Electricity and Magnetism · Problem 10.14
MAGNETIC FIELDS IN MATTER — Edward Purcell & David Morin
Physics — E&M · boundary-conditions-linear-dielectrics
Problem
Find the boundary conditions on , , , and across the interface between two linear dielectrics. Assume that there are no free charges or free currents present.Intuitive explanation
The key is to apply Maxwell's equations at the boundary using Gauss's law and Faraday's law in integral form. For the electric field, imagine a tiny pillbox straddling the interface: the perpendicular component depends on bound surface charge (related to polarization), while the parallel component is continuous because there's no free charge to create a discontinuity. For the magnetic field, use a small loop at the boundary: the parallel component is continuous (no free current), and the perpendicular component must also be continuous since magnetic monopoles don't exist. The trick is recognizing which field components couple to which Maxwell equation.
Intuitive explanation
The key is to apply Maxwell's equations at the boundary using Gauss's law and Faraday's law in integral form. For the electric field, imagine a tiny pillbox straddling the interface: the perpendicular component depends on bound surface charge (related to polarization), while the parallel component is continuous because there's no free charge to create a discontinuity. For the magnetic field, use a small loop at the boundary: the parallel component is continuous (no free current), and the perpendicular component must also be continuous since magnetic monopoles don't exist. The trick is recognizing which field components couple to which Maxwell equation.