A narrow beam of light containing orange (610 nm) and blue (470 nm) wavelengths goes from polystyrene to air, striking the surface at a 28.5° incident angle. What is the angle (in degrees) between the colors when they emerge?

The angle between the two colours when they emerge is 0.4°

Explanation:

The refracting angle for different colors knowing its refractive index can be calculated using Snell's law.

Ni × sin α = Nr × sinβ

Where Ni is the refractive index for light in incident medium

α is the angle the incident makes with normal

Nr is the refractive index for light in the refractive medium

β is the angle the refracted makes with the normal

Making β the subject

β = arcSin ( Ni × sinα)/Nr

For Orange color

The value of refractive index in polystyrene medium is 1.490 ,that is Ni= 1.490

α is 30° and Nr is the refractive index of air = 1

β = arcSin(1.490×sin30°)/1

β= 48.15°

For Blue color

The value of refractive index in polystyrene medium is 1.499,that is Ni= 1.499

α is 30° and Nr is the refractive index of air = 1

β = arcSin(1.499×sin30°)/1

β= 48.55°

The angle between the two colours is the difference in the angles of their refracted rays

48.55-48.15=0.4°

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west