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Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence 3.1 Electric Flux Density • Faraday’s Experiment Electric Flux Density, D • Units: C/m2 • Magnitude: Number of flux lines (coulombs) crossing a surface normal to the lines divided by the surface area. • Direction: Direction of flux lines (same direction as E). • For a point charge: • For a general charge distribution, D3.1 Given a 60-uC point charge located at the origin, find the total electric flux passing through: (a) That portion of the sphere r = 26 cm bounded by 0 < theta < Pi/2 and 0 < phi < Pi/2 D3.2 Calculate D in rectangular coordinates at point P(2,-3,6) produced by : (a) a point charge QA = 55mC at Q(-2,3,-6) 2 P 3 6 0 8.854 10 D 2 3 Q 6 12 QA 4 R r 2 R P Q QA 55 10 r 6.38 10 6 D 9.57 10 6 5 1.914 10 P Q P Q 3 (b) a uniform line charge pLB = 20 mC/m on the x axis (c) a uniform surface charge density pSC = 120 uC/m2 on the plane z = -5 m. Gauss’s Law • “The electric flux passing through any closed surface is equal to the total charge enclosed by that surface.” • The integration is performed over a closed surface, i.e. gaussian surface. • We can check Gauss’s law with a point charge example. Symmetrical Charge Distributions • Gauss’s law is useful under two conditions. 1. DS is everywhere either normal or tangential to the closed surface, so that DS.dS becomes either DS dS or zero, respectively. 2. On that portion of the closed surface for which DS.dS is not zero, DS = constant. Gauss’s law simplifies the task of finding D near an infinite line charge. Infinite coaxial cable: Differential Volume Element • If we take a small enough closed surface, then D is almost constant over the surface. D3.6a 8 x y z4 2 4 D ( x y z) 4 x z 2 3 16 x y z 3 1 2 0 D ( x y 2 ) 10 2 12 d x d y 1.365 10 9 D3.6b 8 x y z4 2 4 D ( x y z) 4 x z 2 3 16 x y z 0 8.854 10 2 P 1 3 10 12 12 E D ( 2 1 3) 0 146.375 146.375 E 195.166 Divergence Divergence is the outflow of flux from a small closed surface area (per unit volume) as volume shrinks to zero. -Water leaving a bathtub -Closed surface (water itself) is essentially incompressible -Net outflow is zero -Air leaving a punctured tire -Divergence is positive, as closed surface (tire) exhibits net outflow Mathematical definition of divergence div D lim v 0 D v dS Surface integral as the volume element (v) approaches zero D is the vector flux density div D D D D x y z y z x - Cartesian Divergence in Other Coordinate Systems Cylindrical div D 1 D 1 D D z z Spherical div D D r 1 r 2 r r 2 1 r sin D sin 1 r sin D Divergence at origin for given vector flux density A A div A div A e x sin ( y ) e x cos ( y ) 2z x e x sin ( y ) e x sin ( y ) e y x e x cos ( y ) sin ( y ) 2 z ( 2 z) 3-6: Maxwell’s First Equation . . A dS Q Gauss’ Law… Q …per unit volume S A dS S v v . A dS Volume shrinks to zero lim v 0 S v lim Q v 0 v Electric flux per unit volume is equal to the volume charge density Maxwell’s First Equation . A dS S lim v 0 v div D lim Q v 0 v v Sometimes called the point form of Gauss’ Law Enclosed surface is reduced to a single point 3-7: and the Divergence Theorem del operator What is ax x del? ay y az z ’s Relationship to Divergence div D D V True for all coordinate systems Other Relationships Gradient – results from operating on a function Represents direction of greatest change Curl – cross product of and Relates to work in a field If curl is zero, so is work Examination of and flux Cube defined by 1 < x,y,z < 1.2 D 2 2 2 2x ya x 3x y a y Calculation of total flux . Q D dS S total . v dv vol left right front back z2 x1 z 1 z2 x2 z 1 y2 y z2 2 2 x1 y d y d z 1 y2 y 1 z2 2 2 x2 y d y d z 1 y1 z y2 z 1 x1 1 x2 1.2 y 1 1 y 2 1.2 z1 1 z2 1.2 x2 x 2 2 3 x y 1 dx dz 1 x2 x 2 2 3 x y 2 dx dz 1 total x1 x2 y1 y2 total 0.103 Evaluation ofV Dat center of cube div D d 2 x2 y dx div D d 3 x2 y 2 dy 2 4 x y 6 x y 2 divD 4 ( 1.1 ) ( 1.1 ) 6 ( 1.1 ) ( 1.1 ) divD 12.826 Non-Cartesian Example Equipotential Surfaces – Free Software Semiconductor Application - Device Charge Field Potential Vector Fields Potential Field Applications of Gauss’s Law