A light ray falls at an angle of `45^(@)` with the surface of transparent slab of thickness 1.5 m. Angle of refraction of light into the slab is `30^(@)`. If speed of light in air is `3 xx 10^(8)` m/s then find the time taken by the light rays to cross the slab.

Options

A. 4.08 ns
B. 8.16 ns
C. 7.07 ns
D. 12.1 ns

Answer

To solve the problem of finding the time taken by a light ray to cross a transparent slab, we can follow these steps:

### Step 1: Understand the Problem
We have a light ray falling at an angle of \(45^\circ\) with the surface of a transparent slab of thickness \(1.5 \, m\). The angle of refraction into the slab is \(30^\circ\). We need to find the time taken by the light to cross the slab.

### Step 2: Calculate the Distance Traveled in the Slab
The thickness of the slab is given as \(t = 1.5 \, m\). When light enters a medium, it bends according to Snell's law. The distance \(x\) that the light travels in the slab can be calculated using the cosine of the angle of refraction.

Using the relationship:
\[ x = \frac{t}{\cos(r)} \]
where \(r = 30^\circ\).

Calculating \(x\):
\[ x = \frac{1.5}{\cos(30^\circ)} = \frac{1.5}{\frac{\sqrt{3}}{2}} = \frac{1.5 \times 2}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \, m \]

### Step 3: Calculate the Speed of Light in the Slab
Using Snell's law:
\[ \frac{\sin(i)}{\sin(r)} = \frac{v_{air}}{v_{glass}} \]
where \(i = 45^\circ\), \(r = 30^\circ\), and \(v_{air} = 3 \times 10^8 \, m/s\).

Calculating \(v_{glass}\):
\[ \frac{\sin(45^\circ)}{\sin(30^\circ)} = \frac{3 \times 10^8}{v_{glass}} \]
Substituting the values:
\[ \frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}} = \frac{3 \times 10^8}{v_{glass}} \]
This simplifies to:
\[ \frac{2}{\sqrt{2}} = \frac{3 \times 10^8}{v_{glass}} \implies v_{glass} = \frac{3 \times 10^8 \times \sqrt{2}}{2} \]

### Step 4: Calculate the Time Taken to Cross the Slab
The time taken \(t\) to cross the slab is given by:
\[ t = \frac{x}{v_{glass}} \]
Substituting the values:
\[ t = \frac{\sqrt{3}}{\frac{3 \times 10^8 \times \sqrt{2}}{2}} = \frac{2\sqrt{3}}{3 \times 10^8 \sqrt{2}} \]

### Step 5: Simplify the Expression
Calculating the numerical value:
\[ t = \frac{2\sqrt{3}}{3\sqrt{2} \times 10^8} \]
Using \(\sqrt{3} \approx 1.732\) and \(\sqrt{2} \approx 1.414\):
\[ t \approx \frac{2 \times 1.732}{3 \times 1.414 \times 10^8} \approx \frac{3.464}{4.242 \times 10^8} \approx 0.817 \times 10^{-8} \, s \]
Converting to nanoseconds:
\[ t \approx 8.17 \, ns \]

### Final Answer
The time taken by the light rays to cross the slab is approximately \(8.17 \, ns\).

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