In certain Young's double slit experiment, the slit separation is 0.05 cm. The slit to screen distance is 100 cm. When blue light is used the distance from central fringe to the fourth order fringe is `0.36 cm`. What is the wavelength of blue light ?

Options

A. 4000 Ã…
B. 4300 Ã…
C. 4400 Ã…
D. 4500 Ã…

Answer

To find the wavelength of blue light used in a Young's double slit experiment, we can follow these steps:

### Step 1: Understand the given values
- Slit separation (d) = 0.05 cm = \(0.05 \times 10^{-2}\) m
- Slit to screen distance (D) = 100 cm = \(100 \times 10^{-2}\) m
- Distance from the central fringe to the fourth order fringe (x_n) = 0.36 cm = \(0.36 \times 10^{-2}\) m
- Order of the fringe (n) = 4

### Step 2: Write the formula for fringe distance
In Young's double slit experiment, the distance of the nth order fringe from the central maximum is given by the formula:
\[ x_n = \frac{n \lambda D}{d} \]
Where:
- \(x_n\) = distance from the central fringe to the nth order fringe
- \(n\) = order of the fringe
- \(\lambda\) = wavelength of light
- \(D\) = distance from the slits to the screen
- \(d\) = slit separation

### Step 3: Rearrange the formula to solve for wavelength (\(\lambda\))
Rearranging the formula to find \(\lambda\):
\[ \lambda = \frac{x_n \cdot d}{n \cdot D} \]

### Step 4: Substitute the known values into the formula
Substituting the values into the rearranged formula:
\[ \lambda = \frac{(0.36 \times 10^{-2} \, \text{m}) \cdot (0.05 \times 10^{-2} \, \text{m})}{4 \cdot (100 \times 10^{-2} \, \text{m})} \]

### Step 5: Calculate the values
Calculating the numerator:
\[ 0.36 \times 10^{-2} \times 0.05 \times 10^{-2} = 0.018 \times 10^{-4} \, \text{m}^2 \]
Calculating the denominator:
\[ 4 \cdot (100 \times 10^{-2}) = 400 \times 10^{-2} \, \text{m} \]
Now substituting back into the equation:
\[ \lambda = \frac{0.018 \times 10^{-4}}{400 \times 10^{-2}} = \frac{0.018 \times 10^{-4}}{4 \times 10^{-1}} = \frac{0.018}{4} \times 10^{-3} = 0.0045 \, \text{m} \]

### Step 6: Convert to Angstroms
To convert meters to Angstroms (1 m = \(10^{10}\) Angstroms):
\[ \lambda = 0.0045 \, \text{m} \times 10^{10} \, \text{Angstroms/m} = 45000 \, \text{Angstroms} \]

### Final Answer
The wavelength of blue light is:
\[ \lambda = 4500 \, \text{Angstroms} \]