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THE GEOMETRY OF STATIC FIELDS
Corinne A. Manogue, Tevian Dray
Contents
Index
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Contents
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Front Matter
Colophon
Acknowledgments
Notation
1
Introduction
Philosophy
Learning Outcomes
2
Static Vector Fields — Prerequisites
Dimensions
Voltmeters
Computer Algebra
3
Coordinates and Vectors
Polar Coordinates
Curvilinear Coordinates
Change of Coordinates
Vectors
Bases
Unit Vectors
The Dot Product
Visualizing the Dot Product
The Law of Cosines
Addition Formulas
Orthonormal Basis Vectors
Polar basis vectors
Orthonormality of Basis Vectors
The Position Vector
The Position Vector in Curvilinear Coordinates
The Position Vector as a Vector Field
The Distance Formula
Scalar Fields
Vector Fields
The Cross Product
Lines and Planes
Linearity of the Dot and Cross Products
4
Potentials due to Discrete Sources
Electrostatic and Gravitational Potentials and Potential Energies
Superposition from Discrete Sources
Visualization of Potentials
Using Technology to Visualize Potentials
Two Point Charges
Series Expansions for Two Point Charges
Using Technology to Visualize Series Expansions
5
Differentials
Review of Single Variable Differentiation
Derivative Notation
Thick Derivatives
Differentials
Rules for Differentials
Properties of Differentials
Substitution
Differentials: Summary
The Multivariable Differential
6
The Vector Differential
The Vector Differential
\(d\rr\)
Finding
\(d\rr\)
on Rectangular Paths
Other Coordinate Systems
Calculating Infinitesimal Distance in Cylindrical and Spherical Coordinates
Calculating
\(d\rr\)
in Curvilinear Coordinates
Scalar Surface Elements
Triple Integrals in Cylindrical and Spherical Coordinates
Using
\(d\rr\)
on More General Paths
Use What You Know
7
Integration
Review of Single Variable Integration
Scalar Line Integrals
Vector Line Integrals
General Surface Elements
Vector Surface Elements
Scalar Surface Integrals
Volume Integrals
8
Delta Functions
Step Functions
The Dirac Delta Function
Properties of the Dirac Delta Function
Representations of the Dirac Delta Function
The Dirac Delta Function in Three Dimensions
The Exponential Representation of the Dirac Delta Function
9
Potentials due to Continuous Sources
Densities
Densities with Step Functions
Total Charge
The Dirac Delta Function and Densities
Potentials from Continuous Charge Distributions
Potential Due to a Uniformly Charged Ring
Potential due to a Finite Line of Charge
Potential due to an Infinite Line of Charge
10
Gradient
The Geometry of Gradient
The Gradient in Rectangular Coordinates
Properties of the Gradient
Visualizing the Geometry of the Gradient
Using Technology to Visualize the Gradient
Contour Diagrams
Directional Derivatives
The Gradient in Curvilinear Coordinates
11
Electric Fields
The Lorentz Force Law
Electric Field
Superposition for the Electric Field
The Geometry of Electric Fields
Using Technology to Visualize the Electric Field
Electric Fields from Continuous Charge Distributions
Electric Field Due to a Uniformly Charged Ring
The electric field of a uniform disk
12
Vector Line Integrals
Flux
Dot Products and Components
Highly Symmetric Surfaces
Less Symmetric Surfaces
13
Gauss's Law (Integral Form)
Flux of the Electric Field
Gauss' Law
Flux through a cube
Gauss's Law and Symmetry
Activity: Gauss's Law on Cylinders and Spheres
Electric Field Lines
14
Derivatives of Vector Fields
The Definition of Divergence
The Divergence in Two Dimensions
Exploring the Divergence
The Divergence in Curvilinear Coordinates
Exploring the Divergence in Polar Coordinates
Visualizing Divergence
The Divergence Theorem
The Geometry of Curl
The Definition of Curl
Exploring the Curl
The Curl in Curvilinear Coordinates
Exploring the Curl in Polar Coordinates
Visualizing Curl
Stokes' Theorem
Second derivatives
The Laplacian
15
Gauss's Law (Differential Form)
Differential Form of Gauss' Law
The Divergence of a Coulomb Field
16
Conservative Fields and Energy
The Relationship between
\(V\text{,}\)
\(\EE\text{,}\)
\(U\text{,}\)
and
\(\FF\)
Electrostatic Energy from Discrete Charges
Electrostatic Energy from a Continuous Source
Independence of Path
Conservative Vector Fields
Visualizing Conservative Vector Fields
Finding Potential Functions
Finding the Potential from the Electric Field
Curl-Free Vector Fields
Divergence-Free Vector Fields
Second derivatives and Maxwell's Equations
17
Current, Magnetic Potentials, and Magnetic Fields
Currents
Magnetic Vector Potential
The Biot–Savart Law
The Magnetic Field of a Straight Wire
Magnetic Field of a Spinning Ring
Comparing
\(\boldsymbol{\vec{B}}\)
and
\(\boldsymbol{\vec{A}}\)
for the spinning ring.
18
Ampère's Law
Ampère's Law
Current in a wire
Ampère's Law and Symmetry
Activity: Ampère's Law on Cylinders
Differential Form of Ampère's Law
19
Conductors and Boundary Conditions
Conductors
Boundary conditions on electric fields
Boundary conditions on magnetic fields
20
Review
The Relationship between
\(\EE\text{,}\)
\(V\text{,}\)
and
\(\rho\)
The Relationship between
\(\boldsymbol{\vec{B}}\text{,}\)
\(\boldsymbol{\vec{A}}\text{,}\)
and
\(\boldsymbol{\vec{J}}\)
Back Matter
A
Formulas
Formulas for Div, Grad, Curl
Product Rules
Integration by parts
B
Notation
Bibliography
Sections from Other Books
Index
Authored in PreTeXt
Section
19.1
Conductors
Please read Section 2.5 in Griffiths.