In Young's double slit experiment the light emitted from source has `lambda = 6500 Å` and the distance between th two slits is 1 mm. Distance between the screen and slit is 1 metre. Distance between third dark and fifth birth fringe will be :

Options

A. 3.2 mm
B. 1.63 mm
C. 0.585 mm
D. 2.31 mm

Answer

To solve the problem, we will follow these steps:

### Step 1: Convert the given wavelength into meters
The wavelength of the light is given as \( \lambda = 6500 \, \text{Å} \).

1 Ångström (Å) = \( 10^{-10} \) meters, so:
\[ \lambda = 6500 \, \text{Å} = 6500 \times 10^{-10} \, \text{m} = 6.5 \times 10^{-7} \, \text{m} \]

### Step 2: Identify the distances
The distance between the two slits is given as \( d = 1 \, \text{mm} \):
\[ d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \]

The distance between the screen and the slits is given as \( D = 1 \, \text{m} \).

### Step 3: Calculate the position of the fifth bright fringe
The position of the \( n \)-th bright fringe is given by the formula:
\[ y_n = \frac{n \lambda D}{d} \]
For the fifth bright fringe (\( n = 5 \)):
\[ y_5 = \frac{5 \times (6.5 \times 10^{-7}) \times 1}{1 \times 10^{-3}} = 5 \times 6.5 \times 10^{-4} = 3.25 \times 10^{-3} \, \text{m} = 3250 \, \text{mm} \]

### Step 4: Calculate the position of the third dark fringe
The position of the \( n \)-th dark fringe is given by the formula:
\[ y_{dark} = \frac{(2n - 1) \lambda D}{2d} \]
For the third dark fringe (\( n = 3 \)):
\[ y_3 = \frac{(2 \times 3 - 1) \times (6.5 \times 10^{-7}) \times 1}{2 \times (1 \times 10^{-3})} = \frac{5 \times (6.5 \times 10^{-7}) \times 1}{2 \times 10^{-3}} = \frac{32.5 \times 10^{-7}}{2 \times 10^{-3}} = 16.25 \times 10^{-4} \, \text{m} = 1.625 \, \text{mm} \]

### Step 5: Calculate the distance between the third dark fringe and the fifth bright fringe
Now, we find the distance between the fifth bright fringe and the third dark fringe:
\[ \text{Distance} = y_5 - y_3 = 3.25 \, \text{mm} - 1.625 \, \text{mm} = 1.625 \, \text{mm} \]

### Final Answer
The distance between the third dark fringe and the fifth bright fringe is \( 1.625 \, \text{mm} \).

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