# Condition of continuity of E field at a boundary

• Gen1111
In summary, the continuity condition for Snell's Law is that the E field tangential to the interface must be continuous. This is because the area for the integral of the E field across the interface shrinks to zero, and the shape of the box that is used to integrate the E field across the interface is the key to ensuring that the line integral becomes zero even if the rotE field is non-zero. If the geometry of the interface is no longer flat, the parallel component of the E field strength will still be continuous, but the phase of the wave will be different depending on the amount of curvature in the interface.
Gen1111
I am trying to understand the derivation of Snell's law using Maxwell's equation and got stuck.

My textbook 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?

Last edited:
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.

Last edited:
Meir Achuz said:
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?

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?

Gen1111 said:
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.

## 1. What is the condition of continuity of electric field at a boundary?

The condition of continuity of electric field at a boundary states that the tangential component of the electric field must be continuous across the boundary between two different materials. This means that the magnitude and direction of the electric field must remain the same as it crosses the boundary.

## 2. Why is the condition of continuity of electric field important?

This condition is important because it ensures that there are no sudden discontinuities or breaks in the electric field at the boundary. This is necessary for the conservation of energy and for the correct calculation of electric field values in a given region.

## 3. How is the condition of continuity of electric field related to Gauss's Law?

The condition of continuity of electric field is related to Gauss's Law, which states that the net electric flux through a closed surface is equal to the charge enclosed by that surface. The continuity of electric field ensures that the electric flux remains constant as it crosses a boundary, allowing for the accurate application of Gauss's Law.

## 4. Does the condition of continuity of electric field apply to all boundaries?

No, the condition of continuity of electric field only applies to boundaries between two different materials. Boundaries between the same material do not require the continuity of electric field, as the electric field will naturally be continuous within the same material.

## 5. What happens if the condition of continuity of electric field is not met at a boundary?

If the condition of continuity of electric field is not met at a boundary, it can result in sudden changes in the electric field, which can lead to inaccuracies in calculations and violate the conservation of energy. This can also cause unexpected behaviors in electric fields, such as reflections or refractions, at the boundary.

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