W1.2 전기역학 방정식 입문(Introduction to Electrodynamics equation)

[커세라] 전기역학: 기초편(Electrodynamics: An Introduction)

1주: 정전기학 입문(Introduction and Basics of Electrostatics)/강의자료
    W1.0 강의안내(Introduction)
    W1.1 전기-자기 입문(Introduction to Electromagnetism)

W1.2 전기역학 방정식  입문(Introduction to Electrodynamics equation)/동영상/영문자막

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전기역학을 기술한 네개의 방정식을 보자. 전기장 E 의 플럭스(flux)와 순환(Circulation), 자기장 B의 플럭스(flux)와 순환(Circulation)을 기술한다. 언뜻 보기만 해도 상당히 난해해 보인다.

* 방정식에 선적분(line integral)과 면적분(surface integral)이 보인다. 장(field)의 개념을 기술 하려고 수학을 전용하고 있다. 고도의 추상적 개념을 그림과 글로 표현하면 수월 하겠으나 여의치 않다. 그림은 삼차원을 평면위에 그림으로 그리기 쉽지 않다. 글은 너무나 모호하다. 과학이 예술이나 문학으로 흘러선 않된다. 글과 그림은 너무나 자유도가 높다. 정의역의 범위가 제한되고 그 작용으로 얻어지는 결과가 명확하게 정의되어야 한다. 지도상에 그려진 산을 보자. 같은 높이를 표시한 등고선의 길이, 그 등고선에서 정상까지의 산의 표면적을 구할 수 있을까. 폐곡선과 그로 둘러 쌓인 표면적은 모두 유한하다. 다만 그 모양새를 기술할 적당한 기하학 함수가 복잡하다. 이 기묘한 모양의 둘레의 길이, 표면적을 수학적으로 표현한 것이 선적분, 면적분이다. 공간 도형에서 선과 면은 두개 이상의 독립변수로 표현되고 직각접선의 함수로 정의된다. 따라서 중적분을 확장하면 선적분과 면적분이 된다. 여러번 적분을 중첩해서 쓰기 번거로우니 선적분과 면적분의 표기를 만들어 냈다. 면적분과 선적분을 추상적 의미를 갖는 적분 연산자라고 이해해 두자. 의미를 이해하도록 노력하고 수학기호에 주눅들지 않도록 하자.

So you can see the surface integral of electric field around a closed loop is equal to the charging side divided by productivity of vacuum(유전율, permittivity).




* 면적 벡터의 방향은 면의 수직이다. 따라서 벡터 E와 벡터 da의 스칼라 곱에 대한 적분은 방사형 전기장 흐름(flux)의 총량이다.

Circulation of E is the line integral of electric field along a closed loop is equal to the negative of d over dt, which is the differential equation of the B da which is the surface integral of magnetic field which is magnetic flux.




Similar equation as the first one, but then on the right side, you have 0. Which means you don't have 'a' in magnetic,




and then the last equation you have the same form as in the second, but on the right side you have one additional term. Which describes the current flow through a wire, which creates a circulation of B field.




[참고] A student's guide to Maxwell's equations, Daniel Fleisch, Cambridge University Press, 2008

아주 난해해 보이는 위의 방정식들은 이 강좌 내내 하나씩 살펴보기로 한다. 수학을 하기 전에 먼저 물리적 의미를  생각해 보기로 하자. 굴곡진 전기장 주변의 순환(what it means to have a circulation of electric field around a curve)이란 무엇일까?

전기장(electric field)이라 하면 포텐셜의 경사(gradient of potential)가 머릿속에 떠오른다. [전기장 뿐만 아니라 모든 '장'이라 하면 3차원 공간상에 분포하는 기하학적 모습을 취한다.] 포텐셜의 경사라고 했으니 고점(high-point)과 저점(low-point)이 있을 것이다. [고점과 저점의 의미는 그 지점의 영향을 미치는 요인(포텐셜)이 있다는 뜻이다. 지형의 높고 낮음을 부여할 수 있었던 것은 중력 포텐셜 때문에 의미를 부여 할 수 있다.]

닫힌 원주를 따라 한점을 고점과 또 한 점을 저점이라 하자. 분명 고점에서 저점으로 경사진다. 그럼 (기하학적 길이를 재는) 선적분은 어느 방향으로 가야할지 고민이 아닐 수 없다. 특히 두번째 방정식의 선적분의 경우 순환 방향을 결정해야 한다. [마치 산의 등고선 고저가 중력 포텐셜의 영향을 받아 생겼듯이 전기장의 순환 방향도 자기장의 변화하는 방향에 영향을 받는다.]

따라서 전기학에서는 자기장이 변하는 상황을 상정해 놓는데 그로인해 아주 복잡한 현상이 나타난다. [두번째 방정식에서 전기장의 선적분(전기장 순환)이 자기장 면적분(자기장 플럭스)의 변화하는 관계를 기술하고 있다.]

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복잡한 전자기 현상(electromagnetic phenomena)은 네개의 방정식으로 요약된다. 이에 더해 로렌츠 힘(Lorentz force)을 합쳐 전기역학(electrodynamics)의 모든 것을 말해준다고 볼 수 있다. [네개의 미적분 방정식과 한개의 힘의 법칙을 배우는 것이 전기역학 강좌의 전부다]

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과학관에서 정전기 장치를 만져봤을 것이다. 양손을 전하를 띈 금속공에 얹어 놓으면 머리카락이 곤두 서는 경험을 해봤을 것이다. 아는 전기장이 머리카락을 들어올린 효과로써 qE 에 해당한다. 머리카락은 두 전하가 밀어내는 평형점을 찾아간다. 전기장이 만드는 힘의 이해는 직관적이다.



이번에는 로렌츠 힘중 자기장 B를 가로지르는 속도 v가 포함된 항을 살펴보자. 이 실험은 일찌기 벤자민 프랭클린에 의해 행해졌었다. 집에서 (교류)전기가 통하는 전깃줄 아래에서 자석을 갖다 댔더니 늘어진 전기줄이 한쪽으로 움직였던 것이다. 전기장과 자기장이 서로 작용하여 힘을 발휘한 것이 분명하다.

Yeah, so let me help Melody to finish this answer. So you can see you have magnetic field pointing upward.




전류(전하)가 흐르는 전선의 수직으로 형성된 자기장은 맥스웰 방정식을 따른다. 그 아래 막대 자석을 갖다 댓더니 늘어졌던 전선이 움직였는데 그 힘을 로렌츠 힘 법칙으로 기술된다. 전신이 움직인 힘은 결국 뉴턴 역학과 같다. 운동방정식은 이미 알고 있던 대로 다음과 같다.



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So, if you look at the wire it is creating a circulation of B field, because of the term on the right side of the force equation in Maxwell's equations.

전류가 흐르는 전선 주위로 자기장 '순환'이 생긴다. 맥스웰 방정식을 보면 이 자기장은 전류의 흐름과 면적 벡터의 스칼라 곱(두 벡터의 정사영)의 면적분(전류 플럭스)에 진공 유전율(permittivity of vacuum)로 나눈 함수다.

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이번에는 자석 대신 전류가 흐르는 다른 전선을 가져가 보자. 자석 대신 전류가 흐르는 전선 주위에 자기장이 형성된다. 두 전선에 흐르는 전류의 방향이 같다면 힘은 인력(attractive)이 될까? 아니면 척력(repulsive)이 될까?



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We just learned About the magnetic field, created by wire carrying current. Also, we also discussed about interaction between current carrying wire and magnets.

자석이란 무엇인가?

Based on our discussion, we're going to think about the essence of magnet. And in order to do that, we're going to revisit the structure property relationship. So if you have the same property, can you expect to have the same structure?

>> Not necessarily, because the mechanism is complex. However, you can kind of
visualize it in a similar way.

Exactly. So if you have the same property, you may not have the exact same structure. However, the important feature of structure will be the same. So, let's take a look at this example.

자석대신 전자석으로 자장을 만들면?

So we know that we can replace this bar of magnets, by electromagnets.



And we know if we take an x-ray picture of magnets, you'll not see this bone inside the magnet. However, they are going to give you the same exact field line. So you may wonder maybe inside magnet you may have a circulating current which is the essential feature of electromagnets.

--------------------

So, we can understand magnets in terms of permanent currents in the atoms, and currents is the motion of charge. Then you can think of planetary motion of Earth orbiting around Sun, or orbit spinning about itself.

전하의 운동으로 전자석이 자장을 형성한다는 점을 알겠다. 그럼 자석이라는 물질에서 전하들이 움직여야 할 것 아닌가?

The same analogy can be applied to the electrons in magnets, where the motion of the electrons in atomic orbits can contribute or make magnetic field. Or the spin motion,
which is the spinning motion of electrons, can also give magnetic field in the magnet.

And we don't have to add extra terms to take care of magnets, right?

Because if we just take all currents, including the circulating currents of the spinning electrons, then the law is right. So then we don't have any more complicate situation in our law.

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All right, Dirac is a big scientist who believed in the presence of monopole. However, none had been found, you can see from the Wikipedia.

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However, some resembling monopoles are recently found. For those of you who are interested in those solid state physics, you can read more in the articles published in Wikipedia.

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We're going to learn about the time varying E field and magnetic field generation, also the big achievement of Maxwell, adding a single term to the force equation. Before Maxwell arranged those equations, people didn't know the existence of the term that is appearing here. They just knew that electric wire carrying current will create circulation B flow, and we call it Lenz's law.

시간에 따라 변하는 전기장(time varying E field)과 그에따라 생성되는 자기장(magnetic field generation)에 대해 알아보자. 맥스웰의 커다란 성과 중 하나로 자기장의 순환에 전기장 관련항을 추가한 것이다. 맥스웰 이전에는 전기장의 항(빨간원)의 존재를 몰랐다. 단지 전류가 흐르면서 자기장이 형성(파란원)된다는 사실만을 알고 있었다. 바로 렌츠의 법칙(Lenz's law)이다.

However if you think about the situation where you have a capacitor like in this case, and you draw a line arbitrary line around the wire, then you find something peculiar.

Let's start with the surface that is drawn here, s1 which is the planar circle where you
are having the loop as the boundary. Then it is easy to understand that the magnetic circulation will be created because you have current carrying wire, according to this equation.

However, if you create another hypothetical plane that is extended like a bowl, but it doesn't cross the wire, then you understand no current will flow through this plain.

So in that case, you will have no circulation of magnetic field if you only have this term.

So knowing this discrepancy or dilemma.

맥스웰은 전자기파(electromagnetic wave)를 예견했다!

Maxwell edit this equation, which means that electric field change as a function of time will also create circulation of B field. With this edition, the equation became complete. Not only it became complete, but with combination of the second equation.  Maxwell was able to predict the existence of electromagnetic wave.

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Now, let's take a look at the moving wire or magnet, the examples that we just discussed, to understand the first term and the second term of the Lenz's Force. So we are going to understand how people came about with the invention of generators and motors. Also, we will think more deeply how the simple law of relativity can sometimes be very complicated.

So let's take a look at this wire hanging over two poles with the magnets underneath and make a magnified view, like in this case.

You have magnets and portion of your wire here. And imagine that you are moving this wire toward the magnets. Moving the wire toward the magnets. Then what happens is, the charge on the wire will have a motion with a velocity of v.

And because of the presence of magnetic field, according to the law of FBI, you will have force along the direction of the wires. So these charge will have a current along the wire. So that's how we know that moving wire with the presence of magnets will create current. That's a generator.

But then, let's take a look from the different perspective.  What if the wire is standing still, and I'm moving the magnets toward the wire. You'd expect from the argument of relativity, you will have the same result. However, from the mathematical point of view, because while you are standing still, there is no velocity of the charge here. So they are standing still. So the magnetic field will not create any force, according to the FBI law. The other things should happen, the electric field should occur inside of wire to make this movement happen.


So that can be understood using the Maxwell's second equation, where you see you will have circulation of an electric field due to the change of magnetic flows.



So, once we extend that idea, then we can understand why, if we were to take this loop inside this horse shoe type of magnets, you will convert mechanical energy into electricity. And this is really the essential part in any power plants, be it fossil fuel, bit it natural gas, be it hydraulic or nuclear power plant, even solar thermal plants. They boil the water, and they turn the turbines like this.

자기장의 시간 변화는 곧 전기장의 순환(circulation)을 부른다는 맥스웰 두번째 방정식이다. 이는 자석을 돌려 전류를 생산해내는 발전기의 이론이다. 운동에너지를 전기 에너지로 바꾸는 변환장치다.

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So with the addition of the term and the force equation in Maxwell, he was able to predict the electromagnetic wave. And you can see, combining those two equations.

If we change this form into differential equations, you will have this very neat equation with relate the spatial. Distribution or change with the temporal change. And you will see they are symmetric, showing that they will have some kind of same form of radiation.

And this is how we are now talking to you through the Internet, through the wireless communication. And thanks to Maxwell, we are living in a world connected and making this world smaller and smaller.

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So we will now try to recap the things that we just discussed. When you think about the fields, what could be the possible complaints,

>> Well, it seems kind of abstract, a little bit.  We don't know exactly, many people don't understand the exact mechanism of how the forces are transmitted.

So intuitively it's easier to understand something that force  is transmitted through a contact. So, if we're not thinking this that way it's becoming more and more difficult.

However, here, the good question you'd have to think about what is the most convenient way to look at electrical effects? So that all the forces or electric fields or magnetic fields, they are just tools for us to understand the phenomenon. So we will also discuss why magnetism is relativistic effect by revisiting the example we thought about in the hanging wire.

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So in this light, we are going to think about why magnetism is a relativistic effect, by revisiting the examples that we covered before, the parallel line example. But before going to that parallel line or parallel wire example, just think about two charges traveling parallel to each other.

So at a glance, this might look similar to the parallel line example because you are creating a current, because current is just the motion of charge. And if you have a parallel charge moving in the same direction, you may first think, like in the power line case, you may have only attractive force, because these are creating magnetic fields that will exert magnetic force that is attractive. However, if you think a little bit more, then if you are looking at those charts and you are moving with them, then those charts will seem stationary and if they are stationary, there is no current. So you will not have any magnetic field. And you will see this is just cross charge sitting on top of each other, so you will have huge repulsive force which is coming from electric phenomenon. So how can we understand this, Melody?



>> So unlike in the case with the wires, there are only positive charges in the wire that are holes in electrons. But here there are only the positive charges moving next to each other. So there will be a large repulsive force between them.

Exactly, so to complement Melody's answer, it is the near perfect cancellation of effects in case of wires, which allowed relativity effects which is magnetism. As you can see, in this case the electric force is much, much larger. So magnetic field will be minor. And as you can see, if Melody is moving with this charge, that effect will disappear. And that's a small correction. So the reason we could understand or see that in parallel wires is because they are perfectly balanced out.

If they're not, then as in this case, the majority will be electric force. And the relativity states that if you move, if the frame is different, then the force will be different because the time frame will change. And that small correction is really represented by presence or absence of magnetism in this specific case, okay?

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All right, so the last but not the least part is about the history of electromagnetism. In a nutshell, there are many texts here, but in a nutshell the most significant event in 19th century, I agree with Richard Feynman that it is Maxwell's discovery of the laws of electrodynamics.

And with this, we're going to wrap the first lecture. And if you have any questions, like we said before, you can email us and we will reply to your email as soon as possible.

Thank you very much, and have a good day, bye.