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ENTC 3331 RF Fundamentals Dr. Hugh Blanton ENTC 3331

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**Gradient, Divergence and Curl: the Basics**

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**We first consider the position vector, l:**

where x, y, and z are rectangular unit vectors. Dr. Blanton - ENTC Gradient, Divergence, & Curl

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

Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**The operator, del: Ñ is defined to be (in rectangular coordinates) as:**

This operator operates as a vector. Dr. Blanton - ENTC Gradient, Divergence, & Curl

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Gradient If the del operator, Ñ operates on a scalar function, f(x,y,z), we get the gradient: Dr. Blanton - ENTC Gradient, Divergence, & Curl

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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 Gradient, Divergence, & Curl

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**a T dl dT ˆ × Ñ = Directional derivatives: l**

Dr. Blanton - ENTC Gradient, Divergence, & Curl

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

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**Divergence 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. Dr. Blanton - ENTC Gradient, Divergence, & Curl

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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). Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Gauss' Theorem (Gaub’s Theorem**

We start with: Surface Areas Dr. Blanton - ENTC Gradient, Divergence, & Curl

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

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**For instance, consider the first term:**

The first part: gives the change in the x-component of A Dr. Blanton - ENTC Gradient, Divergence, & Curl

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

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

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

Dr. Blanton - ENTC Gradient, Divergence, & Curl

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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 Dr. Blanton - ENTC Gradient, Divergence, & Curl

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

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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 Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Now suppose they turn with a constant velocity**

diverges from original direction (degree of) divergence 0 Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Now suppose they turn and speed up.**

diverges from original direction (degree of) divergence >> 0 Dr. Blanton - ENTC Gradient, Divergence, & Curl

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** No divergence from original direction (degree of) divergence = 0**

Current of water No divergence from original direction (degree of) divergence = 0 Dr. Blanton - ENTC Gradient, Divergence, & Curl

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** Divergence from original direction (degree of) divergence ≠ 0**

Current of water Divergence from original direction (degree of) divergence ≠ 0 Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**E-field between two plates of a capacitor.**

+ E-field between two plates of a capacitor. Divergenceless Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**divergenceless solenoidal**

b-field inside a solenoid is homogeneous and divergenceless. divergenceless solenoidal Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 27**

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CURL

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**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 Gradient, Divergence, & Curl

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**The mathematical concept of circulation involves the curl operator.**

The curl acts on a vector and generates a vector. Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**In Cartesian coordinate system:**

Dr. Blanton - ENTC Gradient, Divergence, & Curl

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Example Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Important identities:**

for any scalar function V. Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Closed boundary of that surface.**

Stoke’s Theorem General mathematical theorem of Vector Analysis: Closed boundary of that surface. Any surface Dr. Blanton - ENTC Gradient, Divergence, & Curl

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Given a vector field Verify Stoke’s theorem for a segment of a cylindrical surface defined by: Dr. Blanton - ENTC Gradient, Divergence, & Curl

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z y x Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 37**

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 38**

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 39**

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 40**

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**Note that has only one component:**

Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**The integral of over the specified surface S is**

Dr. Blanton - ENTC Gradient, Divergence, & Curl

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

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z c d b y x a Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**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. Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 46**

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 47**

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 48**

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

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**Dr. Blanton - ENTC 3331 - Gradient, Divergence, & Curl 50**

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**curl or rot place paddle wheel in a river no rotation at the center**

rotation at the edges Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**the vector un is out of the screen right hand rule **

Ds is surface enclosed within loop closed line integral Dr. Blanton - ENTC Gradient, Divergence, & Curl

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**Electric Field Lines 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 Dr. Blanton - ENTC Gradient, Divergence, & Curl

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