In Young's double slit experiment how many maxima 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

Explanation

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

### Step 2: Rearrange the formula
We can rearrange the formula to express \( \sin \theta \):
\[ \sin \theta = \frac{n \lambda}{d} \]

### Step 3: Substitute the given values
Given:
- \( \lambda = 2000 \, \text{Å} = 2000 \times 10^{-10} \, \text{m} \)
- \( d = 7000 \, \text{Å} = 7000 \times 10^{-10} \, \text{m} \)

Substituting these values into the equation:
\[ \sin \theta = \frac{n \cdot (2000 \times 10^{-10})}{7000 \times 10^{-10}} \]
This simplifies to:
\[ \sin \theta = \frac{n}{3.5} \]

### Step 4: Determine the maximum value of \( n \)
Since \( \sin \theta \) must lie between -1 and 1, we can set up the inequality:
\[ -1 \leq \frac{n}{3.5} \leq 1 \]

From the right side of the inequality:
\[ \frac{n}{3.5} \leq 1 \]
This gives:
\[ n \leq 3.5 \]

Since \( n \) must be a non-negative integer, the maximum integer value for \( n \) is 3.

### Step 5: Count the maxima
The values of \( n \) that satisfy the equation are:
- For \( n = 0 \): Central maximum
- For \( n = 1 \): First maximum
- For \( n = 2 \): Second maximum
- For \( n = 3 \): Third maximum

This gives us 4 maxima on one side (including the central maximum).

### Step 6: Consider both sides of the central maximum
Since the maxima are symmetrical on both sides of the central maximum, we have:
- 3 maxima on one side (n = 1, 2, 3)
- 3 maxima on the other side (n = -1, -2, -3)

### Step 7: Calculate the total maxima
Adding the maxima:
- 3 maxima on the positive side + 3 maxima on the negative side + 1 central maximum = 7 maxima in total.

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

---