Question

A vessel of depth \(x\) is half filled with oil of refractive index \(\mu_1\) and the other half is filled with water of refractive index \(\mu_2\). The apparent depth of the vessel when viewed from above is

• A. \(\dfrac{x(\mu_1+\mu_2)}{2\mu_1\mu_2}\)
• B. \(\dfrac{x\mu_1\mu_2}{2(\mu_1+\mu_2)}\)
• C. \(\dfrac{2x\mu_1\mu_2}{\mu_1+\mu_2}\)
• D. \(\dfrac{2x(\mu_1+\mu_2)}{\mu_1\mu_2}\) \bigskip

Hint:

For multiple transparent layers viewed normally, \[ d_{\text{apparent}} = \sum \frac{t_i}{\mu_i}, \] where \(t_i\) is the thickness of the \(i^{\text{th}}\) layer and \(\mu_i\) is its refractive index.

Answer 1

The Correct Option is A

Concept: Apparent depth = real depth / \(\mu\). Total apparent depth of layered liquids is the sum of individual apparent depths.

Step 1: Oil layer thickness = \(x/2\), apparent depth = \(x/(2\mu_1)\). Water layer thickness = \(x/2\), apparent depth = \(x/(2\mu_2)\).

Step 2: Total apparent depth = \(x/(2\mu_1) + x/(2\mu_2) = x(\mu_1+\mu_2)/(2\mu_1\mu_2)\).

Step 3: Write the final answer. \(\boxed{d=\frac{x(\mu_1+\mu_2)}{2\mu_1\mu_2}}\)