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Transcript
Electrodynamics – Maxwell’s Equations in Vacuum Faraday’s law and electric field excited by magnetic field Displacement current and magnetic field excited electric field Maxell’s equations in vacuum 1 Electron Acceleration and Time-Varying Current • Accelerated electron creates time-varying current, timevarying current creates time-varying magnetic field, timevarying magnetic field will create electric field vortex • Why? • Explanation – isolated natural system tend to eliminate any disturbance, with time evolution towards its eigen state: once a time-varying magnetic field is established, a electric field vortex will be induced, to establish yet another magnetic field with an opposite change towards a cancellation of the total field, hence Faraday’s law holds t B ds E dl 0 s l B E t 2 Electric-Magnetic Field Coupling Caused by Time-Varying Source • Accelerated charge causes a time-varying charge distribution, leading to a time-varying current; from Ampere’s law, time-varying magnetic field exists; from Faraday’s law, electric vortex exists • Therefore, not only electric field can be generated in its divergence form by the static charge distribution, it can also be generated in its curl form by the “temporary” charge distribution which • Indicating that the electric field is brought in by both its divergence and curl, following the Helmholtz theorem, the electric field is complete • Hence, time-varied magnetic field excites electric field, electric and magnetic fields become coupled; however, such coupling is unidirectional 3 Illustration of Relation between Electric and Magnetic Fields Static charge Div. Time-varying charge distribution (time-varying current, temporary charge) Moving charge, static charge distribution (constant current) Div. Curl Curl Static Electric Field Time-varying electric field Static Magnetic Field Time-varying magnetic field Curl J t Charge conservation law Gauss’s law E 0 B 0 Ampere’s law B 0 J Faraday’s law B E t (Derived from Coulomb’s law) (Derived from Biot-Savert’s law) 4 Maxwell’s Equations • Contradiction between Ampere’s law and the charge conservation law B 0 J 0 0 t • Maxwell mended Ampere’s law, solved the problem E B 0 ( J ) B 0 J 0 0 t t Gaussian law is called E 0 0 0 5 Introduction of Displacement Current • Significance – likes a current, the time-varying electric field can generate magnetic field • Hence the time-varying rate of the electric displacement vector is equivalent to a current, named as the displacement current; the conventional current caused by the moving charge is then called the conduction current to make a difference • Not only time-varying magnetic field, equivalent to temporary charges, can excite electric field, time-varying electric field, equivalent to a “temporary” current (i.e., the displacement current), can also excite magnetic field • Finally, electric and magnetic fields are fully coupled through mutual excitation 6 Significance of Maxwell’s Equations • Therefore – 1. accelerated charge creates time-varying magnetic field in its neighborhood (through the time-varying conduction current and following Ampere’s law); – 2. time-varying magnetic field excites electric field in its neighborhood (through Faraday’s law) – 3. time-varying electric field excites magnetic field in its neighborhood (through Ampere’s law) • Step 2 and 3 form a sustainable loop, and make electricmagnetic fields propagation! Such an electric-magnetic non-local oscillation is named as the electromagnetic wave 7 Maxwell’s Equations in Vacuum B E t E B 0 J 0 0 t E / 0 B 0 Faraday’s law Maxwell modified Ampere’s law, derived from Biot-Savert’s law Gaussian law, derived from Coulomb’s law Gaussian law, derived from Biot-Savert’s law 8 Home Work 2 1. Give a set of consistent electric and magnetic field expressions for an electron, the observer can either stay still or move at a constant speed. 2. Derive the gravitational wave equation. 9