chap6 - Maxwells Equations James Clerk Maxwell (1831 -...

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Unformatted text preview: Maxwells Equations James Clerk Maxwell (1831 - 1879) November 9, 2001 B ( x , t ) = 0 , H ( x , t ) = 4 c J ( x , t ) + 1 c D ( x , t ) t E ( x , t ) =- 1 c B ( x , t ) t , D ( x , t ) = 4 ( x , t ) Contents 1 Faradays Law of Induction 2 2 Energy in the Magnetic Field 7 2.1 Example: Motion of a permeable Bit in a Fixed J . . . . . . . . . . . 10 2.2 Energy of a Current distribution in an External Field . . . . . . . . . 12 3 Maxwells Displacement Current; Maxwells Equations 15 4 Vector and Scalar Potentials 17 5 Gauge Transformations 19 5.1 Lorentz Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Coulomb Transverse Gauge . . . . . . . . . . . . . . . . . . . . . . . 21 1 6 Greens Functions for the Wave Equation 23 7 Derivation of Macroscopic Electromagnetism 28 8 Poyntings Theorem; Energy and Momentum Conservation 28 8.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.3 Need for Field Momentum . . . . . . . . . . . . . . . . . . . . . . . . 35 8.4 Example: Force on a Conductor . . . . . . . . . . . . . . . . . . . . . 36 9 Conservation Laws for Macroscopic Systems 37 10 Poyntings Theorem for Harmonic Fields; Impedance, Admittance, etc. 37 11 Transformations: Reflection, Rotation, and Time Reversal 42 11.1 Transformation Properties of Physical Quantities . . . . . . . . . . . 43 12 Do Maxwells Equations Allow Magnetic Monopoles? 48 A Helmholtz Theorem 50 2 Our first task in this chapter is to put time into the equations of electromagnetism. There are traditionally two steps in this process. The first of them is to develop Fara- days Law of Induction which is the culmination of a series of experiments performed by Michael Faraday (1791 - 1867) around 1830. Faraday studied the current in- duced in one closed circuit when a second nearby current-carrying circuit either was moved or had its current varied as a function of time. He also did experiments in which the second circuit was replaced by a permanent magnet in motion. The general conclusion of these experiments is that if the magnetic flux through a closed loop or circuit changes with time, an induced voltage or electromotive force (abbreviated by E mf ) appears in the circuit. 1 Faradays Law of Induction Define the magnetic flux through, or linking a closed loop C as F = Z S d 2 x B n (1) where S is an open surface that ends on the curve C and n is the usual unit right-hand normal (see below) to the surface. So long as B = 0, this integral is the same for all such surfaces....
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chap6 - Maxwells Equations James Clerk Maxwell (1831 -...

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