EE3321 ELECTROMAGENTIC FIELD THEORY

EE3321 ELECTROMAGENTIC FIELD THEORYWeek 2Vector OperatorsDivergence and Stokes Theorems

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 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 .

Significance of GradientThe 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)

ExampleAs an example consider the gravitational potential on the surface of the Earth:V(z) = -gzwhere z is the height

The gradient of V would be V = z V/z = -g az

ExerciseConsider the gradient represented by the field of blue arrows. Draw level curves normal to the field.

ExerciseCalculate the gradient off = x2 + y2f = 2xyf = ex sin y

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

Laplacian OperatorThe 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 vector A is a vector.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

ExerciseCalculate the Laplacian of: f = sin 0.1x f = xyzf = cos( kxx ) cos( kyy ) sin( kzz )

10Curl OperatorThe 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.

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.

Line IntegralA line integral is an integral where the function is evaluated along a predetermined curve.

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.

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

Curl in Cartesian CoordinatesIn 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 .

ExerciseFind the curl of F = x ax + yz ay (x2 + z2) az.

17Divergence 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:

F

At every point in the field, the divergence is represented by a scalar.

Definition of Divergencewhere 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.

Surface IntegralIts the integral of a function f(x,y,z) taken over a surface.

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/4a3

First calculate = 4 Fo

Then calculate V = 4 a3/3

Significance of DivergenceThe divergence of a field is the extent to which the vector field flow behaves like a source at a given point.

Divergence in Cartesian CoordinatesIn 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 .

Exercise Determine the following:

divergence of F = 2x ax + 2y ay. divergence of the curl of F = 2x ax + 2y ay.

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.

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.

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 (4/3) b3Show that the flux of F coming out of the spherical surface is (4/3) b3

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:

Stokes Theorem transforms the circulation of the field into a line integral of the field over the contour that bounds the surface.

Significance of Stokes TheoremThe 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.

ExerciseConsider the rectangular surface shown below. Let F = y ax + x ay. Verify Stokes Theorem.

ABHomeworkRead 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