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Boundary Conditions

Knowing how electromagnetic fields change across boundaries is a common goal in undergraduate electricity and magnetism courses. In this sequence, students explore how the components of electric and magnetic fields act at a boundary of a sheet of charge and a sheet of current respectively. 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. Gauss's Law and Ampere's Law activities can be used to provide a foundation in the mathematics and symmetry arguments used with these 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

  • Electric Field Continuity Across a Boundary (Estimated time: 10-20 minutes): Students use Ampere's and Gauss's laws to find the continuity conditions for the electric field's parallel and perpendicular components across a planar boundary with surface charge, $\sigma$. 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 continuity conditions for the magnetic field's parallel and perpendicular components across the planar boundary carrying surface current, $\vec{K}$. 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.

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