Light of wavelength 3000A 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 will follow these steps:

### Step 1: Write down the given data.
- Wavelength of light, \( \lambda = 3000 \) Å (angstroms)
- Refractive index of glass, \( \mu = 1 \)
- Angle of refraction into the plate, \( r = 60^\circ \)
- We need to find the thickness of the plate, \( t \), that will make it appear dark by reflection.

### Step 2: Convert the wavelength into meters.
Since \( 1 \) Å = \( 10^{-10} \) m, we convert \( \lambda \):
\[ \lambda = 3000 \, \text{Å} = 3000 \times 10^{-10} \, \text{m} = 3 \times 10^{-7} \, \text{m} \]

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

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

### Step 5: Substitute the known values into the equation.
Substituting \( n = 1 \), \( \lambda = 3 \times 10^{-7} \, \text{m} \), \( \mu = 1 \), and \( r = 60^\circ \):
\[ t = \frac{1 \times 3 \times 10^{-7}}{2 \times 1 \times \cos 60^\circ} \]

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

### Step 7: Substitute \( \cos 60^\circ \) into the equation.
Now substituting \( \cos 60^\circ \):
\[ t = \frac{3 \times 10^{-7}}{2 \times 1 \times \frac{1}{2}} = \frac{3 \times 10^{-7}}{1} = 3 \times 10^{-7} \, \text{m} \]

### Step 8: Final answer.
Thus, the thickness of the plate that will make it appear dark by reflection is:
\[ t = 3 \times 10^{-7} \, \text{m} = 3000 \, \text{Å} \]

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