist style or is A^\mu really the name of the whole vector? 2. Why do you write the indices of the E up? 3. Taking down the indices of the partial derivatives (while regarding the sign), does it just "work" or is it mathematically correct? Because applying a dual object to a "normal" object isn't the same as applying a "normal" object to another "normal" object. ("normal" = not dual)
Good questions!
1) If the index is free (ie not repeated and summed over), then it is usually interpreted as a vector equation. You can always evaluate the index for mu = (0,1,2 or 3) if you’re interested in a specific component. You just have to do it for both sides of the equation.
2) I have to watch the video again to see what you’re referring to, but it sounds like I’m referring to E as an ordinary vector with upstairs indices. I have a video on contra/co/-variant and ordinary vectors and on the metric tensor which goes into much farther detail on the answers to your questions I think.
3) applying the metric to raise and lower indices both works and is mathematically correct. This is not unique to the partial derivatives, it works with all tensors by construction. The metric acting on a contravariant vector defines a covariant vector. But the scalar product will always involve 1 contravariant and 1 covariant vector, indices summed.
@@AndrewDotsonvideos thank you so much!
This is a great video, it’s very useful for learning about the field strength tensor!
Wow, thanks for a great video! I've wanted to learn about this since I first heard about it!
Lol I have my advanced classical physics exam on this topic in 3 weeks, literally covering all of this video, ty >3 btw consider using natural units, then c=1 ;)
einsteinisbae Explaining with natural units would only make it more difficult
thanks for the video, very helpful 👍
Thank you, Andrew! This video helped me to do well ( I hope 🤞 ) on my electrodynamics final exam!
Awesome video. But why can't the electric and magnetic fields be represented as four-vectors instead of a tensor?
Nice! We're doing this right now. Great timing
Terrific video!
What about squaring it since the tensor squared is part of the maxwell and qed lagrangian
Thank you so much !
This is just what i want !
THANKS YOU!! THAT WAS THAT I NEEDED ❤️
15:38 , Instead of X_(i) , it will be X^(i) . Btw amazing video
Please make a video about Einstein notation for 4-vectors. I tried Griffiths EM book, but I find it very confusing, especially when multiple different indices are used. I just don't get anything of it...
Latin indicies are 1, 2, and 3 (not a single one of those, but all of them), where 1, 2, 3 = x,y,z. greek indicies are 0 through 3, zero being the time dimension, and 1 through three are the three special dimensions.
The actual symbols are dummy indicies, and don't hold much significance. using a mu is the same as using a nu, or a lambda, or a rho. likewise, using an i, j, k, l, m, n, or any other letter all mean the same thing. The only time there is an importance in the letters is if you have a subscript and a superscript of the same letter, that means you sum over that part over the range of 1 through 3 or 0 through 4 depending on if it is a latin or greek index respectively.
You should buy a tensor calculus book or a math methods book to get some good exposure and a better explanation of it than what you will get from Griffiths. Tensors are typically beyond what you learn in undergrad except for a sliver of it at the end of electrodynamics.
In min 7 you said the curl of the electric field is zero if we have static electric field I think its also true if we have static magnetic field because a moving charge induces magnetic field but only change in magnetic field induces electric field
Correct me if im wrong please!
A constant electric current in a wire induces a static magnetic field with a nonzero curl.
So I'm in honors high school physics right now. You know how it is. When we did the whole W=Fdcosx thing we were to assume a constant force. And we just finished up kinematics, where we were told to assume constant acceleration. And I know that could be useful but it rarely is, right? So, I'm really interested in more advanced physics, which is why I watch your channel which inspired me to take AP physics next year. Being in Honors Physics pains me, because I'm in AP Calc right now, so I know a lot of the math skills necessary to do more advanced physics where you do not have to assume forces and accelerations and stuff and other variables as constant. I also spend a lot of time researching, mainly through youtube videos like yours to understand more advanced physics, not just the really cool cosmic stuff like black holes and stuff. But also stuff that is really relevant to Earth and our everyday life/progression of the human species. Its just frustrating not being able to take an integral or a derivative, when I have the necessary skills. But, really cool physics you got on display on this video, I like how I can mostly understand whats happening as a high school junior. That speaks a lot about how well you know the subject and I really appreciate how you take the time to explain what you are doing to me. Thank you.
If it's available to you next semester/year, take AP physics C. It's AP physics but with calculus so you can learn the material with the intended connections using derivatives/integrals!
@@milranduil Yep, that what I'm planning on that for my 2nd semester of senior year!
thanks andrew
Thank you for nice explanation.
Do you need the (1/c)(E^i)? Should it not be be-E^i = (1/c)( d/dt(A^i) ) + ( d/dx(phi) ) ? Just quibbling.
I had my radiation and relativity exam last week, man i should have found out about this vid earlier
Can you give an explanation for the covariant form of this tensor, I tried it actually but I got stuck, the Electric field components get their signs reversed (this part I figured out) but the magnetic field components have the same signs, that I can't figure out
Thank you so much. 🖤
The Faraday tensor resembles the stiffness tensor of an elastic material in Mechanical Engineering
we wunt string theory !!! btw thanks for tensor videos although i think the best way to comprhend tensor is to start by geometric meaning of it ,vectors covervteor and how they transform under change of cooardinates, lowering and raising indesices ones thats done it will be lot easier to deal with any tensor thanks from algeria.
Hey Andrew! Big fan of your videos! Have you ever thought of making pdfs from these topics you talk about and leave a link for them in the description? I think that would be extremely helpful! Thanks
Luis Otávio maybe once I have extra time on my hands
@@AndrewDotsonvideos that would be really great. And maybe that will be the start of a book someday...
@Luis, we're collectively delegating the task to you haha. But seriously, if you skip the video for a pdf, you kind of miss out on the best part 😂 Which is Andrew.
That said, I did take notes on the first video I watched of Andrew's on Maxwell's Equations. They might be useful: ibb.co/V38f6MH
Helped a lot. Is there goong to be a Video on the other field strenght Tensir too?
Do you have a good explanation why we wrote all of physical quantity (space component, 4-potential, 4-current, etc) with contravariant indices naturally?
Very nice 👍
did you mean to write B^3 as B^z as the component F^12?
Hey Andrew, have you seen the Michio Kaku AMA on r/IAMA?
So, Andrew can we expect in the next video on how to tensorically express Maxwell's equations in two equations, one of which is Stress tensor and one is taking dual of it? Because I couldn't well understood it in my EMT course work?(EMT:- Electromagnetic Theory class)
He derived the first one, I hope he'll derive the other one with the dual
Cool
Sir how to obtain Maxwell s equation from Lagrangian..
5:32 My professor gave the same reasoning, but this is simply wrong. That there *exists* a vector potential follows from the poincare lemma. You and it seems many physicists confuse the trivial implication that "B = rotA => divB = 0" with the poincare lemma "divB = 0 => there exists an A such that B = rotA"!
amazing vid boii
🔥👽Awesome. And i have more perspective: We often use 4x4 matrix transformations in 3d computer graphics. The magnetic B-field is located in the field tensor matrix locations mathematically referred to as "shear." 🤯 [CGI uses (x,y,z,w) points and the physics here uses (w, x, y, z) 4-tensor]
The Electric Field wasn't "unified" with the "magnetic field", it was collapsed into the electric field as shearing. It shouldn't be called the "Electro-Magnetic Field"... it would be more accurately called the "Electro-Electroshear Field." A change in the electric field is a perpendicular electric shear, moving at the speed of light. The "magnetic field" doesn't exist and is explainable as the electric field masquerading under shear as the "magnetic field", it's the same field acting in different ways.
I suggest that they thought it was two different fields because those modes of the electric field must be measured differently and so assumed they had their own distinct "field" or "dimension."
A model that i like is that an electric charge is like the displacement of a boat on the field of water. We measure the displacement of boats differently than how we measure the wake of a moving boat. Magnetism is like the "wake" created when that displacement shears with itself (moves across the water). The wake is perpendicular to the direction of change. granted, the the magnetic wake happens across the field itself at each point when a charge is moved. The change of electric potential at each point (from the movement of a charge) shears with itself to create magnetism, so it's 5th+ dimensional. But the boat and wake analogy is an easy way to think of it.
If you shear the electro-shear (aka a changing magnetics), you get back to the direct electrical forces. Exactly as expected.
In a photon, when at the maximum electrical potential, it would also be at the maximum shear/magnetism. They cycle in unison.
This groks with using special relativity to translate between the shear inertial frame and the electro-static inertial frame. there are many videos on TY on the use of special relativity to make magnetism into a relativistic effect of the electric field.
Magnetism is electroshear.
💥
I'm never going to unsub
But how would we know that we have to start from a anti symmetric sensor exactly like that?
Well if you ask how to properly construct a relativistic object (like a four-vector, tensor, etc) from two 3-vectors (E and B), you first note that that is 6 degrees of freedom all together (E_x E_y E_z, B_x, B_y, B_z). A four-vector only has 4 degrees of freedom, and a symmetric 2nd rank tensor has 10 (the 4 diagonal components and then 6 of the off diagonal) which is too many. Because an anti-symmetric tensor has 0's for the diagonal elements, an anti-symmetric 2nd rank tensor only has 6 independent components which is exactly what we need. So that's a roundabout way of arguing why the relativistic object containing info on the electric and magnetic field has to be an anti-symmetric tensor.
I wonder if these videos are making him Ricci.
how can I formulate poynting vector in terms of EM trensor?
Wait, how is a derivative of a tensor equal to a tensor, since A is a rank one tensor, to get another tensor you need to apply the covariant derivative (the big D). I think that you have been doing that all along (defining these 'partial' derivatives as gradients).
Tnx man,it helped me.
Because of antisymmetry you only need to know either the lower or upper triangle in Fuv
I love it
Ahh I love tensor calculus 😁👍🏻👍🏻
Hey Andrew. Miss your daily uploads. Anyway can you please recommend a book on statistical mechanics for undergraduate. Thank you.
It's hard to recommend a good stat mech book for undergrad because you can't really get far without knowing your quantum. I've been using pathria for gradschool, but it already assumes you know thermo:/
Hey I was wondering abt what you mentioned in 2:49, you were talking about how the terms that make up a tensor equation must be tensors themselves. Is that the case in general? I might be misremembering or misinterpreting but I think this is not the case for the covariant derivative, were to variant terms come together to create a tensor. (Coordinate derivatives aren't tensors and chirstoffel isn't a tensor).
Playing Arc I’m saying if we have nothing but tensors characterizing the equation, the result is guaranteed to be a tensor as well. Like the tensor product of two vectors, or the contraction of two tensors. The covariant derivative is a tensor though, at least in the physics context. It transforms covariantly under general coordinate transformations. You’re right about the Christoffel symbols not being tensors. It’s a correcting term in a sense to add to the gradient to make derivatives of tensors still be tensors.
@@AndrewDotsonvideos Yeah yeah I was saying that exactly, covariant derivative is a tensor but it's calculated through objects that aren't (regular coordinate derivatives plus a christoffel correction). But yeah, I get what you were trying to say now, thanks for the response :).
@@PlayingArc cool:)
But isn’t the metric tensor d(-1,1,1,1) instead of d(1,-1,-1,-1)?
@15:39, that should be del/delx^i instead of covariant...I think
Great vid lad
I always made it difficult for the correctors by going for the -,+,+,+ convention instead and stating that at the start of the test/homework >,< It gave me less minus signs to work with.
DrQuatsch It doesn't give you less minus signs to work with, because you can literally just distribute - out of the spatial component and write it as a Pythagorean term. It's still only 1 minus sign. If anything, (+, -, -, -) is even better, because you avoid unnecessary unphysical minus signs in Lorentz invariants such c^2 dτ^2 = - ds^2.
Man this is damn good!!
A gift from my physics professor:
A dipole moment is placed in a uniform electric field oriented along an unknown direction. The
maximum torque applied to the dipole is equal to 0.1 N.m. When the dipole reaches equilibrium its
potential energy is equal to -0.2 J. What was the initial angle between the direction of the dipole moment
and the direction of the electric field?
Probably a piece of cake for you but I cried. Have fun with it :)
hey ! i’m a hs student who is planning on majoring and getting a phd in something with astrophysics but i can’t take a physics class right now because to say the least , the teacher quality at my school is subpar. i was wondering if you had any books that you’d recommended or any resources i could use to learn some before i get it thrown at me in college ?
@@taktoa1 I highly recommend you get the set of four Feynman lectures. You will be way ahead. They are a compilation of physics lectures he taught to undergrads at CalTech. Very clear and comprehensive.
haven't even seen it completely i just know it's gud.
I literally never clicked so fast XD
Nice! We're doing this right now. Great timing.
@@HighTech636 same, we're taking second course on electromagnetism... so this is pretty useful.
Plus, 30 min andrew dotson vid = heaven
Hello Andrew, is there a book on Set Theory that you can recommend me?
Thanks :)
I don't really have any recommendations sorry!
This depends on your goal. Do you want to study the origins and/or the axioms of set theory, or do you just want to know the concepts and terminology that are used in everyday mathematics?
@@EssentialsOfMath The first one :)
(+ - - -) I see you are a man of culture, Andrew.
cool af bro
Much Physics, such wow, much Math
Very comment
you are big smart
Do you have a twitter or something like that ?
"Not too long video". "Just the definition."
*Looks at the video length*
Wha--
But not complaining though lol
Pretty sure you were thinking of Levi Ackerman when you corrected yourself when mentioning the Levi-Civita symbol at 19:41 haha
Good video, down with the mathematical snobs ;-) No need for mathematical precision here...it's physics!
I likes da video, but is very basic for me big brayn.
👍😊😊😊👏👏👏
As someone who just finished learning algebra 1, what the heck?!?!?!?!??!???
B^3 should be B^x