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## Magnetic Field Continuity Across a Boundary: Instructor's Guide

### Main Ideas

Using Ampere's Law and Gauss's Law to find the magnetic field just above and just below an arbitrary plane with surface current density $\Vec K$.

### Students' Task

*Estimated Time: 10 - 20 minutes*

### Prerequisite Knowledge

A knowledge of Ampere's and Gauss's Law

Right hand rule as it applies to line currents

### Props/Equipment

- Tabletop Whiteboard with markers
- A handout for each student

### Activity: Introduction

Students are given a generic plane with some constant surface current density of $\Vec K$. They will be charged with finding and comparing the magnetic field components as they cross the surface boundary. The tools used for this particular job are the integral forms of the Maxwell's equations.

Find the discontinuity in the magnetic field parallel to the surface and parallel to the current, denoted as $B_{\parallel \, \parallel}$

Find the discontinuity in the magnetic field parallel to the surface and perpendicular to the current, denoted as $B_{\parallel \, \perp}$

Find the discontinuity in the magnetic field perpendicular to the surface, denoted as $B_{\perp}$

### Activity: Student Conversations

**Orientation of parallel magnetic field components**–Students may not be able immediately visualize the two orientations of the magnetic field to consider when discussing the magnetic field parallel to the surface.**Ampere's law and Gauss's law**–This activity requires using Ampere's law to show the discontinuity of the parallel components as well as a “Gaussian box” to show the continuity of perpendicular components of magnetic field across the boundary. While it is likely expected to use Ampere's law as it is in the context of magnetic field and a sheet of current, it may not be clear to students why $\oint{\vec{B}\cdot d\vec{a}}=0$ is used for the perpendicular components.**Name the thing you don't know**–If students have completed Electric Field Across a Boundary, some of the difficulties students may have in naming have already been addressed. In addition to the difficulties students have outlined in that activity, there are now two parallel to the surface components: one perpendicular to the current, and one parallel to the current. Students may have difficulty recognizing and naming these components.

### Activity: Wrap-up

### Extensions

This activity is part of a sequence addressing Boundary Conditions in electrostatics and magnetostatics. The other activity which is included in this sequence follows.

- Electric Field Continuity Across a Boundary: Students use Ampere's and Gauss's laws to find the continuity conditions for the electric field's parallel and perpendicular components across the planar boundary with surface charge density, $\sigma$.