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-   -   Condition of continuity of E field at a boundary (http://www.physicsforums.com/showthread.php?t=661331)

 Gen1111 Dec28-12 08:40 PM

Condition of continuity of E field at a boundary

I am trying to understand the derivation of Snell's law using Maxwell's equation and got stuck.

My text book says that "the E field that is tangent to the interface must be continuous" in order to consider refraction of light.
If it were static E field I understand this is true because in electrostatics

rotE = 0

holds. However Snell's law describes how electromagnetic waves change their direction of propagation when going through an interface of two mediums. Since our E filed is changing dynamically, we should use the equation

rotE = -∂B/∂t

in stead. To me it is not obvious why this equation leads to the continuity condition.
How does the continuity condition in Snell's law appears from Maxwell's equations?

 Simon Bridge Dec28-12 10:07 PM

Re: Condition of continuity of E field at a boundary

B also has boundary conditions.
www.cem.msu.edu/~cem835/Lecture03.pdf

 Meir Achuz Dec29-12 07:15 AM

Re: Condition of continuity of E field at a boundary

The continuity of E tangential comes from applying Stokes' theorem to rotE = -∂B/∂t.
The area for $$\int{\bf dS}\partial_t{\bf B}$$ shrinks to zero.

 Gen1111 Dec29-12 11:25 AM

Re: Condition of continuity of E field at a boundary

Quote:
 Quote by Meir Achuz (Post 4211685) The continuity of E tangential comes from applying Stokes' theorem to rotE = -∂B/∂t. The area for $$\int{\bf dS}\partial_t{\bf B}$$ shrinks to zero.
Stokes's theorem for rotE is
$$\int{ rot\bf{E}}{\bf dS}= \oint _{∂S} \bf Edx = \oint _{∂S} \bf \partial_t B dx$$

How does this lead to the continuity condition?

 jtbell Dec29-12 03:34 PM

Re: Condition of continuity of E field at a boundary

http://farside.ph.utexas.edu/teachin...es/node59.html

Note the left-hand portion of the diagram at the top of the page, and start reading around equation (635).

 Gen1111 Dec30-12 05:11 PM

Re: Condition of continuity of E field at a boundary

OK I see. It seems like the continuity condition is something to do with the fact that the interface has zero volume and the planar surface is sufficiently large.
The path of line integration must be an infinitely thin rectangular when the area of the box approaches to 0.
The shape of the box is the key because it will allow the line integration to become 0 even if rotE is non-zero.
Thanks for all the replies.

Will the same rule apply if there is a gradient layer between the two phases?
Let's say the geometry is no longer flat but curved, and the curvature of radius is comparable to the thickness of gradient layer.
I'm pretty sure that the parallel component of the E field strength will still be continuous at any point.
But will the phase still be the same?

 Simon Bridge Jan1-13 08:33 PM

Re: Condition of continuity of E field at a boundary

Quote:
 Quote by Gen1111 (Post 4213256) Will the same rule apply if there is a gradient layer between the two phases? Let's say the geometry is no longer flat but curved, and the curvature of radius is comparable to the thickness of gradient layer. I'm pretty sure that the parallel component of the E field strength will still be continuous at any point. But will the phase still be the same?
You know what the answer has to be already - what usually happens to Snell's Law when the surface is curved or the interface is not sharp?

You could try working it out for a simple setup - like a spherical interface (par-axial) - and see if the general boundary conditions give you the appropriate equations.

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