The bottom of a container is a 6 cm thick glass \( (\mu = 1.5) \) slab. The container contains a thick layer of water \( (\mu_w = 1.33) \), 4 cm on top of the glass. What is the apparent position of a scratch on the outer surface of the bottom of the glass slab when viewed through the container?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3 cm
When an object is seen from above through a layer of thickness \( t \) and refractive index \( \mu \) (nearly normal viewing), it appears raised by the normal shift
\( \displaystyle s = t\left(1 - \frac{1}{\mu}\right) \)
The scratch is seen through both layers, so the shifts add.
\( \displaystyle s_{\mathrm{glass}} = 6\left(1 - \frac{1}{1.5}\right) = 6 \times \frac{1}{3} = 2\,\mathrm{cm} \)
\( \displaystyle s_{\mathrm{water}} = 4\left(1 - \frac{1}{1.33}\right) = 4 \times \frac{0.33}{1.33} \approx 1\,\mathrm{cm} \)
\( \displaystyle s = 2\,\mathrm{cm} + 1\,\mathrm{cm} = 3\,\mathrm{cm} \)
As a check with apparent depths: the scratch is \( 6 + 4 = 10\,\mathrm{cm} \) below the water surface, and appears at a depth of
\( \displaystyle \frac{6}{1.5} + \frac{4}{1.33} \approx 4 + 3 = 7\,\mathrm{cm} \)
So the scratch appears raised by 3 cm from its real position.