In Fresnel's biprism `(mu=1.5)` experiment the distance between source and biprism is `0.3m` and that between biprism and screen is `0.7m` and angle of prism is `1^@`. The fringe width with light of wavelength `6000Å` will be

Options

A. 3 cm
B. 0.011 cm
C. 2 cm
D. 4 cm

Answer

To solve the problem of finding the fringe width in Fresnel's biprism experiment, we will follow these steps:

### Step 1: Identify the given values
- Distance between source and biprism, \( a = 0.3 \, \text{m} \)
- Distance between biprism and screen, \( b = 0.7 \, \text{m} \)
- Wavelength of light, \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} \)
- Refractive index, \( \mu = 1.5 \)
- Angle of prism, \( \alpha = 1^\circ \)

### Step 2: Convert the angle from degrees to radians
To use the angle in calculations, we need to convert it to radians:
\[ \alpha = 1^\circ = \frac{1 \times \pi}{180} \, \text{radians} \]

### Step 3: Use the formula for fringe width
The formula for fringe width \( \beta \) in Fresnel's biprism experiment is given by:
\[ \beta = \frac{(a + b) \lambda}{2a(\mu - 1) \alpha} \]

### Step 4: Substitute the values into the formula
Now we will substitute the known values into the formula:
\[ \beta = \frac{(0.3 + 0.7) \times (6000 \times 10^{-10})}{2 \times 0.3 \times (1.5 - 1) \times \left(\frac{\pi}{180}\right)} \]

### Step 5: Simplify the expression
First, calculate \( a + b \):
\[ a + b = 0.3 + 0.7 = 1.0 \, \text{m} \]

Now substitute and simplify:
\[ \beta = \frac{1.0 \times (6000 \times 10^{-10})}{2 \times 0.3 \times 0.5 \times \left(\frac{\pi}{180}\right)} \]

Calculate the denominator:
\[ 2 \times 0.3 \times 0.5 = 0.3 \]
\[ \frac{\pi}{180} \approx 0.0174533 \, \text{(approximately)} \]
Thus,
\[ 0.3 \times 0.0174533 \approx 0.005236 \]

Now substituting back:
\[ \beta = \frac{6000 \times 10^{-10}}{0.005236} \]

### Step 6: Calculate the final value of fringe width
Calculating the above gives:
\[ \beta \approx 0.0114 \, \text{m} = 0.0114 \times 100 \, \text{cm} = 1.14 \, \text{cm} \]

### Conclusion
The fringe width \( \beta \) is approximately \( 0.011 \, \text{cm} \).

### Final Answer
The correct option is \( 0.011 \, \text{cm} \).

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