Question

A tank is filled with a liquid to a height of \( 12.5 \, \text{m} \). The apparent depth of a needle lying at the bottom of the tank is measured to be \( 9.0 \, \text{m} \). Calculate the speed of light in the liquid.

Hint:

For apparent depth problems:

\( n = \frac{\text{real}}{\text{apparent}} \)
Then use \( v = \frac{c}{n} \)
If apparent depth is smaller, medium is optically denser.

Answer 1

Concept: Refractive index of a medium: \[ n = \frac{\text{Real depth}}{\text{Apparent depth}} \] Also: \[ n = \frac{c}{v} \] Where:

\( c = 3 \times 10^8 \, \text{m/s} \) (speed of light in vacuum)
\( v \) = speed of light in medium

Step 1: Calculate refractive index. \[ n = \frac{12.5}{9.0} \] \[ n = 1.39 \, (\text{approx}) \]
Step 2: Find speed of light in liquid. \[ v = \frac{c}{n} \] \[ v = \frac{3 \times 10^8}{1.39} \] \[ v \approx 2.16 \times 10^8 \, \text{m/s} \] Final Answer: \[ v \approx 2.2 \times 10^8 \, \text{m/s} \]