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1、Chapter 3 Basic Principle of Electrostatics 静电学的基本原理 Chapter 3 Basic Principle of Electrostatics 静电学的基本原理静电学的基本原理 Electrostatics 静电学 is the science of the interactions 相互作用 between electric charges and the electric field in static systems 静止系统 i e systems that do not change in time The theory of ele
2、ctrostatics is important for late understanding of the whole electromagnetism Quasistatic 似稳 approximation 近似 to use the equations of electrostatics for systems that change slowly in time 3 1 Coulomb s Law库仑定律库仑定律 All of electrostatics can be derived 导出 from two principles Coulomb s law and the prin
3、ciple of superposition 叠加原理 Coulomb s law the force between two point like charges 点电荷 is inversely proportional to the square of the distance 与距离的平方成反比 proportional to the product of the charges 与电荷之积成正比 and in the direction of the line joining the charges 沿连线方向 The force on a charge q1 due to a ch
4、arge q2 is F1 K q1q2 r2 r 3 1 where r is the distance between the charges and r is the unit vector pointing from q2 to q1 r r r r r For q1 and q2 with the same sign q1 和q2同号 the force 25 3 1 Coulomb s Law库仑定律 is repulsive 排斥 in the direction of r for charges of opposite signs q1 和 q2异号 the force is
5、attractive 吸引 Another notation 记号 F1 K q1q2 x1 x2 x1 x2 3 where x1 x2 r Figure 3 1 IN SI units 国标单位制 K is defined to be K 1 4 0 10 7c2 Ns2 C2 8 99 109 Nm2 C2 3 3 where C is the unit of charge coulomb 1 C 1 A s 3 1 1 The Superposition Principle 叠加原理叠加原理 The force on a charge q due to a set of charges
6、 q1 q2 q3 qN is the vector sum of the individual Coulomb forces Fq k 1 N qqk 4 0 x xk x xk 3 3 4 3 2 The Electric Field电场电场 3 2 1 Definition 定义定义 The electric field E x is a vector valued function of position 位置的矢量函数 The definition of E x is the force per unit charge that would be exerted 施加 on a sm
7、all test charge 试验电荷 q if it were located at x in the limit q 0 E x lim q 0 F q 3 5 The electric field exerts a force on a charged particle q is Fq qE x 3 6 26 Chapter 3 Basic Principle of Electrostatics 静电学的基本原理 3 2 2 Charge as the source of E 电荷作为场的源电荷作为场的源 The field at x created by a set of stati
8、c point charges q1 q2 q3 qN at xk is E x k 1 N qk 4 0 x xk x xk 3 3 7 The field due to a single point charge q at position xq is E x q 4 0 r r 2 3 8 wherer x xq Example Example 1 Six identical charges one at each vertex 顶点 of a regular hexagon 正六边形 in the x y plane centered at the origin Figure 3 2
9、1 What is the electric field at any point x in the x y plane The position of the kth charge is x k iacos k 3 jasin k 3 The field for arbitrary x is 27 3 2 The Electric Field 电场 E x q 4 0 k 1 6 x acosk 3 i y asink 3 j x acosk 3 2 y asink 3 2 3 2 2 What is the field on the x axis The field on the x ax
10、is Ex x 0 q 4 0 k 1 6 x acosk 3 x2 2axcosk 3 a2 3 2 Ey x 0 qa 4 0 k 1 6 sink 3 x2 2axcosk 3 a2 3 2 0 At the origin E 0 3 What is the field on the x axis for from the origin accurate through order x 4 For x a Ex x 0 1 4 0 6q x2 9qa 2 2x4 4 If the 6th charge is removed 移走 from the group what is the fi
11、eld at the origin for the remaining 5 charges The field at the origin due to the remaining 5 charges is the same as the field of 6 charges superposed of a charge q at a 0 E5 q i 4 0a2 3 2 3 Field of a Charge Continuum 连续电荷的场连续电荷的场 Macroscopic continuous charge distribution 宏观连续电荷分布 in any macroscopi
12、c 宏观的 small volume V there is large number of elementary particles 基本粒子 So the charge distribution can be taken as continuous Volume charge density 电荷密度 x is a function of position x The charge in a small volume dV at x is x dV In Cartesian coordinates dV dxdydz d 3 x E x 1 4 0 x x x x 3 x d 3 x 3 1
13、5 28 Chapter 3 Basic Principle of Electrostatics 静电学的基本原理 or E 1 4 0 rdq r 2 3 16 where r x x Examples Example 2 The electric field on the midplane 中垂面 of a uniformly charged thin wire 细导线 of length 2l The charge per unit length is At z on the charged wire for a line element dz dq dz The distance fr
14、om 0 0 z to x 0 0 is x2 z 2 So the field at x 0 0 due to dq is d E x 0 0 1 4 0 dz x2 z 2 The angle between dE and the x axis is cos x x2 z 2 So Ex x 0 0 1 4 0 l lx dz x2 z 2 3 2 x 4 0 z x2 x2 z 2 1 2 l l That is Ex x 0 0 l 2 0 x x2 l2 3 17 Ey x 0 0 0 On the x y plane Let r x i y j 29 3 2 The Electri
15、c Field 电场 E r l r 2 0r r2 l2 3 18 If the line is infinitely long E r r 2 0r 3 19 Example 3 The electric field on the axis of a circular loop of uniformly charged 均匀带电 thin wire with total charge q For a line element dl ad on the loop dq dl ad The field at 0 0 z due to dq is dE 0 0 z 1 4 0 ad a2 z2
16、The cosine of the angle between dE and z axis is cos z a 2 z2 So dE 0 0 z z 1 4 0 az d a2 z 2 3 2 E 0 0 z z 1 4 0 0 2 az d a2 z2 3 2 2 a 4 0 z a2 z2 3 2 30 Chapter 3 Basic Principle of Electrostatics 静电学的基本原理 Or Ez 0 0 z q z 4 0 a2 z2 3 2 3 20 Example 4 The electric field due to a uniformly charged spherical shell of radius R with total charge Q For the annular ring at polar angle the radius is Rsin the width of the ring is Rd So dQ 2 R2sin d The perpendicular distance from the point P to the ri
