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. (a) `12`
B. (b) `7`
C. (c) `18`
D. (d) `4`

Explanation

To solve the problem of finding the number of maxima in Young's double slit experiment given the wavelength (λ) and the slit separation (d), we can follow these steps:

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

### Step 2: Rearrange the equation for sin θ
From the maxima condition, we can express \( \sin \theta \) as:
\[ \sin \theta = \frac{n \lambda}{d} \]

### 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:
\[ \sin \theta = \frac{2n}{7} \]

### Step 4: Determine the maximum value of n
The sine function has a maximum value of 1. Therefore, we set up the inequality:
\[ \frac{2n}{7} \leq 1 \]

### Step 5: Solve for n
Multiplying both sides by 7 gives:
\[ 2n \leq 7 \]
Dividing by 2:
\[ n \leq \frac{7}{2} \]
Thus, \( n \leq 3.5 \).

### Step 6: Identify the valid integer values of n
Since \( n \) must be a non-negative integer, the possible values of \( n \) are:
- \( n = 0 \)
- \( n = 1 \)
- \( n = 2 \)
- \( n = 3 \)

This gives us a total of 4 maxima on one side of the central maximum.

### Step 7: Count the total maxima
Including the central maximum (n=0), the total number of maxima on both sides of the central fringe is:
- Maxima on one side: 3 (n=1, 2, 3)
- Central maximum: 1 (n=0)
- Maxima on the other side: 3 (n=-1, -2, -3)

Thus, the total number of maxima is:
\[ 3 \, (\text{one side}) + 1 \, (\text{central}) + 3 \, (\text{other side}) = 7 \]

### Final Answer
The total number of maxima (including the central maximum) is **7**.

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