## Static Vector Fields

The Static Vector Field Paradigm continues the discussion of E&M from the Symmetries & Idealizations Paradigm, focusing on electric fields, magnetic fields, and the magnetic vector potential. This course uses a variety of pedagogical techniques (small group activities, computer visualization, kinesthetic activities, and lecture/discussion) to help students build a multifaceted understanding of these ideas. This course emphasizes extending the integral versions of Maxwell's equations (learned in introductory physics) to the local, differential versions; visualizing vector-valued functions in three dimensions using the computer algebra software Maple; and extending the techniques of vector calculus from rectangular to cylindrical and spherical coordinates. (Catalog Description)

### Course Goals

• For students to build conceptual and geometric understanding of current density, magnetic field, and magnetic vector potential and a formal understanding of the relationships between them (using vector calculus)
• For students to understand divergence and curl - formally and geometrically - and the Divergence Theorem and Stoke's Theorem formally and geometrically
• To derive the differential form of Maxwell's equations from the integral form and for students to have link their conceptual understanding with the formalism of Maxwell's equations
• For students to understand Gauss' Law and Ampere's Law and how to make explicit symmetry arguments.
• For students to understand the continuity of electric and magnetic fields across charge/current boundaries.
• For students to understand how energy is stored in electric and magnetic fields, and be able to calculate the energy from sources, fields and potentials.
• For students to come to understand that sources, fields, and potentials are different constructs that address the same phenomena, but are useful in different ways.

Sample Syllabus

Textbook: The Geometry of Vector Calculus—-an introduction to vector calculus, with applications to electromagnetism. One of the Tables of Contents for this online interactive textbook has been specifically designed for this course.

## Course Contents

### Unit: Gauss's Law

#### Gauss's Law (120 minutes)

• Gauss' Law (SGA ) 90 min - students solve for the electric field due to a charged sphere or an infinite cylinder. Emphasis is made on students making symmetry arguments (proof by contradiction) for using Gauss' Law.

#### Divergence (40 min)

• Definition of divergence (Lecture) 20 min
• Visualizing Divergence (Maple Visualization) 20 min Students practice estimating divergence from graphs of various vector fields.

#### Divergence Theorem (20 min)

• Reading: GVC § Divergence Theorem
• Derivation of the Divergence Theorem (lecture). We follow “div, grad, curl and all that”, by Schey. The Divergence theorem is almost a lemma based on the definition of divergence. Draw a diagram of an arbitrary volume divided into lots of little cubes. Calculate the sum of all the fluxes out of all the little cubes (isn't this a strange sum to consider!!) and argue that the flux out of one cube is the flux into the adjacent cube unless the cube is on the boundary.

#### Differential Form of Gauss's Law (10 min)

• Differential Form of Gauss's Law: Maxwell's Eq 1 & 3: $\Vec{\nabla} \cdot \Vec{E} = {\rho \over \epsilon_0}$, $\Vec{\nabla } \cdot \Vec{B} = 0$ (lecture)
• (optional) Divergence of a Coulomb field (requires delta functions) (lecture)
• (optional) Electric field lines (lecture)

### Unit: Current, Magnetic Vector Potential, and Magnetic Field

#### Vector Potentials (Optional)

• Reading: GVC § Magnetic Vector PotentialCurl
• Vector Potential A (lecture) 10 min max This can be just an analogy with electrostatic potential.
• Curl (at least the component definition in rectangular coordinates)

#### Magnetic Fields

• Derivation of the Biot-Savart Law from Magnetic Vector Potential (lecture) 15 min
• (optional) Comparing B and A for spinning ring (class discussion/lecture)

### Unit: Ampère's Law

#### Stokes' Theorem

• Reading: GVC § Stokes' Theorem
• Derivation of Stokes' Theorem (lecture). We follow “div, grad, curl and all that”, by Schey

#### Differential Form of Ampère's Law

• Stokes' Theorem (lecture) (Math 3.12: Stokes' Theorem)
• Differential Form of Ampère's Law: Maxwell Eq. 2 & 4 $\Vec{\nabla } \times \Vec{E} = 0$, $\Vec{\nabla } \times \Vec{B} = \mu_0 \Vec{J}$(lecture) (Physics 41: Differential Form of Ampère's Law)

### Unit: Conductors

#### Conductors (1 hr)

• Conductors (lecture)

### Unit: Conservative Fields

#### Conservative Fields

• Conservative Fields (lecture) (Math 3.5: Independence of Path, Math 3.6: Conservative Vector Fields, Math 3.7: Finding Potential Functions)
• Equivalent Statements (lecture)

### Unit: Energy

#### Energy for Continuous Distributions

• Energy for Continuous Distributions (lecture)

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