### 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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