第八章静电场Chapter 21 potential
Chapter 21
Electric Potential
1,Electric Potential and Potential Difference
2,Relation Between Electric Potential and
Electric Field
3,The Calculation of Electric Potential
4,Equipotential Surfaces
5,Electrostatic Potential Energy
第八章静电场Chapter 21 potential
New words and Expressions
electric potential 电势
potential difference 电势差,电压
circulation theorem 环路定理
electric potential energy 电势能
equipotential surfaces 等势面
electric potential gradient 电势梯度第八章静电场Chapter 21 potential
To study the electric field from the W,we
introduce V (electric potential).
21-1&21-8 Electric Potential and
Potential Difference (p503-505,p515)
1,Features of Electrostatic Force
Coulomb’s Law for
electrostatic force,2 21
04
1
r
qqF

Newton’s Law for
gravitational force,2 21
r
mmGF?
Mathematically identical! Many propertiesfor the gravitational may also be true for
electrostatic forces.
第八章静电场Chapter 21 potential
q
1) Work done by Electrostatic Force
0q
r?
lEqW dd 0
lr
r
qq d
π 4 30
0
c o sdd lrlr rrd?
r
r
qqW d
π 4
d 2
0
0
点电荷的电场
Electric field of a point charge
l?d
rd
Ar
A
Br
B
E?
第八章静电场Chapter 21 potential
B
A
r
r r
rqqW
2
0
0 d
π 4?
0q
r?
l?d
rd
Ar
A
Br
B
E?
)11(
π 4 0
0
BA rr
qq
结果,仅与 的 始末位置 有关,与路径无关,0q
W
任意电荷的电场（视为点电荷的组合）

i
iEE

l lEqW
d
0
l ii
lEq d0
第八章静电场Chapter 21 potential
The work done by electrostatic force is path
independent.
The electrostatic force is a conservative force.
结论,静电场力做功与路径无关,
静电场是保守场第八章静电场Chapter 21 potential
2) Circulation theorem,静电场的环路定理在静电场中，电场强度沿闭合路径的线积分等于零。或电场力移动单位电荷沿闭合路径一周所做的功为 0。
The line integral of the electrostatic field around a
closed loop is zero.
E?

BABA
lEqlEq
2010
dd
0)dd(
210

ABBA
lElEq
0d
l
lE
1
2
A
B
第八章静电场Chapter 21 potential
Because electrostatic force is a conservative force,
an electric potential energy U can be introduced:
2,Electric Potential Energy 电势能 (P515):
静 电场是 保守场,静电场力是 保守力,静电场力所做的功就等于电荷 电势能增量 的 负值,
U
UU
lEqW
AB
AB
BA



)(
d
0

E?0q
A
B
第八章静电场Chapter 21 potential
To ensure U at a point,a reference potential energy
should be chosen:
,0?BU
The U of charge q at arbitrary
point a = the work done by
the during the move of
charge q from a to infinity.e
F?
ABA lEqU d0
试验电荷 在电场中某点的电势能，在数值上就等于把它从该点移到零势能处静电场力所作的功,0q
电势能的 大小 是 相对 的，电势能的 差 是 绝对 的,
E?0q
A
B
第八章静电场Chapter 21 potential
The physical quantity to represent property of
field at any point —— electric potential.
3,Electric Potential (P503):
q
UV A
A?
Electric potential at point a is the potential
energy per unit charge.
lEV
V
A
A

d
0

点
ABA lEqU d0
第八章静电场Chapter 21 potential
电势零点选择方法,有限带电体以无穷远为电势零点,
实际问题中常选择地球电势为零,Choose the potential
to be zero at an infinite distance or the ground.
AA lEV d
物理意义 把单位正试验电荷从点 移到无穷远时，静电场力所作的功,A
Descriptions:?
V is a property of an electric field,regardless of
whether a charged object has been placed in that
field; U is an energy associated with a system (the
charged object plus the charged particles that set
up the electric field).
第八章静电场Chapter 21 potential
4 Electric Potential Difference (p503电势差,电压 ):
bababa lElElEVVV ddd
The?V between any two points a and b equal
to in magnitude work done by the on unit
positive charge during the move from a to b.
eF
The value of electric potential is relative,but
the one of electric potential difference is
absolute,(电势的大小是相对的,电势差是绝对的 )
V is independent on the choice of reference
potential (电势的零点无论选在何处 电势差 不受影响 ).
第八章静电场Chapter 21 potential
)( baab VVqW
A convenient way of calculating work done to
move a charged particle q through a potential
difference?V from a to b (电场力 所做功的计算方法 ):
In any situation,charged particle moves under,
its potential energy U always tends to decreasing.
Thus a positively charged object moves naturally
from a high potential to a low potential,A negative
charge does the reverse.
eF
第八章静电场Chapter 21 potential
21-2 & 21-3 & 21-4 Calculation of Electric
Potential (P505-511)
AA lEV d
bababa lElElEVVV

ddd
第八章静电场Chapter 21 potential
q>0 produces a positive electric potential;
q<0,produces a negative electric potential.
q r
l?d
E?
rdrr
qE?
π4 20?
1,Potential Due to a Point Charge (P507-508)
令
0V

r
lr
r
q
V
d?
π4 20?
r r
rq
2
0 π4
d
r
qV
0π 4?
第八章静电场Chapter 21 potential
正电荷的场中各点电势为正，离电荷越远的地方，
电势越低。
o
V
r
0?q 正电荷沿电力线移动，从高电势到低电势，电势能降低，电场力作功。
负电荷的场中各点电势为负，离电荷越远的地方，
电势越高。
o
V
r
0?q
负电荷沿电力线移动，从高电势到低电势，电势能升高，
电场力做负功。
第八章静电场Chapter 21 potential
2,Potential Due to a Group of Point Charges
Based on the principle
of superposition of,E?
ldEV
AA

ldEEEV n
AA


)( 21
ldEldEldE n
AAA

21
nVVV21 i
n
i
V
1
1q
2q
3q
A
1r
1E
2r
3r
2E
3E

i
iEE

点电荷系第八章静电场Chapter 21 potential
例,在正方形四个顶点上各放置 +q,+q,-q,-q 四个电荷，求正方形中心 O点的电势 V。
q? q?
q? q?
o
解：
)(
4
1
0
qqqq
r


0?
r

4
1 04i i
i
r
qV
由第八章静电场Chapter 21 potential
3,Potential Due to a Continuous Charge
Distribution (p510)
r
dq
dVV
04
1

V
S
l
q
d
d
d
dw h er e
第八章静电场Chapter 21 potential
求电势的方法
r
q
V P
0π 4
d
利用
若已知在积分路径上 的函数表达式，
则
E?
lEV
V
A
A

d
0

点
（利用了点电荷电势,
这一结果已选无限远处为电势零点，即使用此公式的前提条件为 有限大 带电体且选无限远 处为电势零点,）
rqV 0π 4/
讨论第八章静电场Chapter 21 potential
Example,Potential due to a ring of Charge,A
thin circular ring of radius R carries a uniformly
distributed charge q,Determine the electric
potential at a point P on the axis of the ring a
distance x from its center,(P510)
例,正电荷 均匀分布在半径为 的细圆环上,
求 圆环 轴线上距环心为 处点 的电势,
q R
x P
+ +
+
+
+
+
+
++
++
++ +
R
x
Po
x
第八章静电场Chapter 21 potential
Solution:
将圆环分割成无限多个电荷元
+ +
+
+
+
+
+
++
++
++ +
R r
ld
x
P
R
lqlq
π 2
ddd
o
y
z
x
第八章静电场Chapter 21 potential
R
lq
r
V P
π 2
d
π 4
1d
0?
r
q
R
lq
r
V P
00 π 4π 2
d
π 4
1


22
0π 4 Rx
q
Each point on the ring is equidistant from
point P,and this distance is,So the
potential at P is:
2
1
22 )( Rx?
第八章静电场Chapter 21 potential
R
qVx
0
0 π 40,
x
qVRx
P
0π 4?
,
Discussion
R
q
0π4?
xo
V
2122
0 )(π 4 Rx
q

第八章静电场Chapter 21 potential
Example,Potential due to a charged disk,A
thin flat disk,of radius R,carries a uniformly
distributed charge Q,Determine the potential at
a point P on the axis of the disk,a distance x
from its center,(p510)
R
o x
x
P
均匀带电圆盘，半径为 R，带电为 q，求圆盘轴线上一点的电势 V。
第八章静电场Chapter 21 potential
SOLUTION：
电荷面密度 2Rq
Divide the disk into thin rings of radius r and
thickness dr.
R
o x
22 rx?
x
P
rrq dπ 2d
r
rd
第八章静电场Chapter 21 potential
讨论,当 x >> R 时，级数展开

x
RxRx
2
2
22
x
RV
22
2
0?

x
R
0
2
4

x
q
04
带电体距场点很远时，可视为点电荷。
)(
2
22
0
xRx

R
P
rx
rrV
0 22
0
dπ 2
π 4
1?
第八章静电场Chapter 21 potential
Example,Charged conducting sphere.
Determine the potential at a distance r from the
center of a uniformly charged conducting
sphere of radius R for (a) r>R,(b) r=R,(c) r<R.
The total charge on the sphere is Q,(p506)
+ ++
+ +
+
+ +
+
++
Q
R
真空中，有一带电为,半径为 的带电球壳,Q R
试求（ 1）球壳外任意点的电势； （ 2）球壳上点的电势； （ 3）球壳内任意点的电势,
第八章静电场Chapter 21 potential
+ ++
+ +
+
+ +
+
++
Q
R
SOLUTION
r
r
qERr?
π4 202?

，
01 ERr?，
ro r?
（ a） Rr?
r
Q
0π 4?
rr r
Q d
π 4 20?



r rErV o u t
d)(
2
rr
r
QV d
π4 20?

0V
Let V=0 for,r
第八章静电场Chapter 21 potential
（ c） Rr?
R rERr rErV in dd)( 21 RQ
0π 4?
rr
r
QV d
π4 20?

R
QV
Rr
0 π4?

（ b） As r approaches R,we see that
at the surface of the conductor.
The whole conductor,not just its surface,is at
the same potential,
第八章静电场Chapter 21 potential
r
QrV
0π 4
)(
外
R
QrV
0π 4
)(
内
R
Q
0π 4?
R ro
V
r
Q
0π 4?
Plots of both E and V as a function of r are
shown on next slide.
第八章静电场Chapter 21 potential
o Rq
I
II
Ro
E
r
2
04 R
q

o Rq
I
II
Ro
V
r
R
q
04
第八章静电场Chapter 21 potential
Example,Calculating potential V of
spherical shell for region 1,2,and 3.
aQ
aR
bR
bQ?
2
1
3
+
=
aQ
aR
12
3
bRbQ?
12
3
b
b
R
Q
04

r
QV a
04
,4)(
0
RrRQrV
RrrQrV
04
)(
b
b
R
Q
04

a
a
R
QV
04
Region 1:
Region 2:
r
Q b
04

r
QV a
04
Region 3:
Solution:
第八章静电场Chapter 21 potential
无限带电体电势 0 点不宜选无穷远 !
Example,Calculate potential distribution for
infinitely long rod with charges?.电直线线电荷密度为?,求电势分布 。
Solution:
r
E
02

选无穷远为电势 0 点
The electric field of rod is
0V
Let V=0 for,r
o
r
P
r
第八章静电场Chapter 21 potential
ldEV PP
P E d r
r drr
02
o
r
P
r
)ln( l n
2 0
r

无意义对无限带电体电势 0点不宜选无穷远点，也不选在导体上。
第八章静电场Chapter 21 potential
we assume B as the reference point,
o
r
B
Br
P
r
BP rr rEV d
B
r
r
r rrr
d?
π2 0?
r
r Bln
π2 0?

,Brr? 0?PV P点在 B点 左 侧,
,Brr? 0?PV
电势零点位置不同,Vp 也不同，反映了电势的相对性。
P点在 B点 右 侧,
第八章静电场Chapter 21 potential
空间 电势相等的点 连接起来所形成的面称为等势面,为了描述空间电势的分布，规定任意两 相邻 等势面间的 电势差相等,
The electric potential
can be represented
graphically by drawing
equipotential lines or,
in three dimension,
equipotential surfaces.
It is one on which all
points are at the same
potential.
21-5,Equipotential Surfaces 等势面 (P511-512)
点电荷等势面
1,Equipotential lines
第八章静电场Chapter 21 potential
2,Features of equipotential surface
在静电场中，电荷沿等势面移动时，电场力做功为零,
0d)( 00 babaab lEqVVqW
The potential difference between any two
points on the surface is zero,and no work is
required to move a charge from one point to
the other.
第八章静电场Chapter 21 potential
0d0 b
aab
lEqW

lE d
E?在静电场中，电场强度 总是与等势面垂直的，即电场线是和等势面 正交 的曲线簇,
0c o sE d lba
,0c o s 0c o sE d lba
An equipotential surface must be perpendicular
to the electric field at any point.
第八章静电场Chapter 21 potential
按规定，电场中任意两相邻等势面之间的电势差相等，即等势面的 疏密程度 同样可以表示场强的大小．
The place where the equipotential surfaces are
dense has strong electric field.
第八章静电场Chapter 21 potential
点电荷的等势面等势面 密 处电场强度 大 ；等势面疏 处电场强度 小,
Equipotential lines for a point charge
第八章静电场Chapter 21 potential
两平行带电平板的电场线和等势面
+ + + + + + + + + + + +
Equipotential lines between two oppositely charged parallel plates
第八章静电场Chapter 21 potential
一对等量异号点电荷的电场线和等势面
+
Equipotential lines for two equal but oppositely charged particles.
第八章静电场Chapter 21 potential
21-7,Relation between V and (P513-514)E?
1,Electric Potential Gradient (电势梯度 )
电场强度和电势都是描述电场中各点性质的物理量 。
电势是电场强度的线积分,反过来电场强度与电势的关系应该可以用微分形式表示出来 。 即场强等于电势的导数 。
Integral relation —— find V from,E?
零势点P lEV d
Differential relation——find from V? E?
第八章静电场Chapter 21 potential
V
VV
l
E?
lE
A
B
在电场中沿任意方向相距很近的两点 A,B，从 A到 B的微小矢量为 dl，
co s lE
lEVVU ABAB



）（
lEEco s
l
VElEV
ll?
,
l
V
l
VE
ll d
dl i m
0


电场中某一点的 电场强度 沿 某一方向的分量,等于这一点的电势沿该方向单位长度上 电势变化率 的 负 值,
第八章静电场Chapter 21 potential
l
V
l
VE
ll d
dl i m
0


The component of electric field in any direction
is equal to the negative of the rate of change of
electric potential with distance in that direction.
The quantity is called the gradient of V in a
particular direction,
l
V
d
d
第八章静电场Chapter 21 potential
V
VV
E?
l?d
高电势低电势
ne
e
nld
当 时，即 l沿着 E
的方向 n时，变化率有最大值，这时
0
m a xdl
dVE
说明过电场中任意一点,
沿某一方向其电势随距离的变化率最大,此最大值称为该点的 电势梯度 。
If the direction is not specified,the term gradient
refers to that direction in which V changes most
rapidly,This would be the direction of E.
第八章静电场Chapter 21 potential
x
VE
x?

y
VE
y?

z
VE
z?

l
VE
l?

由
)( kji

z
V
y
V
x
VE



在直角坐标系中,(in the right-coordinate) 电场强度沿 3个坐标轴的分量可用电势表示。
Partial derivative of V respect to x,y and z.
第八章静电场Chapter 21 potential
g r a d
★ The operator of gradient in the right-coordinate
直角坐标下 梯度算符,
k
z
j
y
i
x

)()()(



Vk
z
Vj
y
Vi
x
VE


)(
“–”表示 E 的方向为电势降的方向。
is the,partial derivative” of V with respect
to x,with y and z hold constant.x?
第八章静电场Chapter 21 potential
VkzVjyVixVE g r a d) （
VE （ 电势梯度 ）
直角坐标系中为求电场强度 提供了一种新的途径
E?
求 的三种方法
E?
利用电场强度叠加原理利用高斯定理利用电势与电场强度的关系物理意义
（ 1） 空间某点电场强度的大小取决于该点领域内电势 的空间变化率,V
（ 2） 电场强度的方向恒指向电势降落的方向,
讨论第八章静电场Chapter 21 potential
Solution:
x
q
y
x
z
o
R r
lq dd
P?
E?
2122
0 )(π 4 Rx
qV
VE
Example,E for ring and disk,A thin ring of
radius R with a uniform +l around its
circumference,What is the?
pE
x
VEE
x?

第八章静电场Chapter 21 potential
x
q
y
x
z
o
R r
lq dd
P?
E?
2322
0 )(π 4 Rx
qx

2122
0 )(π 4 Rx
q
x?
0 zy EE
It is the the same as what we got before!
第八章静电场Chapter 21 potential
Solution:
Example,A conducting sphere,of radius R and
charge +q,is concentric with a spherical
conducting shell of inner radius R1 and outer
radius R2,This shell has a net charge of Q,(a)
What are the charges on the inner and outer
surfaces of the shell? (b) The potential of the
sphere and the shell?
(a) Qin = -q,(by using G.L)
Spherical symmetry,q uniformly
distributed at inner surfaces o R
q
1R
Q
2R
A
B
qQQ o u t
第八章静电场Chapter 21 potential
(b) Potential of the shell B:
204 R
qQV
B

2010 4
)
11
(
4 R
qQ
RR
q
VVV BABA



Potential of the sphere A:
Question? If A and B are connected by a long
metal wire,what effects will it result?
o
R
q
1R
Q
2R
A
B
第八章静电场Chapter 21 potential
Homework,
P519,11,15,16,17,19,23,30,58