A large glass slab `(mu=5//3)` of thickness 8cm is placed over a point source of light on a plane surface. It is seen that light emerges out of the top surface of the slab from a circular area of radius R cm. What is the value of R?

Explanation

To solve the problem, we need to find the radius \( R \) of the circular area from which light emerges from the top surface of a glass slab. The given parameters are:

- Refractive index of the glass slab, \( \mu = \frac{5}{3} \)
- Thickness of the slab, \( t = 8 \) cm

### Step-by-step Solution:

1. **Understanding the Geometry**:
- The light source is at the bottom of the slab, and the light rays will emerge from the top surface of the slab. The angle of incidence at the bottom surface will determine how wide the emerging light will spread out.

2. **Using Snell's Law**:
- According to Snell's Law, when light travels from one medium to another, the relationship between the angles and the refractive indices is given by:
\[ \mu_1 \sin i = \mu_2 \sin r \]
- Here, \( \mu_1 = 1 \) (for air), \( \mu_2 = \frac{5}{3} \) (for glass), \( i \) is the angle of incidence, and \( r \) is the angle of refraction.

3. **Finding the Critical Angle**:
- The critical angle \( C \) can be found when light travels from glass to air (where it will refract at \( 90^\circ \)):
\[ \frac{5}{3} \sin C = 1 \cdot \sin 90^\circ \]
- This simplifies to:
\[ \sin C = \frac{3}{5} \]
- Therefore, the critical angle \( C \) is:
\[ C = \arcsin\left(\frac{3}{5}\right) \approx 37^\circ \]

4. **Using Geometry to Relate R and Thickness**:
- From the geometry of the situation, we can relate the radius \( R \) of the circular area to the thickness of the slab using the tangent of the angle \( C \):
\[ \tan C = \frac{R}{t} \]
- Substituting \( t = 8 \) cm:
\[ \tan 37^\circ = \frac{R}{8} \]

5. **Finding \( \tan 37^\circ \)**:
- The value of \( \tan 37^\circ \) is known to be approximately \( \frac{3}{4} \):
\[ \frac{3}{4} = \frac{R}{8} \]

6. **Solving for R**:
- Cross-multiplying gives:
\[ 3 \cdot 8 = 4R \]
- Simplifying this:
\[ 24 = 4R \implies R = \frac{24}{4} = 6 \text{ cm} \]

### Final Answer:
The value of \( R \) is \( 6 \) cm.

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