Question

The index of refraction for violet light in silica flint glass is 1.66, and that for red light is 1.62. What is the angular spread of visible light passing through a prism of apex angle 60.0

Answer 1

theta1 is the angle of incidence
theta2 is the first angle of refraction
phi1 is the exiting angle of incidence
phi2 is the emerging angle
psi (the pitchfork at the top) is the apex angle

n1 is the external medium's (air's) refractive index
n2 is the prism medium's refractive index, which will depend on color of light later

First refraction model with Snell's law:
n1*sin(theta1) = n2*sin(theta2)

Second refraction model with Snell's law:
n2*sin(phi1) = n1*sin(phi2)

Geometry relation among angles within the glass:

Angles phi1 and theta1 along with a third angle form a triangle. This third angle is the obtuse angle between both normals (dotted lines). Call this third angle gamma.

Thus:
phi1 + theta1 + gamma = 180 deg

Angles psi, gamma, and two right angles form a quadrilateral.

Thus:
psi + gamma + 2*90deg = 360 deg
psi + gamma = 180 deg
gamma = 180 deg - psi

Substitute:
phi1 + theta1 + 180 deg - psi = 180 deg

Simplify:
phi1 + theta1 = psi

Summarize our system of equations:
n1*sin(theta1) = n2*sin(theta2)
n2*sin(phi1) = n1*sin(phi2)
phi1 + theta1 - psi = 90 deg

Solve our system for theta2, phi1 and phi2 (algebra not displayed):
phi1 = psi - theta1
theta2 = arcsin(n1*sin(theta1)/n2)
phi2 = arcsin(n2*sin(psi - theta1)/n1)

We are only really interested in phi2, the angle at which the light emerges:
phi2 = arcsin(n2*sin(psi - theta1)/n1)

For the red light:
phi2red = arcsin(n2red*sin(psi - theta1)/n1)

For the violet light:
phi2vio = arcsin(n2vio*sin(psi - theta1)/n1)

Data:
n2red:=1.62; n2vio:=1.66; n1:=1.0003; theta1:= 51deg; psi:=60 deg;

Results:
phi2red = 24.78 deg
phi2vio = 25.44 deg

The angular dispersion, difference in the two emerging angles:
phi2vio - phi2red = 0.6549 degrees

Equivalent to
phi2vio - phi2red = 39 arcmin