23. 가우스 법칙 (Gauss’ law)

• Gauss’ Law: Motivation & Definition

• Coulomb’s Law as a consequence of Gauss’ Law

• Charges on Conductors:– Where are they?

• Applications of Gauss’ Law– Uniform Charged Sphere – Infinite Line of Charge– Infinite Sheet of Charge– Two infinite sheets of charge

o

enclqAdEε

=Φ≡⋅∫rr

지난 시간에 …

2o4

1 rq

q

FE

o πε== nEEEE +++= L21

dqp =전기 쌍극자 (Electric dipole)

전기장 (Electric field)

r̂rdVkE e ∫ρ

= 2

rr̂

rdSkE e ∫

σ= 2

rr̂

rdlkE e ∫

λ= 2

r

연속전하의 전기장

Ep×=τ EpU ⋅−=

Fundamental Lawof Electrostatics (정전기학)

• Coulomb’s LawForce between two point charges

OR

• Gauss’ LawRelationship between Electric Fields and charges

Eqrq

πεqF

o02

0

4 ==

o

enclqAdEε

=Φ≡⋅∫rr

23-2. 다발 (flux)

다발 (flux)

Φ = (면에수직인속도성분)×(면의넓이)

( ) AvAvrr

•=≡Φ θcos

• Consider flux through two surfacesthat “intercept different numbers of field lines”

case 1

case 2

θ

case 1

case 2

yEE o ˆ=r

yEE o ˆ=r

2w

E-field surface area E A

2wEo

θcos2 ⋅wEo

Case 2 is smaller!

Flux:

2w

22-3. 전기장 다발 (Electric flux)

22-3. 전기장 다발 (Electric flux)

Gauss 폐 곡면(Gaussian surface)

Electric Flux•What does this new quantity mean?

• The integral is over a CLOSED SURFACE• Since is a SCALAR product, the electric flux is a SCALAR

quantity• The integration vector is normal to the surface and points OUT

of the surface. is interpreted as the component of E which is NORMAL to the SURFACE

• Therefore, the electric flux through a closed surface is the sum of the normal components of the electric field all over the surface.

• The sign matters!!Pay attention to the direction of the normal component as it penetrates the surface… is it “out of” or “into” the surface?

• “Out of” is “+” “into” is “-”

SdErr

•

SdErr

•Sdr

AdErr

•

Adr

∫ ⋅≡Φ AdEr

AdErr

•

dS dS

1 2

A positive charge is contained inside a spherical shell. How does the electric flux dФE through the surface element dS change when the charge is moved from position 1 to position 2?

a) dФE increases

b) dФE decreases

c) dФE doesn’t change

질문

dS dS

1 2

A positive charge is contained inside a spherical shell. How does the flux ФE through the entire surface change when the charge is moved from position 1 to position 2?

a) ФE increases

b) ФE decreases

c) ФE doesn’t change

R

2R

• Consider 2 spheres (of radius R and 2R) drawn around a single charge as shown.– Which of the following statements

about the net electric flux through the 2 surfaces (Φ2R and ΦR) is true?

(a) ΦR < Φ2R (b) ΦR = Φ2R (c) ΦR > Φ2R

• Look at the lines going out through each circle -- each circle has the same number of lines.

• The electric field is different at the two surfaces, because E is proportional to 1 / r 2, but the surface areas are also different. The surface area of a sphere is proportional to r 2.

• Since flux = , the r 2 and 1/r 2 terms will cancel, and the two

circles have the same flux!

∫ •S

AdErr

질문

A cube is placed in a uniform electric field. Find the flux through the bottom surface of the cube.

a) Фbottom < 0b) Фbottom = 0c) Фbottom > 0

질문

Lecture 3, ACT 2Imagine a cube of side a positioned in a region of constant electric field as shown.

Which of the following statements about the net electric flux ΦE through the surface of this cube is true?

(a) ΦE = 0 (b) ΦE ∝ 2a2 (c) ΦE∝ 6a2

a

a

질문

• The electric flux through the surface is defined by:

∫ • SdErr

• on the bottom face is negative. (dS is out; E is in)∫ • SdE

rr• on the top face is positive. (dS is out; E is out)

• Therefore, the total flux through the cube is:

2 20 0E dA Ea Eatopsides bottom•∫Φ ≡ =Φ +Φ +Φ = − + =rr

• is ZERO on the four sides that are parallel to the electric field. ∫ ⋅≡Φ AdEr

∫ ⋅ AdEr

∫ ⋅ AdEr

∫ ⋅ AdEr

보기문제 23-1고른 전기장 E 속에 놓인 반지름 R 인 원통꼴의 가우스 곡면에서의 전기장플럭스 Φ의 값?

보기문제 23-2

그림과 같은 정육면체의 왼쪽, 오른쪽, 위쪽 면에서의전기장 플럭스? 단, 전기장은

E = ( 3.0 x) i + ( 4.0) j (N/C).1) 왼쪽면

2) 오른쪽면

3) 위쪽면

23-4. Gauss 법칙

The net electric flux through any closed surface is proportional to the charge enclosed by that surface.

It is very useful in finding Ewhen the physical situation exhibits massive SYMMETRY.

o

enclqAdEε

=Φ≡⋅∫rr

enclo q=ΦεOR,

23-5. Gauss 법칙과 Coulomb 법칙

가우스 법칙에서 쿨롱 법칙 끌어내기

점전하 q 가원점에있을때반지름 r 인공표면에서의전기장?

구대칭성때문에공표면어디에서나전기장의크기는같고, 방향은밖으로나아가는반지름방향

o

enclqAdEε

=⋅=Φ ∫rr

Gauss 폐곡면의 선택

S1

S2

∫∫ ⋅=⋅=Φ21 SS

C AdEAdErrrr

qkrrqk ee π=π⋅= 44 2

2

041πε

=ek

0ε=Φ

qC

For any empirical surface, ΦC is the same. i.e. ΦC is independent of the surface contour.

23-6 전하를 띤 고립된 도체

고립된 도체에 들어 있는 잉여전하 (excess charge)는

도체의 표면에 퍼져 있게 된다.

◂ 서로 밀어내는 정전기력

◂ 도체 속에서는 전기장이 E = 0 (왜?)

E = 0 E = 0

보기문제 23-4속빈쇠공: 속반지름 R, 전기적으로중성점전하: - 5.0 μC, 중심에서 R/2인곳1) 쇠공의 안팎의 면에 쌓인 전하량?

2) 안팎의 면의 전하분포?

1) 쇠공의안쪽면의전하량:쇠속은전기장이 0⇒쇠속의가우스면에서도전기장이 0⇒가우스면전체의플럭스도 0⇒가우스면속의알속전하는 0⇒쇠공안쪽면에 +5.0 μC 전하분포

2) 쇠공의바깥면의전하량:쇠공은전기적으로중성, 전하는보존⇒쇠공바깥면에 - 5.0 μC전하분포

▸ 전하분포는 공 안쪽은 고르지 않으나 바깥쪽은 고름

23-7 Gauss 법칙의 적용: 선전하 - 원통대칭

o

enclqAdEε

=⋅=Φ ∫rr

(면부터의 거리에 무관하게 일정)

23-8 Gauss 법칙의 적용: 면전하 - 원통대칭

o

enclqAdEε

=⋅=Φ ∫rr

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−=

+== ∫

22

0 2322

12

24

Rzz

drrzrzEE

o

R

oz

εσ

εσ

If R infinite,

o

Eεσ

2≈

Coulomb 법칙

두 도체판• Field outside must be zero.

Two ways to see:

– Superposition

– Gaussian surface encloses zero charge

E=0 E=0σ

+

+

++

++

+

σ

+

++

-

-

--

-

--

-

-

-+

+

---

E

A

AQ σ=

insideoutside AEAESdE +=•∫rr

0εσ

=E

• Field inside is NOT zero:

– Superposition

– Gaussian surface encloses non-zero charge

AA

23-9 Gauss 법칙의 적용: 구 껍질 - 구대칭

구 껍질

o

enclqAdEε

=⋅=Φ ∫rr

q(S2)

(S1)

구대칭 전하분포

Charge density ρ : Qa =ρπ 334

343

aQπ

=ρ⇒

(a) r >a00 ε

=ε

=⋅=Φ ∫QqAdE inc

C

rr

0

24ε

=⋅πQEr

2204

1rQk

rQE e=

πε=⇒ for r > a

(b) r <a 0ε

=⋅=Φ ∫ incC

qAdErr

3

0

0

34

1

r

dV

περ

=

⋅ρε

= ∫

⎟⎠⎞

⎜⎝⎛

π=ρ 34

3aQ

3

3

0

24arQEr

ε=π 33

04 aQrk

arQE e=

πε=⇒ for r < a

Q

Q

(a) r >a

(b) r <a

2204

1rQk

rQE e=

πε=⇒

for r > a

3304 a

QrkarQE e=

πε=⇒

Q

for r < a

23. Summary

Electric Flux ∫ ⋅≡Φ AdEr

Gauss 법칙o

enclqAdEε

=Φ≡⋅∫rr

enclo q=ΦεOR,

S1

S2

s