In Young's double slit experiment how many maximas can be obtained on a screen (including the central maximum) on both sides of the central fringe if `lambda=2000Å` and `d=7000Å`

Options

A. 12
B. 7
C. 18
D. 4

Answer

To solve the problem of how many maxima can be obtained on a screen in Young's double slit experiment, we will follow these steps:

### Step 1: Understand the given parameters
We are given:
- Wavelength (λ) = 2000 Å (angstroms) = \(2000 \times 10^{-10}\) m
- Distance between the slits (d) = 7000 Å = \(7000 \times 10^{-10}\) m

### Step 2: Use the formula for maxima
In Young's double slit experiment, the condition for maxima is given by:
\[ d \sin \theta = n \lambda \]
where:
- \( n \) is the order of the maxima (0, 1, 2, ...).
- \( \theta \) is the angle of the maxima.

### Step 3: Rearranging the formula
We can express \( \sin \theta \) as:
\[ \sin \theta = \frac{n \lambda}{d} \]

### Step 4: Substitute the values
Substituting the values of \( \lambda \) and \( d \):
\[ \sin \theta = \frac{n \times 2000 \times 10^{-10}}{7000 \times 10^{-10}} \]
\[ \sin \theta = \frac{n \times 2000}{7000} \]
\[ \sin \theta = \frac{2n}{7} \]

### Step 5: Determine the maximum value of \( n \)
Since \( \sin \theta \) must be between -1 and 1, we set up the inequality:
\[ -1 \leq \frac{2n}{7} \leq 1 \]

From the upper limit:
\[ \frac{2n}{7} \leq 1 \]
\[ 2n \leq 7 \]
\[ n \leq 3.5 \]

Since \( n \) must be a whole number, the maximum integer value for \( n \) is 3.

### Step 6: Count the maxima
The values of \( n \) can be:
- For \( n = 0 \): Central maximum
- For \( n = 1 \): First maximum on one side
- For \( n = 2 \): Second maximum on one side
- For \( n = 3 \): Third maximum on one side

Thus, we have:
- 3 maxima on one side (n = 1, 2, 3)
- 3 maxima on the other side (n = -1, -2, -3)
- 1 central maximum (n = 0)

### Step 7: Total maxima
Total maxima = 3 (right side) + 3 (left side) + 1 (central) = 7.

### Final Answer
The total number of maxima obtained on the screen, including the central maximum, is **7**.

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