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

§ 13-2 Maxwell’s Equations 麦克斯韦方程组 § 13-1 Displacement Current 位移电流

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 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