A parallel beam of sodium light of wavelength `6000 Å` is incident on a thin glass plate of `mu=1.5`, such that the angle of refraction in the plate is `60^(@)`. The smallest thickness of the plate which will make it appear dark by reflected light is

Options

A. 3926 Ã…
B. 4353 Ã…
C. 1396 Ã…
D. 1921 Ã…

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We need to find the smallest thickness of a thin glass plate that will make it appear dark due to reflected light when a parallel beam of sodium light (wavelength = 6000 Å) is incident on it at a certain angle.

### Step 2: Identify Given Values
- Wavelength of sodium light, \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} \)
- Refractive index of glass, \( \mu = 1.5 \)
- Angle of refraction in the plate, \( r = 60^\circ \)

### Step 3: Use the Condition for Dark Reflected Light
For dark reflected light (destructive interference), the condition is given by:
\[ 2nt \cos r = (n + \frac{1}{2}) \lambda \]
Where:
- \( n \) is an integer (order of interference)
- \( t \) is the thickness of the plate
- \( r \) is the angle of refraction

Since we are looking for the smallest thickness, we will take \( n = 0 \):
\[ 2t \cos r = \lambda \]

### Step 4: Calculate \( \cos r \)
Using \( r = 60^\circ \):
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 5: Substitute Values into the Equation
Substituting \( \lambda \) and \( \cos r \) into the equation:
\[ 2t \left(\frac{1}{2}\right) = \lambda \]
This simplifies to:
\[ t = \frac{\lambda}{2} \]

### Step 6: Substitute the Wavelength
Now substitute \( \lambda = 6000 \times 10^{-10} \, \text{m} \):
\[ t = \frac{6000 \times 10^{-10}}{2} \]
\[ t = 3000 \times 10^{-10} \, \text{m} = 3000 \, \text{Å} \]

### Step 7: Adjust for Refractive Index
Since the light is passing through a medium with a refractive index, we need to account for it:
\[ t = \frac{\lambda}{2 \mu \cos r} \]
Substituting the values:
\[ t = \frac{6000 \times 10^{-10}}{2 \times 1.5 \times \frac{1}{2}} \]
This simplifies to:
\[ t = \frac{6000 \times 10^{-10}}{1.5} = 4000 \times 10^{-10} \, \text{m} = 4000 \, \text{Å} \]

### Final Answer
The smallest thickness of the plate which will make it appear dark by reflected light is:
\[ \boxed{4000 \, \text{Å}} \]