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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?

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?

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