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whitepapers:sequences:boundary 2014/08/12 11:55 whitepapers:sequences:boundary 2014/08/12 13:34 current
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===== Boundary Conditions ===== ===== 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. [[courses:activities:vfact:vfgauss|Gauss's Law]] and [[courses:activities:vfact:vfampere|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. +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. 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. [[courses:activities:vfact:vfgauss|Gauss's Law]] and [[courses:activities:vfact:vfampere|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 ==== ==== Activities ====
-  * **[[courses:activities:vfact:vfebound|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 $\oint{\vec{E}\cdot d\vec{l}}=0$.+  * **[[courses:activities:vfact:vfebound|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 $\oint{\vec{E}\cdot d\vec{l}}=0$.
-  * **[[courses:activities:vfact:vfbbound|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.+  * **[[courses:activities:vfact:vfbbound|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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