A vessel is half filled with a liquid of refractive index `mu`. The other half of the vessel is filled with an immiscible liquid of refractive index `1.5 mu`. The apparent depth of vessel is `50 %` of the actual depth. The value of `mu` is

Options

A. 1.6
B. 1.67
C. 1.5
D. 1.4

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a vessel that is half filled with a liquid of refractive index \( \mu \) and the other half with an immiscible liquid of refractive index \( 1.5\mu \). The apparent depth of the vessel is given as 50% of the actual depth.

### Step 2: Define the variables
Let the actual depth of the vessel be \( h \). Since the vessel is half filled, the depth of each liquid is:
- Depth of liquid with refractive index \( \mu \): \( \frac{h}{2} \)
- Depth of liquid with refractive index \( 1.5\mu \): \( \frac{h}{2} \)

### Step 3: Write the formula for apparent depth
The apparent depth \( h' \) is related to the actual depth \( h \) and the refractive index \( \mu \) of the liquid. The formula for apparent depth in a medium is given by:
\[ h' = \frac{h}{\mu_{\text{relative}}} \]
where \( \mu_{\text{relative}} \) is the effective refractive index of the two layers of liquid.

### Step 4: Calculate the effective refractive index
The effective refractive index \( \mu_{\text{relative}} \) for two layers can be calculated as:
\[ \mu_{\text{relative}} = \frac{h/2}{h/2 \cdot \mu} + \frac{h/2}{h/2 \cdot 1.5\mu} \]
This simplifies to:
\[ \mu_{\text{relative}} = \frac{1}{\mu} + \frac{1}{1.5\mu} \]

### Step 5: Set up the equation with the given condition
Since the apparent depth is 50% of the actual depth, we have:
\[ h' = \frac{h}{2} \]
Substituting into the equation gives:
\[ \frac{h}{2} = \frac{h}{\mu_{\text{relative}}} \]
This implies:
\[ \mu_{\text{relative}} = 2 \]

### Step 6: Substitute and solve for \( \mu \)
Now we can substitute our expression for \( \mu_{\text{relative}} \):
\[ 2 = \frac{1}{\mu} + \frac{1}{1.5\mu} \]
Finding a common denominator gives:
\[ 2 = \frac{1.5 + 1}{1.5\mu} = \frac{2.5}{1.5\mu} \]
Cross-multiplying yields:
\[ 2 \cdot 1.5\mu = 2.5 \]
Thus:
\[ 3\mu = 2.5 \implies \mu = \frac{2.5}{3} = \frac{5}{6} \]

### Step 7: Final answer
The value of \( \mu \) is:
\[ \mu = \frac{5}{6} \approx 0.833 \]