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

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

6. Dr. Champak B. Das ( BITS, Pilani)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 f is
f increases 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. Dr. Champak B. Das ( BITS, Pilani)
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. Dr. Champak B. Das ( BITS, Pilani)
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. Dr. Champak B. Das ( BITS, Pilani)
Prob (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. Dr. Champak B. Das ( BITS, Pilani)
Prob. 1.13
Let rs is the separation vector from (x,y,z) to (x,y,z) .
Dr. Champak B. Das ( BITS, Pilani)

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

14. Dr. Champak B. Das ( BITS, Pilani)
 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. 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) Z X Y dy dx dz A C D B E F G H
Dr. Champak B. Das ( BITS, Pilani)

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

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

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

21. Dr. Champak B. Das ( BITS, Pilani)
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 2 y 4 1 x X
Circulation
Dr. Champak B. Das ( BITS, Pilani)

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

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

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

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

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

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

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

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

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

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

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

48. Dr. Champak B. Das ( BITS, Pilani)
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. Dr. Champak B. Das ( BITS, Pilani)
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)

51. Dr. Champak B. Das ( BITS, Pilani)
Gradient:
Divergence:
Dr. Champak B. Das ( BITS, Pilani)

52. Dr. Champak B. Das ( BITS, Pilani)
Curl:
Dr. Champak B. Das ( BITS, Pilani)

53. Dr. Champak B. Das ( BITS, Pilani)
Laplacian:
Dr. Champak B. Das ( BITS, Pilani)

54. Cylindrical Coordinates
A point is characterized by:
s : distance from z-axis Z
z : cartesian coordinate s P
 : azimuthal angle z Y  X
Dr. Champak B. Das ( BITS, Pilani)

55. Prob. 1.41 : Unit vectors in cylindrical coordinatesZ s z Y  X
Dr. Champak B. Das ( BITS, Pilani)

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

57. Dr. Champak B. Das ( BITS, Pilani)
Ranges of s,  and z
s : 0  
 : 0  2
z : -   
Dr. Champak B. Das ( BITS, Pilani)

58. The Operator  in Cylindrical Coordinates
Dr. Champak B. Das ( BITS, Pilani)

59. Dr. Champak B. Das ( BITS, Pilani)
Gradient:
Divergence:
Dr. Champak B. Das ( BITS, Pilani)

60. Dr. Champak B. Das ( BITS, Pilani)
Curl:
Laplacian:
Dr. Champak B. Das ( BITS, Pilani)

61. Dr. Champak B. Das ( BITS, Pilani)
General expressions for the Derivatives in different coordinate systems
u, v, w
Coordinate System:
Line element :
Dr. Champak B. Das ( BITS, Pilani)

62. Dr. Champak B. Das ( BITS, Pilani)
System u v w f g h
Cartesian x y z 1
Spherical r  
r sin
Cylindrical s
Dr. Champak B. Das ( BITS, Pilani)

63. Dr. Champak B. Das ( BITS, Pilani)
GRADIENT
Dr. Champak B. Das ( BITS, Pilani)

64. Dr. Champak B. Das ( BITS, Pilani)
DIVERGENCE
Dr. Champak B. Das ( BITS, Pilani)

65. Dr. Champak B. Das ( BITS, Pilani)
CURL :
Dr. Champak B. Das ( BITS, Pilani)

66. Dr. Champak B. Das ( BITS, Pilani)
LAPLACIAN
Dr. Champak B. Das ( BITS, Pilani)

67. Recall  Prob. 1.16
Sketch the vector function and compute its Divergence
Dr. Champak B. Das ( BITS, Pilani)

68. Dr. Champak B. Das ( BITS, Pilani)
Calculation of Divergence =>
Divergence theorem =>
Dr. Champak B. Das ( BITS, Pilani)

69. And its integral over ANY volume containing the point r = 0
Note: as r  0; v  ∞
And its integral over ANY volume containing the point r = 0
Dr. Champak B. Das ( BITS, Pilani)

70. Dr. Champak B. Das ( BITS, Pilani)
THE DIRAC DELTA FUNCTION
Dr. Champak B. Das ( BITS, Pilani)

71. The Dirac Delta Function
An infinitely high, infinitesimally narrow “spike” with area 1
 Dirac Delta Function is NOT a Function
Dr. Champak B. Das ( BITS, Pilani)

72. A Generalized Function OR distribution
The Defining Characteristic Integral :
A Generalized Function OR distribution
Dr. Champak B. Das ( BITS, Pilani)

73. Dr. Champak B. Das ( BITS, Pilani)
Delta function is something that is always intended for use under an integral sign.
Let D1(x) & D2(x) are two expressions involving Delta functions and f(x) is any ordinary function
Dr. Champak B. Das ( BITS, Pilani)

74. Dr. Champak B. Das ( BITS, Pilani)
One can show:
………..for a proof, see Example 1.15
Dr. Champak B. Das ( BITS, Pilani)

75. Dr. Champak B. Das ( BITS, Pilani)
The Dirac Delta Function
Shifting the singularity from 0 to a;
Dr. Champak B. Das ( BITS, Pilani)

76. Dr. Champak B. Das ( BITS, Pilani)
The Dirac Delta Function
& the Defining Characteristic Integral :
Dr. Champak B. Das ( BITS, Pilani)

77. Dr. Champak B. Das ( BITS, Pilani)
Prob. 1.43:
Dr. Champak B. Das ( BITS, Pilani)

78. Dr. Champak B. Das ( BITS, Pilani)
Prob :
Dr. Champak B. Das ( BITS, Pilani)

79. Dr. Champak B. Das ( BITS, Pilani)
THE DIRAC DELTA FUNCTION
Dr. Champak B. Das ( BITS, Pilani)

80. Dr. Champak B. Das ( BITS, Pilani)
The Dirac Delta Function
Shifting the singularity from 0 to a;
Dr. Champak B. Das ( BITS, Pilani)

81. Dr. Champak B. Das ( BITS, Pilani)
The Dirac Delta Function
(in three dimension)
Dr. Champak B. Das ( BITS, Pilani)

82. Why such a function ?
Describe very short range forces as nuclear force
Describe a point particle in terms of a mass density
Describe a point charge in terms of a charge density
Dr. Champak B. Das ( BITS, Pilani)

83. Dr. Champak B. Das ( BITS, Pilani)
Prob. 1.46:
Charge density of a point charge q at r :
Charge density of a dipole with -q at 0 and +q at a:
Dr. Champak B. Das ( BITS, Pilani)

84. Dr. Champak B. Das ( BITS, Pilani)
Prob. 1.46: (contd.)
Charge density of a thin spherical shell of radius R and total charge Q:
Dr. Champak B. Das ( BITS, Pilani)

85. Dr. Champak B. Das ( BITS, Pilani)
The Paradox of Divergence of
From calculation of Divergence:
By using the Divergence theorem:
Dr. Champak B. Das ( BITS, Pilani)

86. Dr. Champak B. Das ( BITS, Pilani)
So now we can write:
Such that:
Dr. Champak B. Das ( BITS, Pilani)

87. Theory of Vector Fields
By specifying appropriate boundary conditions, Helmholtz theorem implies that the field can be uniquely determined from its divergence and curl.
Dr. Champak B. Das ( BITS, Pilani)

88. Dr. Champak B. Das ( BITS, Pilani)
Potentials
THEOREM 1:
( For Curl-less fields )
Dr. Champak B. Das ( BITS, Pilani)

89. Conclusions from theorem 1
If curl of a vector field vanishes, (everywhere), then the field can always be written as the gradient of a scalar potential
( not unique )
Dr. Champak B. Das ( BITS, Pilani)

90. Dr. Champak B. Das ( BITS, Pilani)
Potentials
THEOREM 2:
For Divergence-less fields
Dr. Champak B. Das ( BITS, Pilani)

91. Conclusions from theorem 2
If divergence of a vector field vanishes, (everywhere), then the field can always be written as the curl of a vector potential
( not unique )
Dr. Champak B. Das ( BITS, Pilani)

92. Dr. Champak B. Das ( BITS, Pilani)
Helmholtz theorem:
Any vector field F with both source and circulation densities vanishing at infinity may be written as the sum of two parts: one of which is curl-less and the other is divergence-less.
(Always)
Dr. Champak B. Das ( BITS, Pilani)