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)
W1.Q 1주 평가문제(Week 1 Quiz)
2주: 스칼라, 벡터 그리고 미분 연산자(Scalars, Vectors and the ∇ Operator)/강의자료
W2.1 스칼라와 벡터(Scalars and Vectors)
W2.2 ∇ 연산자 활용(Applying the ∇ Operator)
W2.Q 2주 평가문제(Week 2 Quiz)
3주: 가우스 정리, 흐름 그리고 순환(Gauss' Theorem, Flow, and Circulation)/강의자료
W3.1 가우스 정리 유도(Deriving Gauss' Theorem)
W3.2 흐름 그리고 순환(Flow and Circulation)
W3.Q 3주 평가문제(Week 3 Quiz)
4주: 정전기 및 전기 포텐셜의 플럭스(Electrostatics and Flux of Electric Potential)/강의자료
W4.1 정전기 및 전기 포텐셜(Electrostatics and Electric Potential)
W4.2 전기 포텐셜의 플럭스(The flux of an electric potential)
W4.Q 4주 평가문제(Week 4 Quiz)
5주: 정전기장과 차폐(Electrostatic Fields and Shielding)/강의자료
W5.1 정전기 장(Electrostatic Fields)
W5.2 Fields inside of Shell
W5.Q 5주 평가문제(Week 5 Quiz)
W.C Conclusion
"전기역학: 기초편(Electrodynamics: An Introduction)": 수료증
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[W4.2-1]----------------------------------------------------------
[W4.2-2]----------------------------------------------------------
이번에 다루는 전기장의 플럭스(Flux of Electric fields)는 (거리의) 역자승(inverse square)인 힘에 관한 것이다. 전기력(쿨롱 힘)뿐만 아니라 중력(gravitational force)도 거리의 역자승 법칙(inverse square law)이다. [힘이 미치는, 또는 장이 퍼저나가는 범위는 면적에 비례, 거리의 제곱에 반비례]
Inverse Square Law (Hyper physics)
아래에 선보인 빛의 세기(Light Intensity)도 거리 역자승 법칙에 해당한다.
The light intensity(number of photons per unit area) depend on the area of interests, which increase as a function of the radius and which is 1/(r^2).
Electric field represents the flow of something that is conserved, everywhere except at charge. Then, we can prove that electric field is also 1/(r^2).
* 물리법칙을 다룰때 에너지 보존 법칙이 전제되었다. 에너지 '발생원'에서 전방향으로 방출되어 '퍼져나가는 동안' 소멸이나 생성은 없을 경우 성립한다. 빛의 양으로 천체까지 거리 측정에 성간소광(interstellar extinction)을 보정하지 않아서 큰 오차를 빚었었다.
에너지 원(source)에서 퍼져나가는 과정에서 임의의 입체를 통과하는 '흐름(flow)'은 얼마일까? 입체에 들어오는 양과 나가는 양이 동일 해야하는 '보존 법칙'을 생각해 봐도 '흐름'은 0이어야 한다. 이를 좀더 수학으로 설명해 보기로 한다.
다음과 같은 단순한 상황을 설정해보자. 독립적으로 존재하는 한개의 전하가 있다. 이 전하 주변의 구형(sphere)공간을 생각해 보자.
한 전하로부터 방사형으로 반지름이 다른 두 구 사이의 공간을 전기장이 퍼져 나간다.
두 구 사이 공간을 작은 부분으로 자른 이 미소 육면체 공간을 통과하는 흐름, 즉 플럭스(flux)를 따져보자. 플럭스는 면 벡터와 전기장 벡터의 내적이다. 따라서,
- 미소 육면체 중 전기장과 평행인 면에 대한 플럭스는 0이다.
- 전기장과 직각인 내측 구면과 외측 구면을 통과하는 전기장의 곱을 면적분하면 총 플럭스가 된다.
[W4.2-4]----------------------------------------------------------
이번에는 껍질(shell)의 안쪽면과 바깥쪽 면을 전기장 방향에서 약간 꺾어보자. 복사 면적 Δa는 증가하며, 이 복사면의 직각에 해당하는 전기장 E_n은 원래 전기장 벡터보다 작아진다.
area: [Δa ∝ 1/cos(θ)] ; increasing with θ
vector: [E_n ∝ cos(θ)] ; decreasing with θ
플럭스는 면적과 벡터의 스칼라곱 이므로,
flux: [E_n⋅Δa = const]
따라서 전기장이 통과하는 면이 어떤 각도로 기울었든 입사와 방출의 합은 같다는 원칙은 변함없다.
So, you will see the summation of those component will be zero no matter what type of till you give to that part.
[W4.2-4]----------------------------------------------------------
극단적으로 원통(맥주 깡통)처럼 아주 비정형의 입체를 통과하는 벡터 장의 경우에도 이 원칙이 적용된다. [단, 폐포(closed surface)여야 함]
The total flux out of the volume enclosed by any surface(an arbitrary shaped object) is zero.
From the divergence law, no matter how you cut your object, the flux of the arbitrary object will be the same as the summation of the part that you just cut it.
So, each part that you cut will have zero flux and no matter how you scan it, everybody will have zero flux. So, if you sum them up, you have zero flux.
[W4.2-5]----------------------------------------------------------
The surface 'a' has the outward normal, and surface 'b' has also outward normal. The faces parallel to electric field will not contribute to the flux.
We only have to think about the external surface of one pyramid and external surface of the other pyramid. And because both of them has the same polarity for the surface normal, as well as they have the same direction of the electric field, they will have positive number both sides.
If you sum them up, you will have non-zero value. That's why we have non-zero positive flux out of this charge in this case.
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임의의 입체에 대하여 외부에 존재하는 전하에서 생성되는 전기장에 대하여 입사 플럭스와 방출 플럭스의 총합은 같다. 하지만 입체 내부에 전하가 존재할 경우 입체 표면 전체로 전자기장을 방출하므로 플럭스는 0이 아니다. 직관적으로 봐도 당연한 이 사실을 수학적으로 따져보자. 플럭스는 면적당 전기장의 흐름을 따진다. 입체를 어떻게 정의하는지에 따라 총 플럭스 값이 달라진다.
[W4.2-6]----------------------------------------------------------
임의의 입체에서 흘러나오는 전기장 플럭스를 계산해보자. 언뜻보면 아주 어려운 문제 같아보이지만 앞서 배운 다음과 같은 두가지 사실을 토대로 뜻밖에 수월히 구할 수 있다.
Now, so with those two information,
- if the charges outside the arbitrary volume you don't have any flux.
- If the charges inside a volume you have non-zero flux.
임의 입체에서 전하를 중심으로 구형(sphere)을 분리해 냈다고 하자. 분리한 구의 크기가 어떻든 구에서 나오는 전기장 플럭스와 구를 제외한 임의 입체의 플럭스 합은 같다.[에너지 보존]
[E_n= 1/(4πε_0)(q/r^2)]
[Area=4πr^2]
따라서 구의 플럭스는 다음과 같다.
[flux=q/ε_0]
[W4.2-7]----------------------------------------------------------
앞서 임의의 체적을 흐르는 전기장(벡터)의 플럭스에 대해 다뤄 보았다. 전기 장(벡터)의 발산 법칙을 가우스 법칙(Gauss Law,=발산 법칙, divergence law)이라고 하는데 '가우스 정리와는 다소 차이가 있으나 서로 연관을 가지고 있다.
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가우스 정리(Gauss Theorem=발산의 정리):
벡터 미적분학에서, 발산 정리(divergence theorem) 또는 가우스 정리(Gauss' divergence theorem)는 벡터 장(ector field)의 선속(flux)이 그 발산의 삼중 적분(체적, volume)과 같다는 정리이다.
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What is the difference between a law(법칙), a principle(원리) and a theory(정리)?
In science we use the word law(법칙) to mean some observed phenomenon that can be always relied on. Something that happens the same way, every time, without fail.
- The law of gravity(중력법칙) says that there is an attractive force between two massive objects.
That is so, every time we check. It doesn’t explain why. A law is not a proof, or an explanation, just a statement of a repeatable observation that can be measured and used as a firm basis for research.
A principle(원리) is a fundamental mechanism by which some phenomenon is observed to operate.
- Evolution(진화의 원리) operates on two main principles: genetic diversity(돌연변이) and natural selection(자연선택).
- Animal energy production operates on the principle of oxidation of glucose.
- A car engine operates on the principle of internal combustion.
A principle is again not a proof or an explanation - it’s just the straight-out description of a process.
A theory(정리) is more interesting. A theory is an attempt at an explanation, arrived at after exhaustive research and investigation, including critically examining all the laws and principles involved in a process. It is not a tentative guess: that’s a hypothesis.
- The theory (as opposed to the law) of gravity(중력의 정리) is an explanation of how and why objects attract.
A theory is a fully formed account of how we believe something happens. A theory is neither a law nor a proof, and it might even be wrong, but it is much more than just an idea.
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첫번째, 달걀 모양의 입체 외부에 두개의 전하가 존재하는 경우, 플럭스는 0이다.
두번째, 입체 내부에 한 전하 q1이 존재하고 다른 한 전하 q2는 외부에 있는 경우다. 외부의 전하 q2가 입체에 주는 플럭스는 0이다. 입체 밖으로 흐르는 전기장은 오직 내부의 전하 q1에 의한다.
세번째, 두 전하가 모두 입체 내부에 존재하는 경우다. 플럭스는 중첩의 원리에 따른다.
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이제 우리가 공부한 것을 모아 정리해보자.
Imagine you have a sphere of charge, it's smeared out, and it has the same density, charge density Rho. What would be the field inside the sphere, electric field?
From the spherical symmetry of the charge electric field E) should be radial field. If it is a radial field, the electric field that is felt by this point will be shared with all of the sets that makes the sphere inside the sphere.
And the electric field that is experienced with the sets of field is due to the charge inside the sphere, not outside the sphere.
So, if I know the distance from the origin of the center of the sphere to the point that I'm interested in, and denote it as small r, and if I denote the distance from the origin to the outer surface as large R, then I can start to figure out what the electric field here is.
How?
Let's start. What is the charge inside this yellow sphere? You know the density, Rho. Volume is four over three Pi. Is it small r or large R? Small r. Small r to the cube, to the power of three, that's the charge inside the sphere.
[Charge_inside_sphere = ρ*(4/3)* π * r^3]
What is the flux then? You have to put Epsilon naught, that's the flux. How do we define the flux? It's the E field times the area of the sphere.
[flux=E_n * (area_of_sphere)]
and
[area_of_sphere= 4* π * r^2]
By rearranging those two flux, E becomes,
[flux: E_n * (area_of_sphere) = (Charge_inside_sphere)/Epsilon_0]
then,
[E = (ρ*r)/(3*ε_0)] ; electric field inside of 'sphere
[E ∝ r] ; electric field inside volume
[E ∝ 1/(r^2)] ; electric field outside volume(Coulomb Law)
중력: 역자승 법칙
Imagine the gravitational force(gravity) follows the same curve because it's distributed mass and you have the same form of equation.
만일 지구 중심을 관통하는 구멍을 뚫어놓고 공을 떨어뜨리면 이 공은 어떤 운동을 할까?
What it means, if you drill a hole from Seoul to, say, Argentina, you will have linear harmonic oscillator because it is like spring.
답: 스프링 처럼 선형 진동운동
[W4.2-8]----------------------------------------------------------
With Gauss' theorem, you can apply that to many other fields as well.
Gauss' law; the total flux out of a closed surface is equal to the total charge inside divided by Epsilon naught [q/ε_0].
In a mathematical equation, it is the surface integral or the normal component of electric field around a surface is equal to sum of the charge inside divided by Epsilon naught, and you can come up with this law.
Let's think about Gauss' law in terms of derivatives. We can have point function.[미분형 방정식이 적분형보다 다루기 쉽다]
Let's think about the infinitesimal cube again. The Gauss' law says the charge inside a cube, which is ρ times dV, over ε_0 should be equal to the flux, which is divergence of the electric field times the volume of the cube, dV.
[Flux: ∇⋅EdV = ρ*dV/ε_0]
then,
Coulomb's Law:
[∇⋅E = ρ/ε_0] ; The first law of Maxwell equation.
[∇xE = 0] ; Curl-free(electrostatic)
Then if I have the curl-free condition, if I combine them, then it becomes the Coulomb's law of force.
[W4.2-10]----------------------------------------------------------
Now, I just mentioned this before but let's revisit this.
Field of a sphere of charge. What is the electric field, E, at a point, P, anywhere outside the surface of a sphere filled with a uniform distribution of charge?
We figured out the electric field inside but now we're going to think about outside.
Outside the sphere, it doesn't matter what type of distribution you have, if it is radial distribution or spherical distribution. So, we can condense all of the charge into one single point. If you do that, then we know that the flux out of the surface is equal to the charge inside, contained inside the volume, divided by the permittivity(ε), and because we know the distribution is ρ(charge density), then we can come up with the number for the Q,
[E = Q/(4*π*R^2*ε_0] ; Electric field outside of 'sphere'
Remember,
[E = (ρ*r)/(3*ε_0)] ; electric field inside of 'sphere
So, we can replace Q by this one inside the equation but still, our dependence will be inverse R square.
So, it is the same as that for a point charge, Q, and that's why when we think of a gravitational force outside the Earth, we can condense the mass into one point and just think of the distance between the center of mass instead of doing all the integration over the entire volume of the Earth.
[W4.2-11]----------------------------------------------------------
Now, let's think about how we can describe or picture the field lines. So, we can use equi-potential surfaces and that's the geometrical description of the electrostatic field where you have a line of potential, where you have the same potential, then the direction of electric field is always tangent to the lines, that's the gradient direction.
The strength of electric field is represented by the density of lines per unit area through a surface perpendicular to the lines.
So, the more E field lines you have in a smaller area, then you have higher field.
Gauss' law states the lines should start only at plus charges and stop at minus charges, and the number which leave a charge, Q, must equal to Q over Epsilon naught and you can neither create nor annihilate the line in between.
Equipotential surface are at the right angles to the electric field lines and are spheres centered at the charge for a point of charge like this. For the field lines for a dipole, minus and plus, will look like this, and we will use a lot of this picture in the next slides or lectures.
[W4.2-12]----------------------------------------------------------
So, take a look at this and see one or two interesting thing. One is in the center plane, all of the fields are crossing in perpendicular way and the potential here is zero. But potential here, so there's another plane that has zero potential like deferents in addition to the ground and in addition to the infinity plane, it is between the dipole mode. However, that doesn't mean you don't have any field because field is the gradient, not the absolute value of your potential.
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