Light of wavelength 24000A is incident on a thin glass plate of refractive index`1` 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 method outlined in the video transcript.

### Step 1: Write down the given data
- Wavelength of light, \( \lambda = 24000 \, \text{Å} \)
- Refractive index of the glass plate, \( \mu = 1 \)
- Angle of refraction, \( r = 60^\circ \)

### Step 2: Convert the wavelength from Angstroms to meters
1 Angstrom (Å) = \( 10^{-10} \) meters, so:
\[ \lambda = 24000 \, \text{Å} = 24000 \times 10^{-10} \, \text{m} = 24 \times 10^{-7} \, \text{m} \]

### Step 3: Use the formula for dark band in reflected light
The condition for dark bands in the reflected light is given by the formula:
\[ 2 \mu t \cos r = n \lambda \]
where:
- \( t \) is the thickness of the plate,
- \( n \) is the order of the dark band (we will take \( n = 1 \) for the first dark band).

### Step 4: Rearrange the formula to solve for thickness \( t \)
From the formula, we can express thickness \( t \) as:
\[ t = \frac{n \lambda}{2 \mu \cos r} \]

### Step 5: Substitute the known values into the equation
- \( n = 1 \)
- \( \lambda = 24 \times 10^{-7} \, \text{m} \)
- \( \mu = 1 \)
- \( r = 60^\circ \) (and \( \cos 60^\circ = \frac{1}{2} \))

Substituting these values into the equation:
\[ t = \frac{1 \times 24 \times 10^{-7}}{2 \times 1 \times \cos 60^\circ} \]
\[ t = \frac{24 \times 10^{-7}}{2 \times 1 \times \frac{1}{2}} \]
\[ t = \frac{24 \times 10^{-7}}{1} = 24 \times 10^{-7} \, \text{m} \]

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