EE 3321 ELECTROMAGENTIC FIELD THEORY Week 2 Vector

EE 3321 ELECTROMAGENTIC FIELD THEORY Week 2 Vector Operators Divergence and Stoke’s Theorems

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.

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)

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

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

Exercise �Calculate the gradient of �f = x 2 + y 2 �f = 2 xy �f = ex sin y

Exercise �Consider the surface z 2 = 4(x 2 + y 2). Find a unit vector that is normal to the surface at P: (1, 0, 2).

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 vector A is a vector.

Applications �The Laplacian quite important in electromagnetic field theory: �It appears in Laplace's equation 2 f = 0 � �the Helmholtz differential equation 2 f + k 2 f = 0 � �and the wave equation 2 f = (1/c)2 ∂2 f/∂x 2 �

Exercise �Calculate the Laplacian of: �f = sin 0. 1πx �f = xyz �f = cos( kxx ) cos( kyy ) sin( kzz )

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.

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.

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

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.

Exercise �Consider 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 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.

Exercise �Find the curl of F = x ax + yz ay – (x 2 + z 2) az.

Divergence Operator �The 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 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.

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

Example �Consider a field F = Fo/r 2 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 = 3 F o/4 a 3 �First calculate �Then calculate = 4 π Fo V = 4π a 3/3

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.

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.

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

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) b 3 �Show that the flux of F coming out of the spherical surface is (4π/3) b 3

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.

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.

Exercise �Consider the rectangular surface shown below. �Let F = y ax + x ay. Verify Stoke’s Theorem. B A

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