Solution:

Given: Angle of incidence = 45 degree

To find: Minimum index of refraction of glass

Explanation:

When a light ray falls on a square slab at an angle of incidence, it gets refracted and bent at the surface of the slab. If the angle of incidence is greater than the critical angle, then total internal reflection takes place at the vertical surface of the slab.

Critical angle is defined as the angle of incidence at which the angle of refraction becomes 90 degrees.

Formula for critical angle: sin c = 1/n

where c is the critical angle and n is the refractive index of the medium.

If the angle of incidence is greater than the critical angle, then total internal reflection takes place.

Formula for angle of incidence: sin i = n2/n1 * sin r

where i is the angle of incidence, r is the angle of refraction, n1 is the refractive index of the medium in which the incident ray is travelling and n2 is the refractive index of the medium in which the refracted ray is travelling.

If we substitute the angle of incidence as 45 degrees, we get:

sin 45 = n2/n1 * sin r

1/sqrt(2) = n2/n1 * sin r

sin r = n1/(n2 * sqrt(2))

If we substitute the critical angle as 90 degrees, we get:

sin c = 1/n2

1/n2 = sin 90

n2 = 1

Substituting n2 = 1 in the above equation, we get:

sin r = n1/sqrt(2)

sin r = 1/sqrt(2)

r = 45 degrees

This means that if the angle of incidence is 45 degrees and total internal reflection takes place at the vertical surface of the slab, then the angle of refraction is 45 degrees.

Substituting the angle of refraction as 45 degrees in the formula for angle of incidence, we get:

sin 45 = n2/n1 * sin 45

1/sqrt(2) = n2/n1 * 1/sqrt(2)

n1 = n2 = 1

This means that the minimum index of refraction of glass should be equal to 1 in order for total internal reflection to take place at the vertical surface of the slab.

Conclusion:

The minimum index of refraction of glass should be equal to 1 in order for total internal reflection to take place at the vertical surface of the slab.