Question

In my book it says we can use the boundary conditions of electromagnetic waves to derive the refraction law of light. How to derive it?

A) If you have a traveling wave, incident at an angle θ
to the normal to the boundary, then you can express the E-field of the wave as
E⃗ (r⃗ ,t)=E0exp[i(k⃗ ⋅r⃗ −ωt)]u^

B) The form of the transmitted and reflected wave can be written in the same format, but with (potentially) different values for k⃗
and ω.

C) From there, it is easier to use an example - suppose the boundary is the plane y=0
and the normal to the boundary is the y-axis and the incident wave travels in the x−y
plane and has k⃗ =k(sinθ i^+cosθ j^)
and u^=k^

At the boundary, where r⃗ =xi^
then the general form of the incident E-field written above becomes
E⃗ (r⃗ ,t)=E0exp[i(kxsinθ−ωt)]k^ ,
with similar expressions for the reflected and transmitted E-fields, but written with different values of k
, θ
and ω
But the continuity condition that says the tangential component of the E-field must be the same either side of the boundary means that
E⃗ i+E⃗ r=E⃗ t .

D) if this is hold true for any value of x
or t
along the boundary, then the spatial and temporal parts of the arguments of the exponential functions for the incident, reflected and transmitted waves have to be the same.

This leads to the equality
kisinθi=ktsinθt ,
where the subscripts refer to incident and transmitted waves, and
ωi=ωt .
Now, because the speed of light, c/n=ω/k
, where n
is the refractive index, then if ω
is constant, one obtains
nisinθi=ntsinθt .

Answer 1

Snell's law, also known as the law of refraction, explains how light changes direction when it passes from one medium to another with different refractive indices.

Step-by-step explanation:

The law of refraction, also known as Snell's law, explains how light changes direction when it travels across a boundary between two transparent materials with different refractive indices.

When light passes from one medium to another, each wavelet emitted by the wavefront changes direction because the speed of light is slower in the second medium. This change in direction causes the light ray to bend closer to the perpendicular. Snell's law can be derived from the geometry of the wavefront crossing the interface between the media.