Angle of Refraction: Prove Incident Beam is Parallel to Emerging Beam

[Ronnin]

A light beam passes from medium 1 to medium 2 with the latter being a thick slab of material whose index of refraction is n2. Show that the emerging beam is parallel to incident beam.

Am I correct in assuming that the only way this is possible is to have the incident beam making an angle that is 90 degrees with medium 2? Can someone confirm my thinking.

 

[chaoseverlasting]

If there is only one refraction happening, then yeah. If there are two refractions (if its a slab or something so that the light ray emerges back into medium 1) then it will always be parallel to the incident ray.

Mathematically,

$$\mu sinx=k$$ where k is a constant, so for sinx1=sinx2, n1=n2. Since n1 & n2 arent the same, the only angle for which the equality holds is 0, which means that the ray falls normally on surface 2.

 

[m2003]

Yes, your thinking is correct. According to Snell's Law, the angle of refraction (θ2) is related to the angle of incidence (θ1) by the equation n1sinθ1 = n2sinθ2, where n1 and n2 are the indices of refraction of the two media. When θ1 is 90 degrees, sinθ1 = 1 and therefore n1sinθ1 = n1, making the equation n1 = n2sinθ2. This means that the emerging beam will be parallel to the incident beam, as sinθ2 = 0 when θ2 = 0 (meaning the beam is not refracted at all). This is only possible when the incident beam makes an angle of 90 degrees with medium 2. Therefore, in order for the emerging beam to be parallel to the incident beam, the incident beam must be at a 90 degree angle with the second medium.