A ray of light incident normally on a glass slab of refractive index 3/2, travels a distance of 6 cm in glass in time t. What will be the distance travelled by the ray in the same time (t) if it travels in air ?

Options

A. 3cm
B. 6 cm
C. 9cm
D. `3sqrt(3) cm`

Explanation

To solve the problem, we need to find the distance traveled by light in air in the same time \( t \) that it takes to travel 6 cm in glass.

### Step 1: Determine the speed of light in glass

The speed of light in a medium can be calculated using the formula:

\[ v = \frac{c}{n} \]

where:
- \( v \) is the speed of light in the medium,
- \( c \) is the speed of light in vacuum (approximately \( 3 \times 10^8 \) m/s),
- \( n \) is the refractive index of the medium.

Given that the refractive index of glass \( n = \frac{3}{2} \):

\[ v_{glass} = \frac{c}{n} = \frac{3 \times 10^8 \text{ m/s}}{\frac{3}{2}} = \frac{3 \times 10^8 \text{ m/s} \times 2}{3} = 2 \times 10^8 \text{ m/s} \]

### Step 2: Calculate the time \( t \) taken to travel 6 cm in glass

Using the formula:

\[ t = \frac{d}{v} \]

where:
- \( d \) is the distance traveled,
- \( v \) is the speed of light in the medium.

For glass:

\[ t = \frac{6 \text{ cm}}{v_{glass}} = \frac{6 \text{ cm}}{2 \times 10^8 \text{ m/s}} \]

Convert 6 cm to meters:

\[ 6 \text{ cm} = 0.06 \text{ m} \]

Now substituting:

\[ t = \frac{0.06 \text{ m}}{2 \times 10^8 \text{ m/s}} = 3 \times 10^{-10} \text{ s} \]

### Step 3: Determine the speed of light in air

The refractive index of air is approximately \( n \approx 1 \). Therefore, the speed of light in air is:

\[ v_{air} = \frac{c}{n} = \frac{3 \times 10^8 \text{ m/s}}{1} = 3 \times 10^8 \text{ m/s} \]

### Step 4: Calculate the distance traveled by light in air in the same time \( t \)

Using the same formula for distance:

\[ d_{air} = v_{air} \times t \]

Substituting the values:

\[ d_{air} = (3 \times 10^8 \text{ m/s}) \times (3 \times 10^{-10} \text{ s}) \]

Calculating this gives:

\[ d_{air} = 0.09 \text{ m} = 9 \text{ cm} \]

### Final Answer

The distance traveled by the ray in air in the same time \( t \) is **9 cm**.

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