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?