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Dr. Hugh Blanton ENTC 3331

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ENTC 3331

RF Fundamentals

Dr. Hugh Blanton

ENTC 3331

Gradient, Divergence and Curl: the Basics

- We first consider the position vector, l:
- where x, y, and z are rectangular unit vectors.

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- Since the unit vectors for rectangular coordinates are constants, we have for dl:

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- The operator, del: Ñ is defined to be (in rectangular coordinates) as:
- This operator operates as a vector.

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- If the del operator, Ñ operates on a scalar function, f(x,y,z), we get the gradient:

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- We can interpret this gradient as a vector with the magnitude and direction of the maximum change of the function in space.
- We can relate the gradient to the differential change in the function:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 7

dT

=

Ñ

×

ˆ

T

a

l

dl

Directional derivatives:

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- Since the del operator should be treated as a vector, there are two ways for a vector to multiply another vector:
- dot product and
- cross product.

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- We first consider the dot product:
- The divergence of a vector is defined to be:
- This will not necessarily be true for other unit vectors in other coordinate systems.

- The divergence of a vector is defined to be:

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- To get some idea of what the divergence of a vector is, we consider Gauss' theorem (sometimes called the divergence theorem).

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- We start with:

Surface Areas

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- We can see that each term as written in the last expression gives the value of the change in vector A that cuts perpendicular through the surface.

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- For instance, consider the first term:
- The first part:
- gives the change in the x-component of A

- The first part:

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- The second part,
- gives the yz surface (or x component of the surface, Sx) where we define the direction of the surface vector as that direction that is perpendicular to its surface.

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- The other two terms give the change in the component of A that is perpendicular to the xz (Sy) and xy (Sz) surfaces.

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- We thus can write:
- where the vector S is the surface area vector.

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- Thus we see that the volume integral of the divergence of vector A is equal to the net amount of A that cuts through (or diverges from) the closed surface that surrounds the volume over which the volume integral is taken.
- Hence the name divergence for

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- So what?
- Divergence literally means to get farther apart from a line of path, or
- To turn or branch away from.

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Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles:

Goes straight ahead at constant velocity.

(degree of) divergence 0

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Now suppose they turn with a constant velocity

diverges from original direction

(degree of) divergence 0

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Now suppose they turn and speed up.

diverges from original direction

(degree of) divergence >> 0

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Current of water

No divergence from original direction

(degree of) divergence = 0

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Current of water

Divergence from original direction

(degree of) divergence ≠ 0

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 24

+

E-field between two plates of a capacitor.

Divergenceless

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I

b-field inside a solenoid is homogeneous and divergenceless.

divergenceless solenoidal

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CURL

+

+

- Two types of vector fields exists:

Electrostatic Field where the field lines are open and there is circulation of the field flux.

Magnetic Field where the field lines are closed and there is circulation of the field flux.

circulation (rotation) 0

circulation (rotation) = 0

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 29

- The mathematical concept of circulation involves the curl operator.
- The curl acts on a vector and generates a vector.

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- In Cartesian coordinate system:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 31

- Example

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 32

- Important identities:

for any scalar function V.

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 33

- General mathematical theorem of Vector Analysis:

Closed boundary of that surface.

Any surface

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 34

- Given a vector field
- Verify Stoke’s theorem for a segment of a cylindrical surface defined by:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 35

z

y

x

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- Note that has only one component:

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 41

The integral of over the specified surface S is

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 42

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 43

z

c

d

b

y

x

a

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 44

The surface S is bounded by contour C = abcd.

The direction of C is chosen so that it is compatible with the surface normal by the right hand rule.

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- place paddle wheel in a river
- no rotation at the center
- rotation at the edges

Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 51

- the vector un is out of the screen
- right hand rule
- Ds is surface enclosed within loop
- closed line integral

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Rules for Field Lines

- Electric field lines point to negative charges
- Electric field lines extend away from positive charges
- Equipotential (same voltage) lines are perpendicular to a line tangent of the electric field lines

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