Question

In the figure, the optical fibre is l = 2 m long and has a diameter of d = 20 µm. If a ray of light is incident on one end of the fibre at angle θ₁ = 40°, the number of reflections it makes before emerging from the other end is close to:

(Refractive index of fibre is 1.31 and sin 40° = 0.64)

Options

• 55000

• 66000

• 45000

• 57000

Answer 1

57000

Explanation:

Given:

Optical fibre length, l = 2 m

Diameter of the fibre, d = 20 µm

Angle of incidence at the fiber’s entrance θ₁ = 40°

Refractive index of fibre, μ₂ = 1.31

To find:

The total number of reflections inside the fibre.

Calculation:

Using Snell’s law:

`mu_1/mu_2 = (sintheta_2)/(sin40^circ)`,   ...(The refractive index of air is µ₁.)

`sintheta_2 = 1/1.31 sin40^circ`

θ₂ = 29.39

θ = 90 − θ₂

= 90 − 29.39

= 60.6  ...(θ as shown in the diagram)

tan θ = tan 60.6 = `x/d`  ....(x is labelled in the diagram.)

x = (20 × 10⁻⁶) tan 60.6 = 35.5 µm

Number of total reflections;

n = `1/x`

= `(2m)/(35.5mum)`

= 56338

Therefore, the total number of reflections is approximately 56,338. Since this does not exactly match any option, we select the nearest available option, which is 57,000.