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Boundary Conditions

Knowing how electromagnetic fields change across boundaries is a common goal in undergraduate electricity and magnetism courses.

This sequence follows the derivations of boundary conditions found in the Griffiths text–sections 2.3.5 and 5.4.2 for electrostatics and magnetostatics respectively. In order to understand this approach to determining boundary conditions, students must be able to use Gauss's and Ampere's laws. Because the boundary is a sheet with zero thickness, determining the boundary conditions requires taking a limit as the Gaussian surfaces and Amperian loops approach zero thickness.

Activities

  • Gauss's Law (Estimated time: 60 minutes): This small group activity has students use Gauss's law to find the electric field for a cylindrically or spherically symmetric charge density. Students are asked to make explicit symmetry arguments which make use of Proof by Contradiction.
  • Ampere's Law (Estimated time: 45 minutes): Students use Ampere's law in this small group activity to find the magnetic field due to a radially dependent current density in an infinitely long cylindrical shell.
  • Electric Field Continuity Across a Boundary (Estimated time: 10-20 minutes): Students use Ampere's and Gauss's laws to find the electric field just above and just below a plane which has a surface charge density $\sigma$. They find the continuity conditions for the electric field's parallel and perpendicular components across the planar boundary. Gauss's law is used to determine the discontinuity, $\frac{\sigma}{\epsilon_0}$, of the normal component of electric field. Similarly, an Amperian-like loop is used to determine the continuity of the tangential component of electric field by $\oint{\vec{E}\cdot d\vec{l}}=0$.
  • Magnetic Field Continuity Across a Boundary (Estimated time: 10-20 minutes): Students use Ampere's and Gauss's laws to find the magnetic field just above and just below a plane which has a surface current, $\vec{K}$. The students find the continuity conditions for the magnetic field's parallel and perpendicular components across the planar boundary. Ampere's law is used find the two boundary conditions for magnetic field: the component parallel to current is continuous, and the component parallel to the surface but perpendicular to the current has a discontinuity, $\mu_0 K$. Additionally, an analogous form of Gauss's law, $\oint{\vec{B}\cdot d\vec{a}}=0$ is used to determine the continuity of the normal components of the magnetic field across the boundary.

FIXME Add in homework and relationship to limits (going from the Gauss & Ampere activities to a planar boundary)


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