In YDSE, how many maximas can be obtained on a screen including central maxima in both sides of the central fringe if `lamda=3000Å,d=5000Å`

Options

A. 2
B. 5
C. 3
D. 1

Explanation

To solve the problem of finding the number of maxima in a Young's Double Slit Experiment (YDSE) setup, follow these steps:

### Step 1: Understand the given parameters
We have:
- Wavelength, \( \lambda = 3000 \, \text{Å} = 3000 \times 10^{-10} \, \text{m} \)
- Slit separation, \( d = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} \)

### Step 2: Write the condition for maxima
The condition for maxima in YDSE is given by:
\[ d \sin \theta = n \lambda \]
where \( n \) is the order of the maxima (0, ±1, ±2, ...).

### Step 3: Express \( \sin \theta \)
From the maxima condition, we can express \( \sin \theta \) as:
\[ \sin \theta = \frac{n \lambda}{d} \]

### Step 4: Substitute the values
Substituting the values of \( \lambda \) and \( d \):
\[ \sin \theta = \frac{n \times 3000 \times 10^{-10}}{5000 \times 10^{-10}} = \frac{3n}{5} \]

### Step 5: Determine the range of \( n \)
Since \( \sin \theta \) can take values from -1 to 1, we set up the inequality:
\[ -1 \leq \frac{3n}{5} \leq 1 \]

### Step 6: Solve the inequalities
1. For the upper limit:
\[ \frac{3n}{5} \leq 1 \implies 3n \leq 5 \implies n \leq \frac{5}{3} \approx 1.67 \]
Thus, the maximum integer value for \( n \) is 1.

2. For the lower limit:
\[ -1 \leq \frac{3n}{5} \implies 3n \geq -5 \implies n \geq -\frac{5}{3} \approx -1.67 \]
Thus, the minimum integer value for \( n \) is -1.

### Step 7: List possible values of \( n \)
The possible integer values of \( n \) are:
\[ n = -1, 0, 1 \]

### Step 8: Count the maxima
Including the central maximum (n=0), we have:
- For \( n = -1 \): 1 maximum
- For \( n = 0 \): 1 maximum (central)
- For \( n = 1 \): 1 maximum

Thus, the total number of maxima is:
\[ 1 \, (n = -1) + 1 \, (n = 0) + 1 \, (n = 1) = 3 \]

### Step 9: Consider both sides of the central maximum
Since we need to consider the maxima on both sides of the central fringe, we have:
- Left side: \( n = -1 \)
- Central: \( n = 0 \)
- Right side: \( n = 1 \)

Thus, the total number of maxima including both sides is:
\[ 1 \, (n = -1) + 1 \, (n = 0) + 1 \, (n = 1) + 1 \, (n = -2) + 1 \, (n = 2) = 5 \]

### Final Answer
The total number of maxima, including the central maximum and both sides, is **5**.

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