Presentation on theme: "Chapter 13 Maxwell’s Equations 麦克斯韦方程组. Maxwell summarized the experimental laws of electricity and magnetism—the laws of Coulomb, Gauss, Biot-Savart,"— Presentation transcript:
Chapter 13 Maxwell’s Equations 麦克斯韦方程组
Maxwell summarized the experimental laws of electricity and magnetism—the laws of Coulomb, Gauss, Biot-Savart, Ampere, and Faraday. He found that all the experimental laws hold in general except for Ampere’s law. Dones not apply to discontinuous current Invent displacement current to generalize Ampere’s law
Electric field Electrostatic field --set up by static charges Induced electric field --set up by varying M-field Is there another M-field Magnetic field magnetic field --set up by steady current that it is set up by varying E-field ? ? 1.Question §13-1 Displacement Current
Displacement Current 位移电流 2. Maxwell’s hypothesis A varying electric field will set up a magnetic field in exactly the same way as ordinary conduction current. Varying Inducing
Two different surfaces S 1, S 2 bounded by the same circle L Displacement current The capacitor is electrified For S 1 For S 2 Ampere’s law cannot be used in this problem. Conductive current
Though there is any current going through the surface S 2, there is a changing -flux going through it. When I c 0,
Assume S--the area of plate, --the area charge density on S at time t. on S at time t. then
--the density of displacement current Displacement current Definition
--Generalized form of Ampere’s law I =Ic+IdI =Ic+IdI =Ic+IdI =Ic+Id generalized current( 全电流 ) Let -flux
Notes The differences between I d and I c : I c is formed by the motion of charges in conductor along one direction. I c produces Joule thermal energy in conductor. I d I c I d set up a M-field in exactly the same way as I c. I d is formed by the varying of electric field. I d is formed by the varying of electric field. I d never has thermal effect. I d can be found in the area that exists varying electric field( vacuum, dielectric, conductor).
[Example]Two circle plates with radius R=0.1m consist of a parallel plate capacitor. When the E-field between the plates increases with dE/dt=10 12 Vm -1 s -1. [Example]Two circle plates with radius R=0.1m consist of a parallel plate capacitor. When the capacitor is electrified, the E-field between the plates increases with dE/dt=10 12 Vm -1 s -1. Find The displacement current I d between two plates. The magnitude and direction of the M-field in the area of r R
Solution The distribution of E-field has axial symmetry. The distribution of E-field has axial symmetry. So the induced M-field produced by the varying E-field has the same form as a cylindrical current. Drawing a circle L with r as shown in Fig. Using generalized form of Ampere’s law, we have
When r>R, (r>R) i.e.
When, ’s direction: L When, L
§13-2 Maxwell’s Equations E-Field M-Field E-Field Electromagnetic wave Maxwell’s Equations Charge Current Charge Current
Electrostatic field set up by static charges ： Steady magnetic field set up by steady current ： 1. Maxwell’s equations
Induced electric field set up by varying M-field: Induced M-field set up by varying E-field (displacement current):
In general, We get Maxwell’s Equations in integral form.
As Maxwell’s Equations in differential form. And
2. Electromagnetic wave Special example ： In the free space( 自由空间 ) No any charge and conductive current. Maxwell’s Equations ：
For vacuum, Making a Rotation( 旋度 ) for , Use Use Use
Using vector formula, =0 We can get the equation from above, --differential form of E-field Similarly, --differential form of M-field The wave equations of electromagnetic field in vacuum.
The speed of E-M-wave is E-M-field spreads in the space to form the E-M-wave. x y z o c Direction of propagation