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## The Differential Form of Maxwell's Equations

Having a thorough understanding of the differential operations provides students with a concrete means by which to interpret Maxwell's equations and transition from the integral to differential form of the equations. The sequence is split into two parts which separately address divergence and curl to build to the Gauss and Ampere Maxwell equations for both electric and magnetic fields. This sequence is assuming static fields and therefore does not build time dependence into Maxwell's equations.

Geometric understanding of divergence as a flux per unit volume allows students to physically interpret two of Maxwell's equations. For instance, the divergence of the electric field indicates the charge density at a particular location in the electric field by $\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0}$. In the absence of charge, the divergence of the electric field must be zero. Similarly, the divergence of the magnetic field, $\vec{\nabla}\cdot\vec{B}=0$, is always zero because there are no magnetic monopoles.

Geometric understanding of the curl as a circulation per unit area also provides a means for student to physically interpret two of Maxwell's equations. The curl of a static magnetic field is related to the current density by $\vec{\nabla} \times \vec{B} = \mu_0 \vec{J}$ which means the curl is non-zero at the location of a current density and zero where there is no current density. The corresponding equation for electric field states that the curl of a static electric field is zero, $\vec{\nabla}\times \vec{E} = 0$.

### Activities: Gauss's Law

**The Geometry of Flux Sequence***(Estimated time: 20-50 minutes)*: This sequence of activities emphasizes the geometry of flux through three activities and then has an activity where students use Gauss's law in integral form to find the electric field of various charge distributions. Understanding the geometry of flux allows students to understand and effectively use Gauss's law in integral form.

**Definition of Divergence Lecture***(Estimated time: 20 minutes)*: Divergence is introduced as a flux per unit volume at a specific point in a vector field.

**Visualizing Divergence***(Estimated time: 20 minutes)*: Students use a Mathematica notebook to look at various vector fields and use the idea of divergence as a flux per unit volume to determine whether the divergence at a specific point in the vector field is positive, negative, or zero. This activity emphasizes the geometry of divergence by engaging students in thinking about how divergence and flux are related.

**Proof of 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 Lecture**: At this point, following the activities and lectures involving divergence and the Divergence theorem, it is important for students to understand the significance of the differential form of Maxwell's equations which involve divergence. Students should understand $\vec{\nabla}\cdot\vec{E}=\frac{\rho}{\epsilon_0}$ as a way in which to determine a charge density in an electric field. The divergence of an electric field will be zero at a point with no charges, but it will be non-zero at a point where there is a charge density. Although the divergence of a magnetic field, $\vec{\nabla}\cdot\vec{B}=0$, was not explicitly explored in the preceding activities, this is a good point at which to discuss that zero divergence of a magnetic field means there are no magnetic monopoles.

### Activities: Ampere's Law

**Circulation Lecture**: Circulation is introduced.

**Visualizing Curl***(Estimated time: 20 minutes)*: Students use a Mathematica notebook to look at various vector fields to determine the sign of the curl of a vector field at a particular point. Students use the geometric concept of curl as circulation per unit area to determine whether the curl is negative, positive, or zero at a specific point in the vector field. Students should have some familiarity with curl in rectangular coordinates from an earlier brief lecture or from mathematics coursework prior to this activity

**Definition of Curl Lecture**: Curl is introduced as a circulation per unit area at a specific point in a vector field. This lecture serves as a thorough introduction to curl in rectangular, cylindrical, and spherical coordinates by building from the Visualizing Curl activity.

**Derivation of Stokes' Theorem Lecture**: Following “div, grad, curl and all that” by Schey, this lecture derives Stokes' theorem by

**Differential Form of Ampere's Law Lecture**: