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Chapter 13 Maxwell’s Equations 麦克斯韦方程组

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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

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

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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

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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

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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

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Though there is any current going through the surface S 2, there is a changing -flux going through it. When I c 0,

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Assume S--the area of plate, --the area charge density on S at time t. on S at time t. then

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--the density of displacement current Displacement current Definition

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--Generalized form of Ampere’s law I =Ic+IdI =Ic+IdI =Ic+IdI =Ic+Id generalized current( 全电流 ) Let -flux

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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).

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[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

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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

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When r

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When r>R, (r>R) i.e.

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When, ’s direction: L When, L

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§13-2 Maxwell’s Equations E-Field M-Field E-Field Electromagnetic wave Maxwell’s Equations Charge Current Charge Current

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Electrostatic field set up by static charges ： Steady magnetic field set up by steady current ： 1. Maxwell’s equations

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Induced electric field set up by varying M-field: Induced M-field set up by varying E-field (displacement current):

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In general, We get Maxwell’s Equations in integral form.

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As Maxwell’s Equations in differential form. And

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2. Electromagnetic wave Special example ： In the free space( 自由空间 ) No any charge and conductive current. Maxwell’s Equations ：

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For vacuum, Making a Rotation( 旋度 ) for , Use Use Use

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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.

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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

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