Light of wavelength 6000A 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?

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the problem
We need to find the thickness of a thin glass plate that will make it appear dark by reflection when light of wavelength 6000 Å (angstroms) is incident on it at an angle that results in a refraction angle of 60°.

### Step 2: Write down the known values
- Wavelength of light, \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} = 6 \times 10^{-7} \, \text{m} \)
- Refractive index of glass, \( \mu = 0.5 \)
- Angle of refraction, \( r = 60° \)
- Order of the dark band, \( n = 1 \) (for the first dark band)

### Step 3: Use the formula for thickness
For the dark band in reflected light, the formula is given by:
\[ 2 \mu t \cos r = n \lambda \]
We need to rearrange this formula to solve for thickness \( t \):
\[ t = \frac{n \lambda}{2 \mu \cos r} \]

### Step 4: Calculate \( \cos r \)
Since \( r = 60° \):
\[ \cos 60° = \frac{1}{2} \]

### Step 5: Substitute the values into the formula
Now we can substitute the known values into the equation for thickness:
\[ t = \frac{1 \cdot (6 \times 10^{-7} \, \text{m})}{2 \cdot (0.5) \cdot \left(\frac{1}{2}\right)} \]

### Step 6: Simplify the expression
Calculating the denominator:
\[ 2 \cdot 0.5 \cdot \frac{1}{2} = 0.5 \]
Now substituting this back into the thickness equation:
\[ t = \frac{6 \times 10^{-7}}{0.5} = 12 \times 10^{-7} \, \text{m} \]

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