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24장 읽고 있자 ……
24. 전기퍼텐셜 (Electric potential)
• Conservative Forces and Energy Conservation– Total energy is constant and is sum of kinetic and potential
• Electric Potential Energy (U)
• Electric Potential (V)
– A property of the space and sources as is the Electric Field– Potential differences drive all biological & chemical reactions,
as well as all electric circuits
• Path-independence of Potential
• (계산) Electric Potentials: : electric dipole, N point charges, continuous charges
cFU ↔
qEV ↔
지난 시간에 …
Q
Electric Flux ∫ ⋅≡Φ AdEr
Gauss 법칙o
enclqAdEε
=Φ≡⋅∫rr
enclo q=ΦεOR,s
2204
1rQk
rQE e=
πε=⇒
3304 a
QrkarQE e=
πε=⇒
복습 : 일 (Work) 이란 ? • Work done by the force given by:
Positive: Force is in direction movedNegative: Force is opposite direction movedZero: Force is perpendicular to direction moved
• Careful ask WHAT is doing work!
Opposite sign for work done by you!: 즉, 계(system)가한일은외부에서해준일과부호가반대!: ΔW = - ΔWapp
• Conservative Forces
Δ Potential Energy (U) = - Wconservative
θcosFddFW =•=rr
Conservation of Energy of a particle
• Kinetic Energy (K)– non-relativistic
• Potential Energy (U)– determined by force law
• for Conservative Forces: K+U is constant– total energy is always constant
• examples of conservative forces– gravity; gravitational potential energy – springs; coiled spring energy (Hooke’s Law): U(x)=kx2/2– electric; electric potential energy
• examples of non-conservative forces (heat)– friction– viscous damping (terminal velocity)
221 mvK =
),,( zyxU
전기력도 보존력이다 !• Consider a charged particle traveling
through a region of static electric field:
• A negative charge is attracted to the fixed positive charge
• negative charge has more potential energy and less kinetic energy far from the fixed positive charge, and…
• more kinetic energy and less potential energy near the fixed positive charge.
• But, the total energy is conserved
+
-
• 다음에 대해 알아보자
electric potential energy (U)electrostatic potential (V)
• A direct calculation of the work done to move a positive chargefrom point A to point B is not easy.
• Neither the magnitude nor the direction of the field is constant along the straight line from A to B.
• But, you DO NOT have to do the direct calculation.
• Remember: potential difference is INDEPENDENT OF THE PATH!!
• Therefore we can take any path we wish.
A positive charge Q is moved from A to Balong the path shown by the arrow. What is the sign of the work done to move the charge from A to B? A B
(a) WAB < 0 (b) WAB = 0 (c) WAB > 0
0=• ldErrChoose a path along the arc of a circle centered at
the charge. Along this path W = 0 at every point!!
질문
Electric potential energy• Imagine two positive charges, one with charge Q1,
the other with charge Q2:
• Initially the charges are very far apart, so we say that the initial energy Ui of interaction is zero
(we are free to define the energy zero somewhere).
• If we want to push the particles together, this will require work (since they want to repel).
the final energy Uf of the system will increase by the same amount: ΔU = Uf – Ui = ΔWapp
Q2Q1
Q1Q2
Electric potential energy• The initial energy Ui (∞) of interaction is zero
• The final energy Uf (r) of the system will increase by the same amount of the work applied:
ΔU = Uf – Ui ≡ U = ΔWapp
ΔWapp = - ΔWcoulomb= - ΔW∞ ≡ - W
즉, 외부에서해준일과두입자간작용하는힘이한일은
부호가반대 !!!
U = - W
Q2Q1r
확인문제 1, 2, 3
1. (a) 전기장이양성자에한일은 +, - ??
(b) 양성자의퍼텐셜에너지는증가, 감소 ??
2. (a) 힘(f)이양성자에한일은 +, - ??
(b) 양성자는더높은 (낮은) 퍼텐셜쪽으로움직이는가 ??
음의 일
증가
양의 일
높은 쪽으로
Electric potential energy (U)m]N[J ⋅=•−=•−=−= ∫∫ ∞∞
rr
q sdEqsdFWU rrrr
Electric potential (V) : 단위전하당 U
J/C] [V(volt) =•−=−=≡ ∫∞
rsdE
qW
qUV rr
Qr q q
즉, 전하 q가 무한대에서 r 위치까지 이동하는 동안 전기력 (전기장)이 한 일
즉, 단위전하가 무한대에서 r 위치까지 이동하는 동안 전기력 (전기장)이 한 일
F (E)
Q f
전위차 (electric potential difference : ΔV)
( ) ( )
J/C] [V(volt) =•−=−=
−−−=Δ
∫∞∞
f
iif
if
sdEVV
VVVVVrr
두 지점(f, i) 간의 전기 퍼텐셜차 (전위차 : DV)
i
★전자볼트 (electron volt, eV) (아주 작은 에너지의 단위)
≡전자하나의전하량( e ≡ | q전자|)을 1 volt 전위차옮기는데드는에너지
JC -19-19 10 1.60 J/C) 1()10 (1.60 V) (1 e eV 1 ×=××==
★전기장의 또 다른 차원
[ ] [ ] [ ]V/m(J/C)/m(J/m)/C[N/C] [E] ==⇒=
등퍼텐셜면 (등전위면 : Equipotential surface)
WI = 0WII = 0WIII = WIV
등퍼텐셜면 ≡ 전위가 같은 점들로 이루어진 면
각 이동 경로에서전기장이 한 일 ?
등퍼텐셜면은 항상 전기장에 수직 (왜?)
확인문제 3
3. (a) 등퍼텐셜면에대한전기장의방향은?
(b) 각경로에대해한일은 +, - ?
(c) 한일이큰순서대로
------
1, 2, 3, 5 : + , 4 : -
3, 1=2=5, 4
24-6. 점 전하가 만드는 퍼텐셜
rqE
oπε41
=
점전하 q에서임의의 r 만큼떨어진위치에서의전기장
점전하 q에서 R 만큼떨어진위치에서의퍼텐셜 (V)
rqds
ssdErV
rr
o2
o 411
4q- )(
πεπε==•−= ∫∫ ∞∞
rr
rqrV
o41)(πε
=
J/C] [V(volt) =•−=−=≡ ∫∞
rsdE
qW
qUV rr
24-7. 점전하무리가만드는퍼텐셜
∑∑==
==N
n n
N
nn r
qVV1o1 4
1πε
확인문제 4 : P 점에서의알짜전기퍼텐셜의크기순서는?
a = b = c
• The potential energy of an added test charge q0 at point P is just
0
31 20
1 2 3of q at P
p p p
QQ QU q k k kr r r
⎛ ⎞= + +⎜ ⎟⎜ ⎟
⎝ ⎠
Q1
Q2Q3
q0
r1p
r2pr3p
Q1
Q2Q3
P
질문
Which of the following charge distributions produces V (x) = 0 for all points on the x-axis?
(a)
x
+2μC
-2μC
+1μC
-1μC(b)
x
+2μC +1μC
-2μC-1μC (c)
x
+2μC -2μC
+1μC-1μC
The key here is to realize that to calculate the total potential at a point, we must only make an ALGEBRAIC sum of the individual contributions.
Therefore, to make V(x)=0 for all x, we must have the +Q and -Qcontributions cancel, which means that any point on the x-axis must be equidistant from +2μC and -2μC and also from +1μC and -1μC.
This condition is met only in case (a)!
24-8. 전기 쌍극자가 만드는 퍼텐셜
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+==
−+
+−
−+=∑
)()(
)()(
o)()(o
2
1 4q
41
rrrr
rq
rqVV
nn πεπε
P
r >> d )( , cos4
1
cos4
q
cos
2o
2o
2)()(
)()(
qdpr
pr
dV
rrr
drr
=⎟⎠⎞
⎜⎝⎛=
⎟⎠⎞
⎜⎝⎛≈
≈
≈−
−+
+−
θπε
θπε
θ
* 유도 쌍극자 모멘트 (induced dipole moment)
원자 (분자)
E 에의해분극이일어남원자 (분자)에서의유도쌍극자모멘트발생
(참고 : 물질의 굴절률 근원)
E
24-9. 연속 전하분포가 만드는 퍼텐셜
rdqdV
oπε41
= ∫=⇒r
dqVoπε4
1
⎥⎥⎦
⎤
⎢⎢⎣
⎡ ++=
+=
+==
=
∫ ddLL
dxdxV
dxdx
rdqdV
dxdq
o
L
o
oo
22
0 22
22
ln44
41
41
πελ
πελ
λπεπε
λ
L
[ ]zRzRz
dRRV
RzdRR
rdqdV
dRRdq
o
R
o
oo
−+=+
=
+==
=
∫ 22
0 22
22
2'''
2
'''2
41
41
''2
εσ
εσ
πσπεπε
πσ
선 전하
원판 전하
24-10. 퍼텐셜에서 전기장 구하기
sdEVVf
iifrr
⋅−=− ∫sdEdV rr
⋅−=
, , ,
ˆˆˆ
zVE
yVE
xVE
zzVy
yVx
xVV
sddVE
zyx ∂∂
−=∂∂
−=∂∂
−=⇒
⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+∂∂
−=∇−≡−=r
rr
If the E-field has only one component, Ex,
x̂dxdVx̂EE x −==
r
If the charge distribution has spherical symmetry, the E-field is radial, Er,
r̂drdVr̂EE r −==
r
예제 : 쌍극자의 전기퍼텐셜과 전기장
-q +q2a •
x
P
22
2ax
qakax
qax
qkrqkV e
ei
ie −
=⎟⎠⎞
⎜⎝⎛
+−
−== ∑
If x >> a , 2
2x
qakV e≈
( )222
22ax
xqakkdxdVE e
ex−
⋅=−=
3
4x
qakdxdVE e
x ≈−=
24-11. 점 전하계의 전기 퍼텐셜 에너지
• 고정된 점전하계의 전기적 퍼텐셜(위치) 에너지
≡ 주어진 전하분포가 되게 하는데 드는 에너지 (외부에서 해야 하는 일)
rqqVqWU
oapp
212 4
1πε
===
보기문제 24-6
그림과 같은 세 점 전하의 전기적 위치 에너지?
24-12. 고립된 대전 도체의 퍼텐셜
Qr̂rQkE e 2=
r( )Rr >
0= ( )Rr <
rQk
rdEsdErV
e
r
r
=
⋅=⋅−= ∫ ∫∞
∞ rrrr)(
RQk
rdErdEsdErV
e
R
r R
r
=
⋅+⋅=⋅−= ∫∫ ∫∞
∞
rrrrrr)(
( )Rr >
( )Rr <
0
24. Summary
Electric potential energy (U)
m]N[J ⋅=•−=•−=−= ∫∫ ∞∞
rr
q sdEqsdFWU rrrr
Electric potential (V) : 단위전하당 U
J/C] [V(volt) =•−=−=≡ ∫∞
rsdE
qW
qUV rr
∑∑==
==N
n n
N
nn r
qVV1o1 4
1πε
∫=r
dqVoπε4
1
점 전하
연속 전하
⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+∂∂
−=∇−≡−= zzVy
yVx
xVV
sddVE ˆˆˆ
rr
rV E