Light of wavelength 12000A 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, we need to find the thickness of the glass plate that will make it appear dark by reflection when light of a specific wavelength is incident on it. Here’s a step-by-step solution:

### Step 1: Identify the given data
- Wavelength of light, \( \lambda = 12000 \, \text{Å} \)
- Refractive index of glass, \( \mu = 1 \)
- Angle of refraction, \( r = 60^\circ \)

### Step 2: Convert the wavelength into meters
Since \( 1 \, \text{Å} = 10^{-10} \, \text{m} \), we convert the wavelength:
\[ \lambda = 12000 \, \text{Å} = 12000 \times 10^{-10} \, \text{m} = 12 \times 10^{-7} \, \text{m} \]

### Step 3: Use the formula for dark bands 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),
- \( \lambda \) is the wavelength,
- \( \mu \) is the refractive index,
- \( r \) is the angle of refraction.

### Step 4: Rearrange the formula to find thickness \( t \)
Rearranging the formula to solve for \( t \):
\[ t = \frac{n \lambda}{2 \mu \cos r} \]

### Step 5: Substitute the known values into the equation
Substituting \( n = 1 \), \( \lambda = 12 \times 10^{-7} \, \text{m} \), \( \mu = 1 \), and \( r = 60^\circ \):
\[ t = \frac{1 \times 12 \times 10^{-7}}{2 \times 1 \times \cos 60^\circ} \]

### Step 6: Calculate \( \cos 60^\circ \)
We know that:
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 7: Substitute \( \cos 60^\circ \) into the equation
Substituting \( \cos 60^\circ \) into the equation gives:
\[ t = \frac{12 \times 10^{-7}}{2 \times 1 \times \frac{1}{2}} = \frac{12 \times 10^{-7}}{1} = 12 \times 10^{-7} \, \text{m} \]

### Step 8: Final answer
Thus, the thickness of the glass plate that will make it appear dark by reflection is:
\[ t = 12 \times 10^{-7} \, \text{m} = 1.2 \times 10^{-6} \, \text{m} = 1.2 \, \mu m \]