Light of wavelength 24000A is incident on a thin glass plate of refractive index 1.5 such that angle of refraction into plate is 60° .calculate the thickness of plate which will make it appear dark by reflection?

Answer

To solve the problem step by step, we will follow the same logic as outlined in the video transcript.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Wavelength of light, \( \lambda = 24000 \, \text{Å} \)
- Refractive index of glass, \( \mu = 1.5 \)
- Angle of refraction, \( r = 60^\circ \)
- Order of dark band, \( n = 1 \) (since we are looking for the first dark band)

2. **Convert Wavelength to Meters:**
- \( \lambda = 24000 \, \text{Å} = 24000 \times 10^{-10} \, \text{m} = 24 \times 10^{-7} \, \text{m} \)

3. **Calculate \( \cos r \):**
- \( r = 60^\circ \)
- \( \cos 60^\circ = \frac{1}{2} \)

4. **Use the Formula for Dark Band in Reflection:**
- The condition for a dark band in reflected light is given by:
\[ 2 \mu t \cos r = n \lambda \]
- Rearranging the formula to find the thickness \( t \):
\[ t = \frac{n \lambda}{2 \mu \cos r} \]

5. **Substitute the Values into the Formula:**
- Substitute \( n = 1 \), \( \lambda = 24 \times 10^{-7} \, \text{m} \), \( \mu = 1.5 \), and \( \cos r = \frac{1}{2} \):
\[ t = \frac{1 \cdot 24 \times 10^{-7}}{2 \cdot 1.5 \cdot \frac{1}{2}} \]

6. **Simplify the Expression:**
- The denominator simplifies as follows:
\[ 2 \cdot 1.5 \cdot \frac{1}{2} = 1.5 \]
- Therefore:
\[ t = \frac{24 \times 10^{-7}}{1.5} \]

7. **Calculate the Thickness:**
- Performing the division:
\[ t = 16 \times 10^{-7} \, \text{m} \]

8. **Final Answer:**
- The thickness of the glass plate that will make it appear dark by reflection is:
\[ t = 16 \times 10^{-7} \, \text{m} = 1.6 \times 10^{-6} \, \text{m} = 1.6 \, \mu\text{m} \]