Question

A vessel contains water up to a height of 20 cm and above it an oil up to another 20 cm. The refractive indices of the water and the oil are 1.33 and 1.30 respectively. Find the apparent depth of the vessel when viewed from above.

Answer 1

The apparent depth of the vessel when viewed from above is 30.42 cm

Given, the refractive index of water is μ = 1.33, and depth of water, d = 20cm

So, shift due to water, t = (1 - 1/μ)*d

Substituting the values, we get t as,

t = (1 - 1/1.33)*20

= 4.96 cm

Again, the refractive index of oil is μ' = 1.30, and depth of oil, d' = 20cm

So, shift due to water, t' = (1 - 1/μ')*d'

Substituting the values, we get t' as,

t' = (1 - 1/1.30)*20

= 4.62 cm

Total shift = t + t' = 4.96cm + 4.62 cm

= 9.58cm

Total height = 20 cm + 20cm = 40cm

Thus, apparent depth = 40cm - 9.58cm

= 30.42 cm

This is the required answer.

Answer 2

The apparent depth of the vessel when viewed from above is 30.4 cm.

Explanation:

To find the apparent depth of the vessel :

Step 1:

Given data :

Water and oil refractive indices: 1.33 and 1.30

Water height  =20 cm

Oil height = 20 cm

Step 2:

Move due to water    $\Delta t w=\left(1-\frac{1}{\mu}\right) d$

                                          $=\left[1-\left(\frac{1}{1.33}\right)\right] \times 20$    

                                          $=\left[\left(\frac{1.33-1}{1.33}\right)\right] \times 20$

                                          $=\left[\left(\frac{0.33}{1.33}\right)\right] \times 20$

                                          = 0.248×20

                                          = 4.96 ≈ 5cm

Step 3:        

Shift due to oil :    

                              $=\left(1-\frac{1}{\mu}\right) d$

                              $=\left(1-\frac{1}{1.30}\right) \times 20$

                              $=20-\frac{20}{1.30}$

                              = 20- 15.3

                              = 4.6 cm

Total Change = 5 + 4.6 = 9.6 cm

Apparent depth = 40 - 9.6

                           = 30.4 cm.

Thus, the apparent depth of the vessel is 30.4 cm from the surface.