There is a small air bubble inside a glass sphere (\( \mu = 1.5 \)) of radius 10 cm. The bubble is 4.0 cm below the surface and is viewed normally from outside as shown. The apparent depth of the bubble will be
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3.0 cm below the surface
Light from the bubble is refracted at the curved surface of the sphere, from glass into air, so use the formula for refraction at a spherical surface.
\( \displaystyle \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \)
Sign convention: distances are measured from the top of the surface, positive in the direction of the light (upward, towards the eye). The bubble and the centre C lie below the surface, against the light.
\( n_1 = 1.5\ (\text{glass}) \)
\( n_2 = 1\ (\text{air}) \)
\( u = -4.0\,\mathrm{cm} \)
\( R = -10\,\mathrm{cm} \)
\( \displaystyle \frac{1}{v} - \frac{1.5}{-4} = \frac{1 - 1.5}{-10} \)
\( \displaystyle \frac{1}{v} = 0.05 - 0.375 = -0.325\,\mathrm{cm^{-1}} \)
\( \displaystyle v = -3.08\,\mathrm{cm} \approx -3.0\,\mathrm{cm} \)
The negative sign means the image is on the glass side, so the bubble appears about 3.0 cm below the surface.