Question 142: Ray Optics

Question

An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is $$12 \mathrm{~cm}$$. When viewed from the opposite side, the apparent distance of the bubble is observed as $$4 \mathrm{~cm}$$. If the side of the ice cube is $$24 \mathrm{~cm}$$, the refractive index of the ice cube is

Accepted Answer

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:
$$n = \frac{	ext{True depth}}{	ext{Apparent depth}}$$
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 $$\frac{3}{2}$$.

Answer

$$\frac{3}{2}$$

Answer

$$\frac{4}{3}$$

Answer

$$\frac{2}{3}$$

Answer

$$\frac{6}{5}$$

Ray Optics

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