A circular beam of light of diameter `d = 2 cm` falls on a plane refractive of glass. The angle of incidence is `60^(@)` and refractive index of glass is `mu = 3//2`. The diameter of the refracted beam is

Options

A. `4.00 cm`
B. `3.0 cm`
C. `3.26 cm`
D. `2.52 cm`

Answer

To find the diameter of the refracted beam of light when it passes through a plane refractive surface of glass, we can follow these steps:

### Step 1: Understand the Given Information
- Diameter of the incident beam, \( d = 2 \, \text{cm} \)
- Angle of incidence, \( i = 60^\circ \)
- Refractive index of glass, \( \mu = \frac{3}{2} \)

### Step 2: Use Snell's Law
Snell's Law states that:
\[ \mu_1 \sin(i) = \mu_2 \sin(r) \]
Assuming the refractive index of air (or vacuum) is \( \mu_1 = 1 \), we can write:
\[ 1 \cdot \sin(60^\circ) = \frac{3}{2} \cdot \sin(r) \]
Calculating \( \sin(60^\circ) \):
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Substituting this into Snell's Law:
\[ \frac{\sqrt{3}}{2} = \frac{3}{2} \cdot \sin(r) \]
Now, solving for \( \sin(r) \):
\[ \sin(r) = \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}} \]

### Step 3: Calculate \( \cos(r) \)
Using the identity \( \cos^2(r) + \sin^2(r) = 1 \):
\[ \cos^2(r) = 1 - \sin^2(r) = 1 - \left(\frac{1}{\sqrt{3}}\right)^2 = 1 - \frac{1}{3} = \frac{2}{3} \]
Thus,
\[ \cos(r) = \sqrt{\frac{2}{3}} = \frac{\sqrt{6}}{3} \]

### Step 4: Relate the Diameter of the Refracted Beam
The diameter of the refracted beam \( d' \) can be related to the diameter of the incident beam \( d \) using the cosine of the angle of refraction:
\[ d' = d \cdot \cos(r) \]
Substituting the known values:
\[ d' = 2 \cdot \cos(r) = 2 \cdot \frac{\sqrt{6}}{3} \]

### Step 5: Calculate the Diameter of the Refracted Beam
Calculating \( d' \):
\[ d' = \frac{2\sqrt{6}}{3} \, \text{cm} \]
To find the numerical value:
\[ \sqrt{6} \approx 2.45 \Rightarrow d' \approx \frac{2 \cdot 2.45}{3} \approx \frac{4.9}{3} \approx 1.63 \, \text{cm} \]

### Step 6: Final Answer
Thus, the diameter of the refracted beam is approximately:
\[ d' \approx 1.63 \, \text{cm} \]