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Dr. Hugh Blanton ENTC 3331

ENTC 3331 RF Fundamentals. 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. Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles:.

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Dr. Hugh Blanton ENTC 3331

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  1. ENTC 3331 RF Fundamentals Dr. Hugh Blanton ENTC 3331

  2. Gauss’s Law

  3. 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

  4. 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

  5. Now suppose they turn with a constant velocity  diverges from original direction (degree of) divergence  0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 5

  6. Now suppose they turn and speed up.  diverges from original direction (degree of) divergence >> 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 6

  7. Current of water  No divergence from original direction (degree of) divergence = 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 7

  8. Current of water  Divergence from original direction (degree of) divergence ≠ 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 8

  9. Source • Place where something originates. • Divergence > 0. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 9

  10. Sink • Place where something disappears. • Divergence < 0. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 10

  11. Derivation of Divergence Theorem • Suppose we have a cube that is infinitesimally small. y Vector field, V(x,y,z) x one of six faces z Dr. Blanton - ENTC 3331 - Gauss’s Theorem 11

  12. Need the concept of flux: • water through an area • current through an area • water flux per cross-sectional area (flux density implies • (total) flux = = scaler. A Dr. Blanton - ENTC 3331 - Gauss’s Theorem 12

  13. Let’s assume the vector, V(x,y,z), represents something that flows, then • flux through one face of the cube is: • For example might be: • and Dr. Blanton - ENTC 3331 - Gauss’s Theorem 13

  14. The following six contributions for each side of the cube are obtained: Dr. Blanton - ENTC 3331 - Gauss’s Theorem 14

  15. Now consider the opposite faces of the infinitesimally small cube. • This holds equivalently for the two other pairs of faces. vector magnitude on the input side. y differential change of Vxover dx x z Dr. Blanton - ENTC 3331 - Gauss’s Theorem 15

  16. Flux in the x-direction. y and x z Dr. Blanton - ENTC 3331 - Gauss’s Theorem 16

  17. Divergence Theorem • Divergence Theorem • Gauss’s Theorem • Valid for any vector field • Valid for any volume, • Whatever the shape. Note that the above only applies to the Cartesian coordinate system. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 17

  18. Since Gauss’s law can be applied to any vector field, it certainly holds for the electric field, and the electric flux density, . • The use of in this context instead of is historical. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 18

  19. + • 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. • must be radially outward along the unit vector, . q Dr. Blanton - ENTC 3331 - Gauss’s Theorem 19

  20. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 20

  21. What about the volume integral? • only has a component along the radius vector Dr. Blanton - ENTC 3331 - Gauss’s Theorem 21

  22. What is this? Dr. Blanton - ENTC 3331 - Gauss’s Theorem 22

  23. Throw in some physics! integration and differentiation cancel out Dr. Blanton - ENTC 3331 - Gauss’s Theorem 23

  24. So what? • Coulomb’s law and Gauss’s law are equivalent for a point charge! Dr. Blanton - ENTC 3331 - Gauss’s Theorem 24

  25. divergence theorem Dr. Blanton - ENTC 3331 - Gauss’s Theorem 25

  26. Gauss’s Law Dr. Blanton - ENTC 3331 - Gauss’s Theorem 26

  27. 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. • Any surface of an arbitrary volume. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 27

  28. Note • which infers definition of charge distribution Gauss’s Law Differential form of Gauss’s Law Dr. Blanton - ENTC 3331 - Gauss’s Theorem 28

  29. Maxwell Equation • One of two Maxwell equations for electrostatics. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 29

  30. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 30

  31. Maxwell's Equations Electric flux density or Displacement Field [C/m2] Charge Density [C] Magnetic Induction [Weber/m2 or Tesla]] Time [s] Electric Field [V/m] Magnetic Field [A/m] Current Density [A/m2] Dr. Blanton - ENTC 3331 - Gauss’s Theorem 31 Page 139

  32. Maxwell's Equations Dr. Blanton - ENTC 3331 - Gauss’s Theorem 32 Page 139

  33. Use Gauss’s law to obtain an expression for the E-field from an infinitely long line of charge. 0 X Dr. Blanton - ENTC 3331 - Gauss’s Theorem 33

  34. Symmetry Conditions • Infinite line of charge Dr. Blanton - ENTC 3331 - Gauss’s Theorem 34

  35. 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. 0 Dr. Blanton - ENTC 3331 - Gauss’s Theorem 35

  36. The total charge inside the Gaussian volume is: • The integral is: • The right and left surfaces do not contribute since. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 36

  37. and Dr. Blanton - ENTC 3331 - Gauss’s Theorem 37

  38. Two infinite lines of charge. • Each carrying a charge density, rl. • Each parallel to the z-axis at • x = 1 and x = -1. • What is the E-field at any point along the y-axis? Dr. Blanton - ENTC 3331 - Gauss’s Theorem 38

  39. For a single line of constant charge • Using the principle of superposition of fields: Dr. Blanton - ENTC 3331 - Gauss’s Theorem 39

  40. y x x 1 -1 z Dr. Blanton - ENTC 3331 - Gauss’s Theorem 40

  41. Only interested in the y-component of the field Dr. Blanton - ENTC 3331 - Gauss’s Theorem 41

  42. + • A spherical volume of radius a contains a uniform charge density rV. • Determine for • and Note: Charge distribution for an atomic nucleus where a = 1.210-15 m  A⅓ (A is the mass number) q Dr. Blanton - ENTC 3331 - Gauss’s Theorem 42

  43. Outside the sphere (R  a), use Gauss’s Law • To take advantage of symmetry, use the spherical coordinates: • and Dr. Blanton - ENTC 3331 - Gauss’s Theorem 43

  44. Field is always perpendicular for any sphere around the volume. • The left hand side of Gauss’s Law is Dr. Blanton - ENTC 3331 - Gauss’s Theorem 44

  45. Recall that Dr. Blanton - ENTC 3331 - Gauss’s Theorem 45

  46. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 46

  47. Inside the sphere (R  a), use Gauss’s Law previously calculated Dr. Blanton - ENTC 3331 - Gauss’s Theorem 47

  48. Dr. Blanton - ENTC 3331 - Gauss’s Theorem 48

  49. Thin spherical shell • Find E-field for • and Dr. Blanton - ENTC 3331 - Gauss’s Theorem 49

  50. Inside ( ) • Gauss’s Law • This is only possible if . Dr. Blanton - ENTC 3331 - Gauss’s Theorem 50

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