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D. K. Cheng Field and Wave Electromagnetics Chapter 5 Steady Electric Currents Chapter 5 Steady Electric Currents §5-1 Introduction 由运动电荷导致的电流可分三类:传导电流、电解电流、对流(运流)电流 (D. K. Cheng, p. 198) Conduction current: Drift motion of conduction electrons and/or holes in conductors and semiconductors; Electrolytic current: Migration of positive and negative ions; Convection current: Motion of electrons and/or ions (positively or negatively charged particles) in vacuum or in rarefied gas. Convection current, the result of hydrodynamic motion involving a mass transport, are not governed by Ohm’s law. (D. K. Cheng, p. 199) When an external electric field is applied on a conductor, an organized motion of the conduction electrons will result, producing an electric current. The average drift velocity of the electrons is very low (1/1000 m/s) even for very good conductors because they collide with the atoms, dissipating part of their kinetic energy as heat. (D. K. Cheng, p. 199) §5-2 Current density and Ohm’s law We’d like to point out… The explicit expression for the electron mobility in a metallic conductor The physical meanings of the mobility μ: The electron mobility μ depends on q/m and τ(relaxation time). If the electrons are scattered (collided) very frequently by ions, the τ (relaxation time) is small. The large relaxation time will lead to large mobility μ. We have presented the explicit expression for the electron mobility in a metallic conductor. Now we discuss the relevant concepts in more details: Count Alessandro Giuseppe Antonio Anastasio Volta, 1745--1827, Italy André-Marie Ampère, 1775--1836, France Georg Simon Ohm, 1789—1854, German The relation between microscopic and macroscopic Ohm’s laws Fig. 5-3 Homogeneous conductor with a constant cross section (J=σE) Note: J=σE is the microscopic form of Ohm’s law, and I=U/R is the macroscopic form of Ohm’s law. They can be related by Eq. (5-27). §5-3 Electromotive force and Kirchhoff’s voltage law Eq. (5-32) tells us that a steady current cannot be maintained in the same direction in a closed circuit by an electrostatic field. (D. K. Cheng, p. 206) steady: 稳态的; static: 静态的 区别:稳态,可以有运动,但运动恒速(x的一阶时间 导数不为零,二阶导数为零); 静态,没有运动(x的一阶导数也为零)。 例如:一个稳恒的电流可以产生一个静态的磁场 A steady current can lead to a static magnetic field (magnetostatic field). Fig. 5-4 Electric fields inside an electric battery Ei is the impressed electric field (non-conservative electric field) caused by chemical action. Note: The line integral of the impressed field Ei from the negative to the positive electrode (from electrode 2 to electrode 1 in figure) inside the battery is “electromotive force” (电动势)。(D. K. Cheng, p. 206) (voltage rise) Note: E is the conservative electrostatic field, and Ei is the impressed electric field (non-conservative electric field) §5-4 Equation of continuity and Kirchhoff’s current law The current leaving the region is the total outward flux of the current density vector through the surface S (D. K. Cheng, p. 208) The charge conservation law can also be given as follows Thus, ρcan be considered zero in the interior of a conductor. (D. K. Cheng, p. 210) 由于静电平衡(导体是 一个等势体),金属内 部不可能有多余的自由 电荷。一旦有多余电荷, 就立即跑到金属表面, 尤其积聚到曲率大的地 方(如尖端处)。 §5-5 Power dissipation and Joule’s law The power provided by an electric field E in moving a charge q is This is the (electric) power density (power per unit volume). §5-6 Boundary conditions for current density In the absence of non-conservative energy source, we shall have We can obtain the boundary conditions for J (as in Fig. 3-23 and in Sec. 3-9): Eq. (5-59) states that the ratio of the tangential components of J at two sides of an interface is equal to the ratio of the electric conductivities. (D. K. Cheng, p. 212) §5-7 Resistance calculations The basic formula for capacitance can be written as Fig. 5-7 Two conductors in a lossy dielectric medium Since the metallic conductors are equipotential media, you can choose anyone of the integral paths for calculating the electric potential difference. Note: The dimension of RC and ε/σ is [time].