Name: Snell's Window
Uploaded: 2019-07-19
Duration: 4 min 53 s
Description: Snell's window is a phenomenon by which an underwater viewer sees everything above the surface through a cone of light of width of about 96 degrees. Let's le...

Question

There is a small air bubble inside a glass sphere $(µ = 1.5)$ of radius 5 cm. The bubble is at $O$ at $7.5 cm$ below the surface of the glass. The sphere is placed inside water $(µ = \frac{4}{3})$ such that the top surface of glass is $10 cm$ below the surface of water. The bubble is viewed normally from air. Find the apparent depth of the bubble.

Accepted Answer

Case 1: Refraction from the glass surface

According to question, refractive index of incident ray, $μ_{1} = 1.5 = \frac{3}{2}$ , refractive index of refracted ray , $μ_{2} = \frac{4}{3}$ , radius of curvature, $R = - 5 cm$ and object distance, $u = - 7.5 cm$ .

Let the position of the image from the surface be $v$ .

Using the formula of refraction of light through a spherical surface, we get,

$\frac{μ_{2}}{v} - \frac{μ_{1}}{u} = \frac{μ_{2} - μ_{1}}{R}$

$\Rightarrow \frac{\frac{4}{3}}{v} - \frac{1.5}{- 7.5} = \frac{\frac{4}{3} - \frac{3}{2}}{- 5} \Rightarrow \frac{4}{3 v} = - \frac{1}{5} + \frac{1}{30} = - \frac{1}{6} \Rightarrow v = - 8 cm$

Hence, the image $I_{1}$ forms at a distance $8 cm$ below the top surface of the glass.

Case 2: Refraction from plane surface

Distance of $I_{1}$ from the plane surface $= 8 + 10 = 18 cm$ . $I_{1}$ behaves as object for refraction from plane surface.

Real depth $= 18 cm$ , refractive index of water $μ = \frac{4}{3}$ ,

As $μ = \frac{real depth}{apparent depth}$

$\Rightarrow$ apparent depth $= \frac{real depth}{μ} = \frac{18}{\frac{4}{3}} = \frac{27}{2} cm = 13.5 cm$

Hence, the apparent depth of the bubble is $13.5 cm$ .

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