QJEEM23Ph0077: JEE Main 2023 Physics - Class 12 JEEM23Ph0077 Solution: An ice cube has a bubble inside. When viewed from one side the…

Let's denote the true distance of the bubble from one side of the ice cube as $x$ and the refractive index of the ice cube as $n$ . We will use the formula for apparent depth, which states that the ratio of the true depth to the apparent depth is equal to the refractive index:

$n = \frac{\text{True depth}}{\text{Apparent depth}}$

When viewing the bubble from one side, the true depth is $x$ and the apparent depth is $12 \mathrm{~cm}$ . Using the formula:

$n = \frac{x}{12}$

When viewing the bubble from the opposite side, the true depth is $24 - x$ (since the side of the ice cube is $24 \mathrm{~cm}$ ) and the apparent depth is $4 \mathrm{~cm}$ . Using the formula:

$n = \frac{24 - x}{4}$

Now we have a system of two equations with two variables:

1) $n = \frac{x}{12}$

2) $n = \frac{24 - x}{4}$

We can solve this system by setting the two expressions for $n$ equal to each other:

$\frac{x}{12} = \frac{24 - x}{4}$

To solve for $x$ , first multiply both sides by $12$ :

$x = 3(24 - x)$

$x = 72 - 3x$

Add $3x$ to both sides:

$4x = 72$

Divide by $4$ :

$x = 18$

Now that we have the value of $x$ , we can find the refractive index $n$ using either equation 1 or 2. Using equation 1:

$n = \frac{18}{12} = \frac{3}{2}$

Therefore, the refractive index of the ice cube is $\frac{3}{2}$ .