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1、Engineering ElectromagneticsW.H. Hayt Jr. and J. A. Buck,Chapter 3:Electric Flux Density, Gauss Law,and Divergence,Faraday Experiment,He started with a pair of metal spheres of different sizes; the larger one consisted of two hemispheres that could be assembled around the smaller sphere,Faraday Appa
2、ratus, Before Grounding,The inner charge, Q, inducesan equal and opposite charge, -Q, on the inside surface of theouter sphere, by attracting free electrons in the outer material towardthe positive charge. This means that before the outer sphere is grounded, charge +Q resides on the outside surface
3、of the outer conductor.,Faraday Apparatus, After Grounding,Attaching the ground connects theouter surface to an unlimited supplyof free electrons, which then neutralize the positive charge layer. The net charge on the outer sphere is then the charge on the inner layer, or -Q.,Interpretation of the F
4、araday Experiment,Electric Flux Density,Vector Field Description of Flux Density,A vector field is established which points in the direction of the “flow”or displacement. In this case, thedirection is the outward radial direction in spherical coordinates. At each surface,we would have:,Radially-Depe
5、ndent Electric Flux Density,r,D(r),Point Charge Fields,If we now let the inner sphere radius reduce to a point, while maintaining the same charge, and let the outer sphere radius approach infinity, we have a point charge. The electric flux density is unchanged, but is defined over all space:,C/m2 (0
6、 r ),We compare this to the electric field intensity in free space:,V/m (0 r ),.and we see that:,Finding E and D from Charge Distributions,Gauss Law,The electric flux passing through any closed surface is equal to the total charge enclosed by that surface,Development of Gauss Law,We define the diffe
7、rential surface area (a vector) as,where n is the unit outwardnormal vector to the surface, and where dS is the area of thedifferential spot on the surface,Mathematical Statement of Gauss Law,Using Gauss Law to Solve for D Evaluated at a Surface,Example: Point Charge Field,Begin with the radial flux
8、 density:,and consider a spherical surface of radius a that surrounds the charge, on which:,On the surface, the differential area is:,and this, combined with the outward unit normal vector is:,Point Charge Application (continued),Another Example: Line Charge Field,Line Charge Field (continued),Anoth
9、er Example: Coaxial Transmission Line,We have two concentric cylinders, with the z axis down their centers. Surface charge of density S existson the outer surface of the inner cylinder.,A -directed field is expected, and this should vary only with (like a line charge). We therefore choosea cylindric
10、al Gaussian surface of length L and of radius , where a b.,The left hand side of Gauss Law is written:,and the right hand side becomes:,Coaxial Transmission Line (continued),Coaxial Transmission Line: Exterior Field,Electric Flux Within a Differential Volume Element,Taking the front surface, for exa
11、mple, we have:,Electric Flux Within a Differential Volume Element,Charge Within a Differential Volume Element,Divergence and Maxwells First Equation,Mathematically, this is:,and when the vector field is the electric flux density:,Divergence Expressions in the Three Coordinate Systems,The Del Operator,Divergence Theorem,Statement of the Divergence Theorem,
