A glass slab consists of thin uniform layers of progressively decreasing refractive indices refractive index such that the refractive index of any layer is `mu-mDeltamu`. Here `mu and Deltamu` denotes the refractive index of 0th layer and the difference in refractive index betweeen any two consecutive layers, resepctively. The integer `m=0,1,2,3`... denotes the number of the suscessive layers. A ray of light from the 0th layers enters the 1st layer at an angle of incidence of `30^(@)`. After undergoing the mth refraction, the ray emerges paralllel to the interaface. If `mu=1.5 and Deltamu=0.015`, then the value of m is

Options

A. 20
B. 30
C. 40
D. 50

Explanation

To solve the problem, we need to determine the value of \( m \) for which a ray of light emerges parallel to the interface after undergoing \( m \) refractions in a glass slab with layers of progressively decreasing refractive indices.

### Step-by-Step Solution:

1. **Understanding the Refractive Index**:
The refractive index of the \( m \)-th layer is given by:
\[ \mu_m = \mu - m \cdot \Delta \mu \]
where \( \mu = 1.5 \) and \( \Delta \mu = 0.015 \).

2. **Using Snell's Law**:
When the ray of light enters the first layer at an angle of incidence \( \theta_1 = 30^\circ \), we apply Snell's law:
\[ \mu_0 \cdot \sin(\theta_1) = \mu_m \cdot \sin(\theta_m) \]
Here, \( \mu_0 = 1.5 \) (the refractive index of the 0-th layer), \( \theta_1 = 30^\circ \), and \( \theta_m \) is the angle in the \( m \)-th layer.

3. **Calculating the Sine Values**:
We know:
\[ \sin(30^\circ) = \frac{1}{2} \]
Therefore, substituting into Snell's law gives us:
\[ 1.5 \cdot \frac{1}{2} = \mu_m \cdot \sin(\theta_m) \]
Simplifying this, we find:
\[ 0.75 = \mu_m \cdot \sin(\theta_m) \]

4. **Emerging Parallel to the Interface**:
For the ray to emerge parallel to the interface after \( m \) refractions, the angle \( \theta_m \) must be \( 90^\circ \) (since it is parallel to the interface). Thus, \( \sin(90^\circ) = 1 \).

5. **Substituting Values**:
Substituting \( \sin(90^\circ) = 1 \) into the equation gives:
\[ 0.75 = \mu_m \cdot 1 \implies \mu_m = 0.75 \]

6. **Finding \( m \)**:
Now we substitute \( \mu_m \) back into the equation for the refractive index:
\[ 0.75 = 1.5 - m \cdot 0.015 \]
Rearranging gives:
\[ m \cdot 0.015 = 1.5 - 0.75 \implies m \cdot 0.015 = 0.75 \]
Dividing both sides by \( 0.015 \):
\[ m = \frac{0.75}{0.015} = 50 \]

### Final Answer:
The value of \( m \) is \( 50 \).

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