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PHYSICS-II (PHY C132)

PHYSICS-II (PHY C132). ELECTRICITY & MAGNETISM. Introduction to Electrodynamics: by David J. Griffiths (3 rd Ed.). VECTOR ANALYSIS. Differential Calculus. Integral Calculus. Revisited. Curvilinear Coordinates. The Dirac Delta Function. Theory of Vector Fields.

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PHYSICS-II (PHY C132)

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  1. PHYSICS-II (PHY C132) ELECTRICITY & MAGNETISM Introduction to Electrodynamics: by David J. Griffiths (3rd Ed.) Dr. Champak B. Das ( BITS, Pilani)

  2. VECTOR ANALYSIS • Differential Calculus • Integral Calculus Revisited • Curvilinear Coordinates • The Dirac Delta Function • Theory of Vector Fields Dr. Champak B. Das ( BITS, Pilani)

  3. Differential Calculus Derivative of any function f(x,y,z): Dr. Champak B. Das ( BITS, Pilani)

  4. Change in a scalar function f corresponding to a change in position : Gradient of function f  f is a VECTOR Dr. Champak B. Das ( BITS, Pilani)

  5. Geometrical interpretation of Gradient Z P Q dl Y change in f : X =0 => f  dl Dr. Champak B. Das ( BITS, Pilani)

  6. Z Q dl P Y X Dr. Champak B. Das ( BITS, Pilani)

  7. The rate of change of f is max. for • The max. value of rate of change of fis • fincreases in the direction of • Grad f is in the direction of the normal to the surface of constant f Dr. Champak B. Das ( BITS, Pilani)

  8. Gradient of a function slope of the function along the direction of maximum rate of change of the function Dr. Champak B. Das ( BITS, Pilani)

  9. If  f= 0 at some point (x0,y0,z0) => df = 0 for small displacements about the point (x0,y0,z0) (x0,y0,z0) is a stationary point of f(x,y,z) Dr. Champak B. Das ( BITS, Pilani)

  10. Prob. 1.12 The height of a certain hill (in feet) is: h(x,y) = 10(2xy – 3x2 -4y2 -18x + 28y +12) where x is distance (in mile) east and y north of Pilani. (a) Where is the top located ? Ans: 3 miles North & 2 miles West Dr. Champak B. Das ( BITS, Pilani)

  11. Prob. 1.12 (contd.) h(x,y) = 10(2xy – 3x2 -4y2 -18x – 28y +12) (b) How high is the hill ? Ans: 720 ft (c) How steep is the slope at 1 mile north and 1 mile east of Pilani? In what direction the slope is steepest, at that point ? Ans: 311 ft/mile, direction is Northwest Dr. Champak B. Das ( BITS, Pilani)

  12. Prob. 1.13 Let rsis the separation vector from (x,y,z) to (x,y,z) . Dr. Champak B. Das ( BITS, Pilani)

  13. The Operator   is NOT a VECTOR, but a VECTOR OPERATOR Satisfies: • Vector rules • Partial differentiation rules Dr. Champak B. Das ( BITS, Pilani)

  14.  can act: • On a scalar function f :f GRADIENT • On a vector function F as:.F DIVERGENCE • On a vector function F as: ×F CURL Dr. Champak B. Das ( BITS, Pilani)

  15. Divergence of a vector Divergence of a vector is a scalar. Dr. Champak B. Das ( BITS, Pilani)

  16. Geometrical interpretation of Divergence .F is a measure of how much the vector F spreads out/in (diverges) from/to the point in question. Dr. Champak B. Das ( BITS, Pilani)

  17. Z H G D C dz E F dx Y A B dy X Physical interpretation of Divergence Flow of a compressible fluid: (x,y,z) density of the fluid at a point (x,y,z) v(x,y,z) velocity of the fluid at (x,y,z) Dr. Champak B. Das ( BITS, Pilani)

  18. Net rate of flow out through all pairs of surfaces (per unit time): Dr. Champak B. Das ( BITS, Pilani)

  19. Net rate of flow of the fluid per unit volume per unit time: DIVERGENCE Dr. Champak B. Das ( BITS, Pilani)

  20. Example: Calculate, Dr. Champak B. Das ( BITS, Pilani)

  21. Prob. 1.16 Sketch the vector function and compute its divergence. Explain the answer ! ! Dr. Champak B. Das ( BITS, Pilani)

  22. Curl Curl of a vector is a vector Dr. Champak B. Das ( BITS, Pilani)

  23. Geometrical interpretation of Curl ×F is a measure of how much the vector F “curls around” the point in question. Dr. Champak B. Das ( BITS, Pilani)

  24. Physical significance of Curl Circulation of a fluid around a loop about a point : Y 3 y 2 4 1 x X Circulation Dr. Champak B. Das ( BITS, Pilani)

  25. Circulation per unit area z-component of CURL Dr. Champak B. Das ( BITS, Pilani)

  26. Sum Rules For Gradient: For Divergence: For Curl: Dr. Champak B. Das ( BITS, Pilani)

  27. Rules for multiplying by a constant For Gradient: For Divergence: For Curl: Dr. Champak B. Das ( BITS, Pilani)

  28. Product Rules For a Scalar from two functions: Gradients: Dr. Champak B. Das ( BITS, Pilani)

  29. Product Rules For a Vector from two functions: Divergences: Dr. Champak B. Das ( BITS, Pilani)

  30. Product Rules Curls: Dr. Champak B. Das ( BITS, Pilani)

  31. Prob. 1.21 (a) What does the expression mean ? Prob. 1.21 (b) Compute: Ans: 0 Dr. Champak B. Das ( BITS, Pilani)

  32. Quotient Rules Dr. Champak B. Das ( BITS, Pilani)

  33. Second Derivatives Of a gradient: Divergence : Laplacian Curl : ( Prob. 1.27: Prove it ! ) Dr. Champak B. Das ( BITS, Pilani)

  34. Second Derivatives Of a divergence: Gradient : Dr. Champak B. Das ( BITS, Pilani)

  35. Second Derivatives Of a Curl: Divergence : Prob. 1.26: Prove it ! Curl : Dr. Champak B. Das ( BITS, Pilani)

  36. Integral Calculus Line Integral: Surface Integral: Volume Integral: Dr. Champak B. Das ( BITS, Pilani)

  37. Fundamental theorem for gradient Line integral of gradient of a function is given by the value of the function at the boundaries of the line. Dr. Champak B. Das ( BITS, Pilani)

  38. Corollary 1: Corollary 2: Dr. Champak B. Das ( BITS, Pilani)

  39. Fundamental theorem for Divergence The integral of divergence of a vector over a volume is equal to the value of the function over the closed surface that bounds the volume. Gauss’ theorem, Green’s theorem Dr. Champak B. Das ( BITS, Pilani)

  40. Fundamental theorem for Curl Integral of a curl of a vector over a surface is equal to the value of the function over the closed boundary that encloses the surface. Stokes’ theorem Dr. Champak B. Das ( BITS, Pilani)

  41. Corollary 1: Corollary 2: Dr. Champak B. Das ( BITS, Pilani)

  42. Curvilinear coordinates: used to describe systems with symmetry. Spherical Polar coordinates (r, , ) Cylindrical coordinates (s, , z) Dr. Champak B. Das ( BITS, Pilani)

  43. Spherical Polar Coordinates A point is characterized by: r : distance from origin Z  : polar angle P r   : azimuthal angle Y  X Dr. Champak B. Das ( BITS, Pilani)

  44. Cartesian coordinates in terms of spherical coordinates: Z P r  Y  X Dr. Champak B. Das ( BITS, Pilani)

  45. Spherical coordinates in terms of Cartesian coordinates: Z P r  Y  X Dr. Champak B. Das ( BITS, Pilani)

  46. Prob. 1.37 : Unit vectors in spherical coordinates Z r  Y  X Dr. Champak B. Das ( BITS, Pilani)

  47. Line element in spherical coordinates: Volume element in spherical coordinates: Dr. Champak B. Das ( BITS, Pilani)

  48. Area element in spherical coordinates: on a surface of a sphere (r const.) on a surface lying in xy-plane (const.) Dr. Champak B. Das ( BITS, Pilani)

  49. Ranges of r,  and  r: 0    : 0    : 0  2 Dr. Champak B. Das ( BITS, Pilani)

  50. The Operator in Spherical Polar Coordinates Dr. Champak B. Das ( BITS, Pilani)

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