A vessel of depth '$d$' is half filled with oil of refractive index $n_{1}$ and the other half is filled with water of refractive index $n_{2}$. The apparent depth of this vessel when viewed from above will be-

To find the apparent depth of the vessel when viewed from above, we can calculate the apparent depths of the oil and water separately and then add them together. <br/><br/> The formula to find the apparent depth ($h_{apparent}$) is: <br/><br/> $h_{apparent} = \frac{h_{real}}{n}$ <br/><br/> Where $h_{real}$ is the actual depth, and $n$ is the refractive index of the medium. <br/><br/> For the oil (with depth $\frac{d}{2}$ and refractive index $n_1$): <br/><br/> $h_{oil} = \frac{\frac{d}{2}}{n_{1}}$ <br/><br/> For the water (with depth $\frac{d}{2}$ and refractive index $n_2$): <br/><br/> $h_{water} = \frac{\frac{d}{2}}{n_{2}}$ <br/><br/> Now, add the two apparent depths together to find the total apparent depth: <br/><br/> $$h_{total} = h_{oil} + h_{water} = \frac{\frac{d}{2}}{n_{1}} + \frac{\frac{d}{2}}{n_{2}}$$ <br/><br/> Combine the terms: <br/><br/> $h_{total} = \frac{d}{2}\left(\frac{1}{n_{1}} + \frac{1}{n_{2}}\right)$ <br/><br/> Now, find a common denominator for the fractions: <br/><br/> $h_{total} = \frac{d}{2}\left(\frac{n_{1} + n_{2}}{n_{1}n_{2}}\right)$ <br/><br/> Multiply the fractions: <br/><br/> $h_{total} = \frac{d(n_{1} + n_{2})}{2n_{1}n_{2}}$ <br/><br/> The correct answer is: <br/><br/> $\frac{d\left(n_{1}+n_{2}\right)}{2 n_{1} n_{2}}$