EE3321 ELECTROMAGENTIC FIELD THEORY

1. EE3321 ELECTROMAGENTIC FIELD THEORY Week 2 Vector Operators Divergence and Stoke’s Theorems

2. Gradient Operator • The gradient is a vector operator denoted and sometimes also called “del.” It is most often applied to a real function of three variables. • In Cartesian coordinates, the gradient of f(x, y, z) is given by grad (f) =  f = x ∂f/∂x + y ∂f/∂ + z ∂f/∂z • The expression for the gradient in cylindrical and spherical coordinates can be found on the inside back cover of your textbook .

3. Significance of Gradient • The direction of grad(f) is the orientation in which the directional derivative has the largest value and |grad(f)| is the value of that directional derivative. • Furthermore, if grad(f) ≠ 0, then the gradient is perpendicular to the “level” curve z = f(x,y)

4. Example • As an example consider the gravitational potential on the surface of the Earth: V(z) = -gz where z is the height • The gradient of V would be  V = z ∂V/∂z = -g az

5. Exercise • Consider the gradient represented by the field of blue arrows. Draw level curves normal to the field.

6. Exercise • Calculate the gradient of • f = x2 + y2 • f = 2xy • f = ex sin y

7. Exercise • Consider the surface z2 = 4(x2 + y2). Find a unit vector that is normal to the surface at P:(1, 0, 2).

8. Laplacian Operator • The Laplacian of a scalar function f(x, y , z) is a scalar differential operator defined by • 2 f = [∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 ]f • The expression for the Laplacian operator in cylindrical and spherical coordinates can be found in the back cover of your textbook . • The Laplacian of a vectorA is a vector.

9. Applications • The Laplacianquite important in electromagnetic field theory: • It appears in Laplace's equation 2 f = 0 • the Helmholtz differential equation 2 f + k2 f = 0 • and the wave equation 2 f = (1/c)2∂2 f/∂x2

10. Exercise • Calculate the Laplacian of: • f = sin 0.1πx • f = xyz • f = cos( kxx ) cos( kyy ) sin( kzz )

11. Curl Operator • The curl is a vector operator that describes the rotation of a vector field F:  x F • At every point in the field, the curl is represented by a vector. • The direction of the curl is the axis of rotation, as determined by the right-hand rule. • The magnitude of the curl is the magnitude of rotation.

12. Definition of Curl where the right side is a line integral around an infinitesimal region of area A that is allowed to shrink to zero via a limiting process and n is the unit normal vector to this region.

13. Line Integral • A line integral is an integral where the function is evaluated along a predetermined curve.

14. Significance of Curl • The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space.

15. Exercise • Consider the field shown here. • If we stick a paddle wheel in the first quadrant would it rotate? • If so, in which direction?

16. Curl in Cartesian Coordinates • In practice, the curl is computed as • The expression for the curl in cylindrical and spherical coordinates can be found on the inside back cover of your textbook .

17. Exercise • Find the curl of F = x ax + yzay – (x2 + z2) az.

18. Divergence Operator • The divergenceis a vector operator that describes the extent to which there is more “flux” exiting an infinitesimal region of space than entering it:  ·F • At every point in the field, the divergence is represented by a scalar.

19. Definition of Divergence where the surface integral is over a closed infinitesimal boundary surface A surrounding a volume element V, which is taken to size zero using a limiting process.

20. Surface Integral • It’s the integral of a function f(x,y,z) taken over a surface.

21. Example • Consider a field F = Fo/r2ar. Show that the ratio of the flux coming out of a spherical surface of radius r=a to the volume of the same sphere is = 3Fo/4a3 • First calculate = 4 πFo • Then calculate V = 4π a3/3

22. Significance of Divergence • The divergence of a field is the extent to which the vector field flow behaves like a source at a given point.

23. Divergence in Cartesian Coordinates • In practice the divergence is computed as • The expression for the divergence in cylindrical and spherical coordinates can be found on the inside back cover of your textbook .

24. Exercise • Determine the following: • divergence of F = 2x ax + 2y ay. • divergence of the curl of F = 2x ax + 2y ay.

25. Divergence Theorem • The volume integral of the divergence of F is equal to the flux coming out of the surface A enclosing the selected volume V : • The divergence theorem transforms the volume integral of the divergence into a surface integral of the net outward flux through a closed surface surrounding the volume.

26. Example • Consider the “finite volume” electric charge shown here. • The divergence theorem can be used to calculate the net flux outward and the amount of charge in the volume. • Requirement: the field must be continuous in the volume enclosed by the surface considered.

27. Exercise • Consider a spherical surface of radius r = b and a field F = (r/3) ar. • Show that the divergence of F is 1. • Show that the volume integral of the divergence is (4π/3) b3 • Show that the flux of F coming out of the spherical surface is (4π/3) b3

28. Stokes' Theorem • It states that the area integral of the curl of F over a surface A is equal to the closed line integral of F over the path C that encloses A: • Stoke’s Theorem transforms the circulation of the field into a line integral of the field over the contour that bounds the surface.

29. Significance of Stoke’s Theorem • The integral is a sum of circulation differentials. • The circulation differential is defined as the dot product of the curl and the surface area differential over which it is measured.

30. Exercise • Consider the rectangular surface shown below. • LetF = y ax + x ay. Verify Stoke’s Theorem. B A

31. Homework • Read book sections 3-3, 3-4, 3-5, 3-6, and 3-7. • Solve end-of-chapter problems • 3.32, 3.35, 3.49, 3.39, 3.41, 3.43, 3.45, 3.48