Question

An ice cube has a bubble inside. When viewed from one side, the apparent distance of the bubble is \( 12 \, \text{cm} \). When viewed from the opposite side, the apparent distance of the bubble is \( 4 \, \text{cm} \). If the side of the ice cube is \( 24 \, \text{cm} \), the refractive index of the ice cube is:

• A. \( \frac{4}{3} \)
• B. \( \frac{3}{2} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{6}{5} \)

Hint:

To find the refractive index of an object with apparent depths viewed from two sides, equations for the real and apparent depths set up and solve for \( \mu \).

Answer 1

The Correct Option is B

The refractive index \( \mu \) is defined as the ratio of the real depth to the apparent depth: \[ \mu = \frac{\text{Real Depth}}{\text{Apparent Depth}}. \] Let the real depth of the bubble be \( x \, \text{cm} \). The apparent depth when viewed from one side is \( 12 \, \text{cm} \), and when viewed from the opposite side, it is \( 24 - x \, \text{cm} \). Using the refractive index formula, we have two expressions for \( \mu \): \[ \mu = \frac{x}{12} \quad \text{(from one side)}, \] \[ \mu = \frac{24 - x}{4} \quad \text{(from the opposite side)}. \] Equating these two expressions: \[ \frac{x}{12} = \frac{24 - x}{4}. \] Simplifying the equation: \[ 4x = 12(24 - x). \] \[ 4x = 288 - 12x. \] \[ 16x = 288, \] which yields \( x = 18 \, \text{cm} \). Substituting \( x = 18 \, \text{cm} \) into the first expression for \( \mu \): \[ \mu = \frac{18}{12} = \frac{3}{2}. \] \[ \boxed{\frac{3}{2}} \]