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Dr. Hugh Blanton ENTC 3331 Gauss’s Law • Recall • Divergence literally means to get farther apart from a line of path, or • To turn or branch away from. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 3 • Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles: Goes straight ahead at constant velocity. (degree of) divergence 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 4 Now suppose they turn with a constant velocity diverges from original direction (degree of) divergence 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 5 Now suppose they turn and speed up. diverges from original direction (degree of) divergence >> 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 6 Current of water No divergence from original direction (degree of) divergence = 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 7 Current of water Divergence from original direction (degree of) divergence ≠ 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 8 • Source • Place where something originates. • Divergence > 0. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 9 • Sink • Place where something disappears. • Divergence < 0. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 10 • Derivation of Divergence Theorem • Suppose we have a cube that is infinitesimally small. y Vector field, V(x,y,z) n̂i x z Dr. Blanton - ENTC 3331 - Gauss’s Theorem one of six faces 11 • Need the concept of flux: • water through an area • current through an area A Â ĵ • water flux per cross-sectional area (flux density implies • (total) flux = ˆj A ˆ = scaler. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 12 • Let’s assume the vector, V(x,y,z), represents something that flows, then • flux through one face of the cube is: V nˆ i • For example n̂ i might be: • and nˆ yz dydz xˆ Vx xˆ dydz xˆ Vx dydz Dr. Blanton - ENTC 3331 - Gauss’s Theorem 13 • The following six contributions for each side of the cube are obtained: Vx dydz Vx dydz V y dxdz V y dxdz Vz dxdy Vz dxdy Dr. Blanton - ENTC 3331 - Gauss’s Theorem 14 • Now consider the opposite faces of the infinitesimally small cube. vector magnitude on the input side. Vx1 Vx 2 Vx1 dx x y n̂i Vx1 z differential change of Vx over dx Vx 2 x dx • This holds equivalently for the two other pairs of faces. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 15 • Flux in the x-direction. Vx Vx 2 Vx1 dx x y and n̂i Vx1 z Vx xˆ dydz xˆ Vx dydz Vx 2 x dx Vx1 Vx Vx1 x dx dydz Vx 2 x dx dydz Dr. Blanton - ENTC 3331 - Gauss’s Theorem 16 Divergence Theorem • Divergence Theorem • Gauss’s Theorem • Valid for any vector field • Valid for any volume, • Whatever the shape. divV V Vx V y Vz x y z Note that the above only applies to the Cartesian coordinate system. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 17 • Since Gauss’s law can be applied to any vector field, it certainly holds for the electric field, Ex, y, z and the electric flux density, D x, y, z . DdV D d sˆ V S • The use of D in this context instead of E is historical. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 18 • If Gauss’s law is true in general, it should be applicable to a point charge. • Constuct a virtual sphere around a positive charge with radius, R. + q dŝ D • D must be radially outward along the unit vector, R̂ . Dr. Blanton - ENTC 3331 - Gauss’s Theorem 19 ˆ R ˆ d s Dd s ˆ D d s D R S S Dd s D d s D R S DR S 2 0 2DR 2 S 2 sin dd S 2 sin d d 0 0 sin d 2DR cos 0 2DR 2 1 1 4DR 2 2 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 20 • What about the volume integral? 1 1 1 D 2 D dV R D D sin R V R 2 R R sin R sin D 1 2 1 D sin 1 D 2 R DR R R R sin R sin DR 0, D 0, D 0 • D only has a component along the radius vector Dr. Blanton - ENTC 3331 - Gauss’s Theorem 21 1 D 2 R 2 DR R R R 1 2 2 2 V DdV 0 R2 R R DR R dR0 sin d 0 d 4 4 R 0 R 1 2 2 2 R D R dR 4 R R 2 0 R DR dR R R What is this? DR o ER o 1 q q 2 D R R 4o R 2 4 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 22 • Throw in some physics! DR ER 4 R 0 1 q 4 R 2 R R 2 q R dR q dR q 1 q 2 0 0 R 4R R integration and differentiation cancel out Dr. Blanton - ENTC 3331 - Gauss’s Theorem 23 • So what? 4DR D dsˆ D dV q 2S V 4 oER q 1 q E 2 4 o R 2 • Coulomb’s law and Gauss’s law are equivalent for a point charge! Dr. Blanton - ENTC 3331 - Gauss’s Theorem 24 4DR D dsˆ D dV q 2 S V divergence theorem Dr. Blanton - ENTC 3331 - Gauss’s Theorem 25 4DR D dsˆ D dV q 2 S V Gauss’s Law Dr. Blanton - ENTC 3331 - Gauss’s Theorem 26 • Because of its greater mathematical versatility, Gauss’s law rather than Coulomb’s law is a fundamental postulate of electrostatics. • A postulate is believed to be true, although no proof may be possible. D dsˆ D dV Q S V • Any surface of an arbitrary volume. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 27 • Note V dV Q D dV D dsˆ V V definition of charge distribution S Gauss’s Law • which infers D V Dr. Blanton - ENTC 3331 - Gauss’s Theorem Differential form of Gauss’s Law 28 • Maxwell Equation D V • One of two Maxwell equations for electrostatics. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 29 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 30 Electric flux density or Displacement Field [C/m2] Charge Density [C] D Magnetic Induction [Weber/m B 0 or Tesla]] B E Time [s] t Electric Field [V/m] D H J t Magnetic Field [A/m] 2 Current Density [A/m2] Dr. Blanton - ENTC 3331 - Gauss’s Theorem 31 Page 139 ( 0 r E ) ( 0 r H ) 0 ( 0 r H ) E t ( 0 r E ) H J t Dr. Blanton - ENTC 3331 - Gauss’s Theorem 32 Page 139 • Use Gauss’s law to obtain an expression for the E-field from an infinitely long line of charge. l constant 0 E(r ) X Dr. Blanton - ENTC 3331 - Gauss’s Theorem 33 • Symmetry Conditions • Infinite line of charge • D E 0 • Dz Ez 0 • Dr Dr r̂ Dr. Blanton - ENTC 3331 - Gauss’s Theorem 34 • Gauss’s law considers a hypothetical closed surface enclosing the charge distribution. • This Gaussian surface can have any shape, but the shape that minimizes our calculations is the shape often used. Ddsˆ Q l constant dŝ D 0 S h Dr. Blanton - ENTC 3331 - Gauss’s Theorem 35 • The total charge inside the Gaussian volume is: Q l h • The integral is: 2 h D dsˆ Dr rˆ rˆrddz 0 o S • The right and left surfaces do not contribute since. Dz 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 36 Dr r 2 0 h o ddz 2hDr r and 2hDr r l h l Dr o Er 2r Dr. Blanton - ENTC 3331 - Gauss’s Theorem 37 • Two infinite lines of charge. • Each carrying a charge density, l. • Each parallel to the z-axis at • x = 1 and x = -1. • What is the E-field at any point along the y-axis? l constant x 1 z l constant Dr. Blanton - ENTC 3331 - Gauss’s Theorem 1 38 • For a single line of constant charge l Er 2o r • Using the principle of superposition of fields: Etot E1 E2 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 39 r1 rˆ1 0 12 y 02 1 y2 r2 rˆ2 xˆ yyˆ r1 y x r2 -1 0 12 y 02 1 y2 xˆ yyˆ r2 xˆ yyˆ xˆ yyˆ Etot l 2 2o 1 y 1 y 2 E (0, y,0) r1 1 x z Dr. Blanton - ENTC 3331 - Gauss’s Theorem 40 • Only interested in the y-component of the field l Etot 2o yyˆ yyˆ 1 y 2 1 y 2 l Etot 2o 2 yyˆ 1 y 2 l yyˆ Etot o 1 y 2 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 41 • A spherical volume of radius a contains a uniform charge density V. • Determine D E for • R a and • Ra Note: Charge distribution for an atomic nucleus where a = 1.210-15 m A⅓ (A is the mass number) Dr. Blanton - ENTC 3331 - Gauss’s Theorem + q dŝ D 42 • Outside the sphere (R a), use Gauss’s Law D dsˆ S • To take advantage of symmetry, use the spherical coordinates: ds R sin dd 2 • and D Dr rˆ Dr. Blanton - ENTC 3331 - Gauss’s Theorem 43 • Field is always perpendicular for any sphere around the volume. • The left hand side of Gauss’s Law is 2 2 2 ˆ ˆ D dsˆ DR R RR sin dd DR R sin dd 0 S 0 S 4 Q 4DR R Q DR 4R 2 2 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 44 • Recall that D V DdV V dV V V V dV V 0 2 a 0 0 R 2 sin dRd d V 2 a 0 0 0 2 dV sin d d R V V dR V 4 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 45 4V a 0 3 a R 2 dR 4V Q 3 4V a 3 V a 3 Q DR 2 2 4R 4 3R 3R 2 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 46 • Inside the sphere (R a), use Gauss’s Law D dsˆ V dV Q S V 4DR R 2 previously calculated Dr. Blanton - ENTC 3331 - Gauss’s Theorem 47 V dV R 0 2 0 0 V R 2 sin dRd d V R V 4 R 2 dR 0 V 4R 3 3 4DR R 2 V 4R 3 3 V R DR 3 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 48 • Thin spherical shell • Find E-field for • R a and • Ra S 0 a S constant Dr. Blanton - ENTC 3331 - Gauss’s Theorem 49 • Inside (R a) • Gauss’s Law D dsˆ Q 0 S S 0 a S constant • This is only possible if D 0. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 50 • Outside (R a) • Gauss’s Law D dsˆ Q S dS S S 0 a S constant S 4DR R 2 previously calculated S dS 0 S S 2 0 S a 2 sin dd dS 4 S a 2 S Dr. Blanton - ENTC 3331 - Gauss’s Theorem 51 4DR R 2 4 S a 2 2 a DR S 2 R Dr. Blanton - ENTC 3331 - Gauss’s Theorem 52 • An electric field is given as 1 Ex, y, z xˆ 2x y yˆ 3x 2 y V m • Determine • V • Q in a 2m 2m 2m cube. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 53 • Maxwell’s equation of Electrostatics div D V div E y div E 2 x y 3x 2 y x y div E 2 2 0 V 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem x z 54 D dsˆ Q y S 1 For the surface 1 directed in the x-direction. 2x y xˆ xˆdydz 2 2 0 0 x x y dydz 0 0 2 2 2 z 2 y 2 x y z dy 4 x y dy 4 xy 0 0 2 0 2 2 0 2 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 2 55 y 2 2 y 4 xy 8 x 8 2 0 1 For the surface 2 directed in the -x-direction. 2x y xˆ xˆdydz 2 2 0 0 2 0 2x y dydz 8x 8 2 2 x z 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 56 y For the surface 3 & 4 directed in the z- & -z directions. D dsˆ 0 4 3 S x z Dr. Blanton - ENTC 3331 - Gauss’s Theorem 57 y For the surface 5 directed in the y-direction. 5 3x 2 y yˆ yˆ dxdz 2 2 0 0 3x 2 y dxdz 2 2 0 0 3xz 2 yz 2 2 0 0 x dx 6 x 4 y dx 2 z 0 2 6x 4 xy 12 8 y 2 0 2 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 58 y For the surface 6 directed in the -y-direction. 3x 2 y yˆ yˆ dxdz 2 2 0 0 2 0 6 3x 2 y dxdz 2 x 0 12 8 y Dr. Blanton - ENTC 3331 - Gauss’s Theorem z 59 • By superposition D dsˆ 0 S • Indeed, there is no charge in the cube. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 60 • Find D in all regions of an infinitely long cylindrical shell. • Inner shell( r 1 ) • Cylindrical volume. D dsˆ Q 0 V constant V 0 1 S 3 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 61 • Shell itself ( 1 r 3 ) • Cylindrical coordinates. V constant 2 h D dsˆ Dr rˆ rˆ rd dz 0 V 0 0 S 1 r dŝ D r 3 h Dr. Blanton - ENTC 3331 - Gauss’s Theorem 62 2 h 0 0 Dr rˆ rˆ rd dz 2 0 Dr r 2 0 h 0 h 0 Dr rd dz 2 d dz Dr rh d 2Dr rh 0 • Top and bottom face of cylinder do not contribute to D . r 2 h 0 0 1 V rdrd dz Dr. Blanton - ENTC 3331 - Gauss’s Theorem 63 r 2 h 0 0 1 r V rdrd dz 2 V h rdr 1 r 2 V h 2 2 r 2 V h r 1 1 D dsˆ V dV S V 2Dr rh V h(r 2 1) Dr. Blanton - ENTC 3331 - Gauss’s Theorem 64 2Dr rh V h(r 2 1) Dr V (r 2 1) 2 r Dr. Blanton - ENTC 3331 - Gauss’s Theorem 65