Light of wavelength 3000A is incident on a thin glass plate of refractive index 1.5 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, we need to calculate the thickness of the glass plate that will make it appear dark by reflection. Here’s a step-by-step solution:

### Step 1: Understand the Condition for Dark Reflection
For destructive interference (dark appearance) in reflected light, the condition is given by the formula:
\[ 2 \mu t \cos r = n \lambda \]
where:
- \( \mu \) = refractive index of the material (glass in this case)
- \( t \) = thickness of the plate
- \( r \) = angle of refraction
- \( n \) = order of the dark fringe (for the first dark fringe, \( n = 1 \))
- \( \lambda \) = wavelength of the incident light

### Step 2: Convert Wavelength to Meters
The given wavelength is in Angstroms. We need to convert it to meters.
\[ \lambda = 3000 \text{ A} = 3000 \times 10^{-10} \text{ m} = 3 \times 10^{-7} \text{ m} \]

### Step 3: Identify Given Values
From the problem statement, we have:
- \( \mu = 1.5 \)
- \( r = 60^\circ \)
- \( n = 1 \)
- \( \lambda = 3 \times 10^{-7} \text{ m} \)

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

### Step 5: Substitute Values into the Formula
Now we can substitute the values into the formula for dark reflection:
\[ 2 \mu t \cos r = n \lambda \]
Substituting the known values:
\[ 2 \times 1.5 \times t \times \frac{1}{2} = 1 \times 3 \times 10^{-7} \]

### Step 6: Simplify the Equation
The equation simplifies to:
\[ 1.5 t = 3 \times 10^{-7} \]

### Step 7: Solve for Thickness \( t \)
Now, we can solve for \( t \):
\[ t = \frac{3 \times 10^{-7}}{1.5} \]
\[ t = 2 \times 10^{-7} \text{ m} \]

### Final Answer
The thickness of the glass plate that will make it appear dark by reflection is:
\[ t = 2 \times 10^{-7} \text{ m} \]

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