Presentation on theme: "EE3321 ELECTROMAGENTIC FIELD THEORY"— Presentation transcript:
1 EE3321 ELECTROMAGENTIC FIELD THEORY Week 2Vector OperatorsDivergence and Stoke’s Theorems
2 Gradient OperatorThe 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 bygrad (f) = f = x ∂f/∂x + y ∂f/∂ + z ∂f/∂zThe 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 ExampleAs an example consider the gravitational potential on the surface of the Earth:V(z) = -gz where z is the heightThe gradient of V would be V = z ∂V/∂z = -g az
5 ExerciseConsider the gradient represented by the field of blue arrows. Draw level curves normal to the field.
6 ExerciseCalculate the gradient off = x2 + y2f = 2xyf = ex sin y
7 ExerciseConsider the surface z2 = 4(x2 + y2). Find a unit vector that is normal to the surface at P:(1, 0, 2).
8 Laplacian OperatorThe Laplacian of a scalar function f(x, y , z) is a scalar differential operator defined by2 f = [∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 ]fThe expression for the Laplacian operator in cylindrical and spherical coordinates can be found in the back cover of your textbook .The Laplacian of a vector A is a vector.
9 ApplicationsThe Laplacian quite important in electromagnetic field theory:It appears in Laplace's equation2 f = 0the Helmholtz differential equation2 f + k2 f = 0and the wave equation2 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 OperatorThe curl is a vector operator that describes the rotation of a vector field F: x FAt 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 Curlwhere 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 IntegralA line integral is an integral where the function is evaluated along a predetermined curve.
14 Significance of CurlThe 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 asThe expression for the curl in cylindrical and spherical coordinates can be found on the inside back cover of your textbook .
17 ExerciseFind the curl of F = x ax + yz ay – (x2 + z2) az.
18 Divergence OperatorThe divergence is a vector operator that describes the extent to which there is more “flux” exiting an infinitesimal region of space than entering it: · FAt 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 IntegralIt’s the integral of a function f(x,y,z) taken over a surface.
21 ExampleConsider a field F = Fo/r2 ar. 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/4a3First calculate = 4 π FoThen 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 asThe 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 TheoremThe 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 ExerciseConsider 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) b3Show that the flux of F coming out of the spherical surface is (4π/3) b3
28 Stokes' TheoremIt 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. Let F = y ax + x ay. Verify Stoke’s Theorem.BA