Question

A point source of light is placed at the centre of the bottom of a jar having a liquid of refractive index `5/3`. An opaque disc of radius 1.0 cm is placed on the liquid surface with its centre vertically above the source. What is the maximum height of the liquid for which the source is not visible from above?

Answer 1

Given:

Refractive index of liquid: μ = `5/3`​

Radius of opaque disc: r = 1.0 cm

To Find:

Maximum height h of the liquid such that no light escapes (i.e., total internal reflection occurs at all angles).

Concept:

Only the light rays within the critical angle can emerge.

To prevent any light from emerging, the opaque disc must cover the circular patch formed by rays at a critical angle.

Use geometry:

tan ⁡C = `r/h`

⇒ h = `r/(tan⁡C)`

Step 1: Find critical angle:

sin C = `1/μ`

sin C = `1/(5/3)`

sin C = `3/5`

sin C = 0.6

C = sin⁻¹ (0.6)

C ≈ 36.87°

Step 2: Apply geometry:

h = `r/(tan C)`

= `1.0/(tan (36.87°))`

= `1.0/0.75`

= `4/3` cm