Two thin parallel slits that are 0.012 mm apart are illuminated by a laser beam of wavelength 650 nm. On a very large distant screen, the total number of bright fringes including the central fringe and those on both sides of it is

Options

A. 38
B. 37
C. 40
D. 39

Explanation

To solve the problem of determining the total number of bright fringes formed by two thin parallel slits illuminated by a laser beam, we can follow these steps:

### Step 1: Understand the given data
- Distance between the slits (d) = 0.012 mm = 0.012 × 10^-3 m = 1.2 × 10^-5 m
- Wavelength of the laser beam (λ) = 650 nm = 650 × 10^-9 m = 6.5 × 10^-7 m

### Step 2: Use the formula for fringe formation
The condition for bright fringes in a double-slit experiment is given by the formula:
\[ d \sin \theta = n \lambda \]
where:
- \( d \) = distance between the slits
- \( \theta \) = angle of the fringe
- \( n \) = order of the fringe (0, ±1, ±2, ...)

### Step 3: Determine the maximum order of fringe
For a very large distant screen, we can consider the maximum angle \( \theta \) to be 90 degrees (since the fringes will spread out). At this angle, \( \sin \theta = 1 \). Thus, the equation simplifies to:
\[ d = n \lambda \]
From this, we can solve for \( n \):
\[ n = \frac{d}{\lambda} \]

### Step 4: Substitute the values
Now substitute the values of \( d \) and \( \lambda \):
\[ n = \frac{1.2 \times 10^{-5}}{6.5 \times 10^{-7}} \]

### Step 5: Calculate \( n \)
Calculating the above expression:
\[ n = \frac{1.2 \times 10^{-5}}{6.5 \times 10^{-7}} \approx 18.46 \]

### Step 6: Determine the number of bright fringes
Since \( n \) must be a whole number (you cannot have a fraction of a fringe), we take the integer part of \( n \), which is 18. This means there are 18 bright fringes on one side of the central maximum.

### Step 7: Count the total number of fringes
The total number of bright fringes includes:
- 18 fringes on one side
- 18 fringes on the other side
- 1 central fringe

Thus, the total number of bright fringes is:
\[ \text{Total fringes} = 18 + 18 + 1 = 37 \]

### Final Answer
The total number of bright fringes, including the central fringe, is **37**.

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