Skip to content Skip to navigation

Connexions

You are here: Home » Content » Electromagnetism at an Interface

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Recently Viewed

This feature requires Javascript to be enabled.

Electromagnetism at an Interface

Module by: Paul Padley

Summary: We derive the properties of E and B fields at an interface between two media.

EM at an interface

We want to understand with Electromagnetism what happens at a surface. From Maxwell's equations we can understand what happens to the components of the E ⃗ E ⃗ and B ⃗ B ⃗ fields: First lets look at the E ⃗ E ⃗ field using Gauss' law. Recall ∮ ε E ⃗ ⋅ d s ⃗ = ∫ ρ ⅆ V ∮ ε E ⃗ ⋅ d s ⃗ = ∫ ρ ⅆ V

Figure 1
Figure 1 (Gauss-E-at-interface_small.png)
Consider the diagram, the field on the incident side is E i ⃗ + E r ⃗ E i ⃗ + E r ⃗ . On the transmission side, the field is E t ⃗ E t ⃗ . We can collapse the cylinder down so that it is a pancake with an infinitely small height. When we do this there are no field lines through the side of the cylinder. Thus there is only a flux through the top and the bottom of the cylinder and we have; ∮ ε E ⃗ ⋅ d s ⃗ = ∮ [ ε i ( E i ⊥ + E r ⊥ ) − ε t E t ⊥ ] ⅆ s = 0. ∮ ε E ⃗ ⋅ d s ⃗ = ∮ [ ε i ( E i ⊥ + E r ⊥ ) − ε t E t ⊥ ] ⅆ s = 0. I have set ∫ ρ ⅆ V = 0 ∫ ρ ⅆ V = 0 since we will only consider cases without free charges. So we have ε i E i ⊥ + ε i E r ⊥ = ε t E t ⊥ ε i E i ⊥ + ε i E r ⊥ = ε t E t ⊥ if u ̂ n u ̂ n is a unit vector normal to the surface this can be written ε i u ̂ n ⋅ E ⃗ i + ε i u ̂ n ⋅ E ⃗ r = ε t u ̂ n ⋅ E ⃗ t ε i u ̂ n ⋅ E ⃗ i + ε i u ̂ n ⋅ E ⃗ r = ε t u ̂ n ⋅ E ⃗ t

Similarly Gauss' law of Magnetism ∮ B ⃗ ⋅ d s ⃗ = 0 ∮ B ⃗ ⋅ d s ⃗ = 0 gives B i ⊥ + B r ⊥ = B t ⊥ B i ⊥ + B r ⊥ = B t ⊥ or u ̂ n ⋅ B ⃗ i + u ̂ n ⋅ B ⃗ r = u ̂ n ⋅ B ⃗ t u ̂ n ⋅ B ⃗ i + u ̂ n ⋅ B ⃗ r = u ̂ n ⋅ B ⃗ t

Amperes law can also be applied to an interface.Then

Figure 2
Figure 2 (Ampere-B-at-interface_small.png)
∫ B ⃗ μ ⋅ d l ⃗ = ∫ j ⃗ ⋅ d s ⃗ + ⅆ ⅆ t ∫ ε E ⃗ ⋅ d s ⃗ ∫ B ⃗ μ ⋅ d l ⃗ = ∫ j ⃗ ⋅ d s ⃗ + ⅆ ⅆ t ∫ ε E ⃗ ⋅ d s ⃗ (note that in this case d s ⃗ d s ⃗ is perpendicular to the page)

Now we will not consider cases with surface currents. Also we can shrink the vertical ends of the loop so that the area of the box is 0 so that ∫ ε E ⃗ ⋅ d s ⃗ = 0 ∫ ε E ⃗ ⋅ d s ⃗ = 0 . Thus we get at a surface B i ∥ + B r ∥ μ i = B t ∥ μ t B i ∥ + B r ∥ μ i = B t ∥ μ t or u ̂ n × B ⃗ i μ i + u ̂ n × B ⃗ r μ r = u ̂ n × B ⃗ t μ t u ̂ n × B ⃗ i μ i + u ̂ n × B ⃗ r μ r = u ̂ n × B ⃗ t μ t

Similarly we can use Faraday's law ∫ E ⃗ ⋅ d l ⃗ = − ⅆ ⅆ t ∫ B ⃗ ⋅ d s ⃗ ∫ E ⃗ ⋅ d l ⃗ = − ⅆ ⅆ t ∫ B ⃗ ⋅ d s ⃗ and play the same game with the edges to get E i ∥ + E r ∥ = E t ∥ E i ∥ + E r ∥ = E t ∥ or u ̂ n × E ⃗ i + u ̂ n × E ⃗ r = u ̂ n × E ⃗ t u ̂ n × E ⃗ i + u ̂ n × E ⃗ r = u ̂ n × E ⃗ t (notice ε ε does not appear)

In summary we have derived what happens to the E ⃗ E ⃗ and B ⃗ B ⃗ fields at the interface between two media: ε i u ̂ n ⋅ E ⃗ i + ε i u ̂ n ⋅ E ⃗ r = ε t u ̂ n ⋅ E ⃗ t ε i u ̂ n ⋅ E ⃗ i + ε i u ̂ n ⋅ E ⃗ r = ε t u ̂ n ⋅ E ⃗ t

u ̂ n ⋅ B ⃗ i + u ̂ n ⋅ B ⃗ r = u ̂ n ⋅ B ⃗ t u ̂ n ⋅ B ⃗ i + u ̂ n ⋅ B ⃗ r = u ̂ n ⋅ B ⃗ t

u ̂ n × B ⃗ i μ i + u ̂ n × B ⃗ r μ i = u ̂ n × B ⃗ t μ t u ̂ n × B ⃗ i μ i + u ̂ n × B ⃗ r μ i = u ̂ n × B ⃗ t μ t

u ̂ n × E ⃗ i + u ̂ n × E ⃗ r = u ̂ n × E ⃗ t u ̂ n × E ⃗ i + u ̂ n × E ⃗ r = u ̂ n × E ⃗ t

Comments, questions, feedback, criticisms?

Send feedback