A vessel of depth \(\mathrm{t}\) is half filled with oil of refractive index \(\mathrm{n}_{1}\) and the other half is filled with water (refractive index \(\mathrm{n}_{2}\) ). The apparent depth of the vessel when viewed from above is \(\ldots \ldots\) (A) $\left[\left\\{2 t\left(n_{1}-\mathrm{n}_{2}\right)\right\\} /\left(\mathrm{n}_{1} \mathrm{n}_{2}\right)\right]$ (B) $\left[\left\\{2 \mathrm{t}\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\right\\} /\left(\mathrm{n}_{1} \mathrm{n}_{2}\right)\right]$ (C) $\left[\left\\{t\left(n_{1}-n_{2}\right)\right\\} /\left(2 n_{1} \mathrm{n}_{2}\right)\right]$ (D) $\left[\left\\{t\left(n_{1}+n_{2}\right)\right\\} /\left(2 n_{1} \mathrm{n}_{2}\right)\right]$

Step by step solution

01

Calculate the actual depths of oil and water

As the vessel is half-filled with oil and half-filled with water, the actual depths of oil and water in the vessel can be calculated as follows: Actual depth of oil: \(t_{1} = \frac{t}{2}\) Actual depth of water: \(t_{2} = \frac{t}{2}\)

02

Calculate apparent depths of oil and water

The formula for calculating the apparent depth, given the actual depth and refractive index, is: Apparent depth = \(\frac{\text{Actual Depth}}{\text{Refractive Index}}\) Using this formula, we'll calculate the apparent depths of oil and water: For oil: Apparent depth \(p_{1} = \frac{t_{1}}{n_{1}} = \frac{\frac{t}{2}}{n_{1}}\) For water: Apparent depth \(p_{2} = \frac{t_{2}}{n_{2}} = \frac{\frac{t}{2}}{n_{2}}\)

03

Calculate the total apparent depth of the vessel

To find the total apparent depth of the vessel, we will add up the apparent depths of oil and water: Total apparent depth \(p_{t} = p_{1} + p_{2}\) Plugging in the values we found in Step 2: \(p_{t} = \frac{\frac{t}{2}}{n_{1}} + \frac{\frac{t}{2}}{n_{2}}\) By finding a common denominator, we can simplify this expression further: \(p_{t} = \frac{t(n_{1} + n_{2})}{2n_{1}n_{2}}\) Comparing this expression with the given options, we find that the apparent depth of the vessel when viewed from above is: \[\boxed{\text{(D)}\ \left[\frac{t\left(n_{1}+n_{2}\right)}{2 n_{1}n_{2}}\right]} \]

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