Light of wavelength 12000A is incident on a thin glass plate of refractive index 0.5 such that angle of refraction into plate is 60° .calculate the thickness of plate which will make it appear dark by reflection?

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Identify the Given Data
We start by noting the values provided in the question:
- Wavelength of light, \( \lambda = 12000 \, \text{Å} \)
- Refractive index of the glass plate, \( \mu = 0.5 \)
- Angle of refraction into the plate, \( r = 60^\circ \)

### Step 2: Convert Wavelength to Meters
The wavelength is given in angstroms, and we need to convert it to meters for our calculations:
\[ \lambda = 12000 \, \text{Å} = 12000 \times 10^{-10} \, \text{m} = 12 \times 10^{-7} \, \text{m} \]

### Step 3: Use the Formula for Thickness
To find the thickness \( t \) of the plate that will make it appear dark by reflection, we use the formula:
\[ t = \frac{n \lambda}{2 \mu \cos r} \]
where:
- \( n \) is the order of the dark band (for the first dark band, \( n = 1 \))
- \( \lambda \) is the wavelength in meters
- \( \mu \) is the refractive index
- \( r \) is the angle of refraction

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

### Step 5: Substitute the Values into the Formula
Now we can substitute the values into the formula:
\[ t = \frac{1 \times (12 \times 10^{-7})}{2 \times 0.5 \times \cos 60^\circ} \]
Substituting \( \cos 60^\circ = \frac{1}{2} \):
\[ t = \frac{12 \times 10^{-7}}{2 \times 0.5 \times \frac{1}{2}} = \frac{12 \times 10^{-7}}{2 \times 0.5 \times 0.5} = \frac{12 \times 10^{-7}}{0.5} = 24 \times 10^{-7} \, \text{m} \]

### Step 6: Final Answer
Thus, the thickness of the plate that will make it appear dark by reflection is:
\[ t = 24 \times 10^{-7} \, \text{m} = 2.4 \times 10^{-6} \, \text{m} \]