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**EE3321 ELECTROMAGENTIC FIELD THEORY**

Week 2 Vector Operators Divergence and Stoke’s Theorems

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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 .

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**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)

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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

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

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

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

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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.

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Applications The Laplacian quite 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

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**Exercise Calculate the Laplacian of: f = sin 0.1πx f = xyz**

f = cos( kxx ) cos( kyy ) sin( kzz )

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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.

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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.

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

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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.

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**Exercise Consider the field shown here.**

If we stick a paddle wheel in the first quadrant would it rotate? If so, in which direction?

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**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 .

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

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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.

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**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.

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

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Example Consider 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 π Fo Then calculate V = 4π a3/3

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**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.

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**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 .

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

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

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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.

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**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.

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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

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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.

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**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.

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**Exercise Consider the rectangular surface shown below.**

Let F = y ax + x ay. Verify Stoke’s Theorem. B A

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**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

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