Notes 13 ECE 2317 Applied Electricity and Magnetism Prof. D. Wilton ECE Dept. Notes 13 Notes prepared by the EM group, University of Houston.

Divergence -- Physical Concept Start by considering a sphere of uniform volume charge density The electric field is calculated using Gauss's law: r < a: z v = v0 y r x a

Divergence -- Physical Concept (cont.) r > a: z v = v0 y a r x

Divergence -- Physical Concept (cont.) Flux through a spherical surface: (r < a) (r > a) (r < a) (r > a)

Divergence -- Physical Concept (cont.) Observation: More flux lines are added as the radius increases (as long as we stay inside the charge region). The net flux out of a small volume V inside the charge region is not zero. V S Divergence is a mathematical way of describing this.

Gauss’s Law -- Differential Form Definition of divergence: V Note: the limit exists independent of the shape of the volume (proven later).

Gauss’s Law -- Differential Form Apply divergence definition to small volume inside a region of charge V v (r)

Gauss’s Law -- Differential Form (cont.) Alternatively, The electric Gauss law: This is one of Maxwell’s equations.

Example v = v0 r Choose V to be small sphere of radius r: a z Verify that the differential form of Gauss’s law gives the correct result at the origin for the example of a sphere of uniform volume charge density. y r Choose V to be small sphere of radius r: x a

Calculation of Divergence x y z (0,0,0) x y z Assume point of interest is at the origin for simplicity. The integrals over the 6 faces are approximated by “sampling” the integrand at the centers of the faces.

Calculation of Divergence (cont.)

Calculation of Divergence (cont.) For arbitrary origin, just add x,y,z to coordinate quantities in parentheses! Hence

Calculation of Divergence (cont.) The divergence of a vector is its “flux per unit volume”

“del operator” The “del” operator is a vector differential operator Examples of derivative operators: scalar scalar -> scalar vector scalar -> vector vector->scalar vector->vector

Example Find V

“del operator” (cont.) Now consider: Hence so  Note: No unit vectors appear! Hence so

Summary of Divergence Formulas Rectangular: Cylindrical: Spherical: The divergence of a vector is its “flux per unit volume”

Example r < a : v = v0 r a Evaluate the divergence of the electric flux vector inside and outside a sphere of uniform volume charge density, and verify that the answer is what is expected from the electric Gauss law. r < a : z v = v0 y r x a Note: This agrees with the electric Gauss law.

Example (cont.) r > a : v = v0 a z y x Note: This agrees with the electric Gauss law.

Maxwell’s Equations Faraday’s law Ampere’s law electric Gauss law magnetic Gauss law

Divergence Theorem S V In words, for a vector : The volume integral of “flux per unit volume” equals the total flux!

Divergence Theorem (cont.) Proof: V rn is the center of cube n

Divergence Theorem (cont.) From the definition of divergence: Hence:

Divergence Theorem (cont.) Consider two adjacent cubes: is opposite on the two faces 1 2 Hence: the surface integral cancels on all INTERIOR faces.

Divergence Theorem (cont.) Hence: Therefore: (proof complete) The vol. integral of the “flux per unit volume” is the “flux”

Example Given: Verify the divergence theorem using the box. 1 [m] z Given: Verify the divergence theorem using the box. 1 [m] y 3 [m] 2 [m] x

Validity of Divergence Definition Is this limit independent of the shape of the volume? S r V From the divergence theorem: Hence, the limit is the same regardless of the shape of the limiting volume.

Gauss’s Law (Conversion between forms) Divergence theorem: S This is valid for any volume, so let V  V (a small volume inside the original volume) v V 0 Hence:

Gauss’s Law (Summary of two forms) Integral (volume) form of Gauss’s law Divergence definition Divergence theorem Differential (point) form of Gauss’s law