Electromagnetic Field Theory (EMT) Lecture # 7 Vector Calculus (Continued)

Topics to be Covered: Vector Calculus Differential Length, Area, and Volume Line, Surface, and Volume Integrals Del Operator Gradient of a Scalar Divergence of a Vector and Divergence Theorem Curl of a Vector and Stokes's Theorem Laplacian of a Scalar Classification of Vector Fields

Laplacian of a Scalar Field A useful operator which is the composite of gradient and divergence operators The Laplacian of a scalar field V, written as 2 V, is the divergence of the gradient of V In Cartesian coordinates, Laplacian is: OR

Laplacian of a Scalar Field In cylindrical coordinates: In spherical coordinates:

Laplacian of a Scalar Field

Laplacian of a Vector Field A version of the Laplacian that operates on vector functions is known as the vector Laplacian, and a tensor Laplacian can be similarly defined.

Problem

Classification of Vector Fields A vector field is uniquely characterized by its divergence and curl. Neither the divergence nor curl of a vector field is sufficient to completely describe the field. All vector fields can be classified in terms of their vanishing or non-vanishing divergence or curl as follows:

Gradient of a Scalar The vector derivative of a scalar field f is called the gradient. The gradient of a scalar field V is a vector that represents both the magnitude and the direction of the maximum space rate of increase of V

MATLAB Section Example for Gradient: Assume that there exists a surface that can be modeled with the equation z = e x2 +y 2. Calculate z at the point (x = 0, y = 0). In addition, use MATLAB to illustrate the profile and to calculate and plot this field. Solution: 2xe x2 +y 2 a x 2ye x2 +y 2 a y Commands to Learn: 1) meshgrid 2) whitebg 3) surf 4) axis off and axis equal 5) colorbar, colormap, caxis 6) view 7) contour: contour(x,y,z,n) draw N contour lines 8) quiver 9) gradient

Divergence of a Vector Physical Interpretation of Divergence: The divergence operator is useful in determining if there is a source or a sink at a certain location in space in a region where a vector field exists. For electromagnetic fields, these sources and sinks will turn out to be positive and negative electrical charges. Example: Imagine a vector field represents water flow. Then if the divergence is a positive number, this means water is flowing out of the point (like a water spout - this location is considered a source). If the divergence is a negative number, then water is flowing into the point (like a water drain - this location is known as a sink).

Divergence of a Vector Physically, we may regard the divergence of the vector field A at a given point as a measure of how much the field diverges or emanates from that point (a) Positive (b) Negative (c) Zero

MATLAB Section Example for Divergence: Find the divergence of the two-dimensional vector field A = e r α 2 r, where r = xa x + ya y and r 2 = x 2 + y 2 by application of the MATLAB divergence function (α = const). Solution: A = xe r α 2 a x + ye r α 2 a y A = x xe r α 2 + x xe r α 2 Commands to Learn: 1) length 2) divergence

Curl of a Vector The curl is a vector operation that can be used to state whether there is a rotation associated with a vector field. This is visualized most easily by considering the experiment of inserting a small paddle wheel in a flowing river as shown in Figure below. If the paddle wheel is inserted in the center of the river, it will not rotate since the velocity of the water a small distance on either side of the center will be the same.

MATLAB Section Example for Curl: Find the curl of the two-dimensional vector field A = e r α 2 ω r, where r = xa x + ya y, r 2 = x 2 + y 2, α = constant and ω = ωa x by application of the MATLAB curl function. Solution: Commands to Learn: 1) curl A = xe r α 2 ω a x +ye r α 2 ω a y

MATLAB Section Example: The open surface =2.0 m and = 4.0 m, z = 3.0 m and z = 5.0 m, and =20 0 and = 60 0 identify a closed surface. Find a) the enclosed volume, b) the total area of the enclosed surface. Write a MATLAB program to verify your answers.