Study the properties and laws of electric field, magnetic field and electromagnetic field that they are stimulated by charges and currents. Volume 2 Electromagnetism

Chapter 8 Electrostatic Field in Vacuum stimulated by static charges with respect to observer. Inertial reference frame

§8-1 Coulomb’s Law 库仑定律 §8-2 The Electric Field 电场 电场强度 §8-4 Gauss’ Law 高斯定理 §8-5 Electric Potential 电势 §8-3 Electric Field Line and Flux 电力线 电通量 §8-6 Equipotential Surface and Potential Gradient 等势面 电势梯度 §8-7 The Electric Force Exerted on a Moving Particle 运动带电粒子所受电场力

§8-1 Coulomb’s Law 1. Two kinds of electric charges positive charge Like charges repel each other Negative charge Unlike charges attract each other

2. quantization of charge Electron is the smallest negative charge in nature. experiments show ： e= × C(Coulomb) q=  Ne Proton is the smallest positive charge in nature. The magnitude of electric charge possessed by a body is not continuous. integer

 Electrification by rubbing 3. Conservation of charge Positive charges and negative charges have same magnitude.

 Electrification by induction ： inducible charges have same magnitude.

The conservation of charge ： in any interaction the net algebraic amount of electric charge remains constant.

or  Coulomb’s Law ： 4. Coulomb’s Law  Point charge ： The size of charged bodies << their distance

In SI ：  0 =8.85  C 2 /N  m permittivity of vacuum （真空介电系数）

5. Superposition principle of electrostatic forces Assume there are many point charges in space:q 0 、 q 1 、 q 2 、 q 3 … q n ， the resultant force acting on q 0 : ---vector addition

§8 -2 The Electric Field  Viewpoint of action-at-a distance:  Viewpoint of field: 1.Viewpoints of the interaction about electric charges charge charge charge field charge Field is a kind of matter.  

The behavior of electric field as a kind of matter:  force ： E-field exerts a force on the charges in it.  work ： E-field does work on charges during the charges move in it.  induction and polarization ： in the field, conductor and dielectric produce induction and polarization.

2. Electric field Test charge:  small size--point charge  small charge magnitude—no influence for original field. Test results:  same q 0 is put on different points in space,

 Put different test charges on same point, the electric forces that the test charges suffer change. the direction and the magnitude of the force that q 0 suffers is different at different points ---E-field is different at different points the ratio =constant vector at same point.

 the electric field is defined ： SI unit ：牛顿 / 库仑 (N/C) or 伏特 / 米 (V/m)

3. the superposition principle of electric field There are q 1 、 q 2 、 q 3 … q n in space, q 0 is put on the point P, the force acting on q 0 : At point P, the E-field is set up by q 1 、 q 2 、 q 3 … q n :

.point charge: Put q o on point P ， using Coulomb Law ， q o suffers --the E-field of a point charge 4. The distributions of electric field about several different charged bodies

The distribution is spherical symmetry

There are q 1 ， q 2 ， …, q n in space Each charge set up its field at point P ： .the point charge system P

the total field at P ：

.A continuously distributed charged body At point P, element charge dq produces : The total field at P produced by entire charged body:

 According to the distribution of charge, dq is written as follow:  for Cartesian coordinate system ： line distribution area distribution volume distribution

5. Examples of calculating E-field Steps ：  divide charged body into many small charge elements.  write out produced by dq at point P  set up a coordinate system, write components of, such as

 total E-field  calculate the components of, such as

[Example] Calculate the E-field at point P produced by a charged line. P d lqlq 11 22 L 、 q 、 d 、  1 、  2 are known

Solution . divide q  dq . Any dq produces at P P d lqlq 11 22 dq r The magnitude of : The direction of shows in Fig.

x y o . set up Cartesian coordinate, . calculate E x 、 E y P d lqlq 11 22 dq r 

In the figure ： x y o P d lqlq 11 22 dq r 

Same as . the total

（ 1 ） If P locates on the mid-perpendicular plane of the line, i.e. Discussion

（ 2 ） If P is very close to the line --the length of the charged line tends to infinity 、 E-field distribution of the infinite line with uniform charge

（ 3 ） If P is far away from the line The charged line can be regarded as a point charge.

Question ： If P locates on the elongating line of the charged line shown as in figure, How do we calculate E x 、 E y ? l a P q

Caution ！ (1) If the charged body is not a point charge, we can not use the formula (2) If the directions of for different are not same, we can not integrate directly. directly, use only for point charge. integrating its components

[Example] Find the E-field of an uniform charged ring on its axis. （ q 、 R 、 x are known) x x r  P q R ·

Solution Divide q dq direction x x r  P q R · dq

Direction: along x axis

Discussion （ 1 ） at x=0 ， E=0. When x   ，  E has extreme values on x axis. let We get （ 2 ） when x>>R ， can be regarded as a point charge.

E x 0

[Example] thin round plate with uniform charge area density , radius R. find its field on the axis.

（ 1 ） when x « R, discussion The E-field set up by uniform sheet charge of a infinite plane. （ 2 ） when x » R, as a point charge

§8-3 Electric field line and flux 1. E-field line （ line ） --area element perpendicular to line. d  e – the number of line crossing.  the tangential direction of line at any point gives the direction of at that point.  the density of line gives the magnitude of

 line originate on positive charges and terminate on negative charges (or go on infinity). They never originate or terminate on a no-charge point in finite space. 2. Electric flux The properties of line.  Two lines never intersect at a point. – the number of line crossing any area.

  The plane S is at right angle to the uniform E-field.  The plane S is at any angle with :the unit vector at the normal direction of the plane.

The total E-flux crossing S ： Take any dS on S : An arbitrary surface S is placed in a no-uniform E-field.

E d  e < 0 If S is a closed surface ： d  e >0 Stipulation ： the direction of is outward. n

relate to the charge in the surface. If there is no any charge in the closed surface, the number of line entering it equals the number of line going out it. If there are charges in the closed surface, the number of line entering it does not equal the number of line going out it.

1. The E-flux crossing a sphere surface with a point charge q in its center. q S §8-4 Gauss’s Law of electrostatic field

q S Discussion   e does not relate with r. S’  If S’is an arbitrary closed surface surrounding q, then

2. If the charge is outside the closed surface S, + q S  e = 0

3.  e crossing any closed surface S with point charge system. Inside S : Outside S: Then S

q 4.  e crossing any closed surface S with any charged body S Inside S

Summary above results Gauss’Law In any electrostatic field, the electric flux crossing any closed surface equals the algebraic sum of the charges enclosed by the surface divided by The closed surface S –Gaussian surface or S内S内

Notes: Œ is the algebraic sum of the charges enclosed by the Gaussian surface i.e., the  e crossing S depends on the charges enclosed by S only, and has nothing to do with the charges outside S.  the total at any point of S is concerned with all the charges (inside and outside S).  indicates that the electrostatic field is a 有源场

If the distribution of charges and its E-field has some symmetry, Gauss’s Law can be used to calculate the E-field. Common steps:  Analyze the symmetry of charges and its E-field.  Choose a suitable enclosed surface as Gaussian surface S.  Calculate  e crossing S.  Calculate the algebraic sum of charges inside S.  Use Gauss’s Law to calculate E 。 4. Applications of Gauss’s Law

[Example 1] Calculate the E-field distribution of a infinite line with uniform charges. (Assume the linear density of charges is ) Solution ：  Analyze the symmetry --axial symmetry λ  Choose a suitable Gaussian surface --cylinder surface

 Calculate  e crossing S ：  Calculate the algebraic sum of charges inside S  Use Gauss’s Law ：

E-field distribution of the infinite line with uniform charge

Similar problem: the charge distribution on a infinite cylinder surface with radius R. The charges per meter of length of the cylinder isλ.  axial symmetry  Gaussian surface – cylinder surface （1）（1） r R >

下 上 侧  ： ：  use G-surface 

（2）（2） r  R （ r > R ） Distribution of E-field of the cylinder charges <

[Example 2] Calculate the E-field distribution of a infinite plane with uniform charges. (Assume the area density of charges is σ) Solution: Analyze the distribution character of E-field P  E-field :area symmetry

σ SS σ. P  Make a cylinder surface through point P as Gaussian surface. G-surface

左 右 侧  Use ：： 

[Example 3] Calculate the E-field distribution of a sphere surface with uniform charge q. (1) r > R  spherical symmetry  Make a sphere as Gaussian surface through P. R q r PP G-surface 

  use

(2) r < R G-surface  E-field—spherical symmetry  Gaussian surface--sphere P.P. r R q  

 use The distribution of E-field for charged spherical surface: 0 （ r>R ） （ ） r < R 1 2 r r E 0 R

R q （1）（1） < r R r G-sueface  Similar question ： uniform charged spherical body

（2）（2） r R > G-surface R r. P

The distribution of E-field with charged spherical body: （ r > R ） < r R E o