The ratio of intensities of consecutive maxima in the diffraction pattern due to a single slit is

Options

A. (a) `1:4:9`
B. (b) `1:2:3`
C. (c) `1:(4)/(9pi^2):(4)/(25pi^2)`
D. (d) `1:(1)/(pi^2):(9)/(pi^2)`

Explanation

To find the ratio of the intensities of consecutive maxima in the diffraction pattern due to a single slit, we can follow these steps:

### Step 1: Understand the Diffraction Pattern
The diffraction pattern produced by a single slit consists of a central maximum and several secondary maxima on either side. The intensity of these maxima decreases as we move away from the center.

### Step 2: Intensity Formula
The intensity \( I \) of the maxima in a single slit diffraction pattern can be expressed using the formula:
\[ I(\theta) = I_0 \left( \frac{\sin(\beta)}{\beta} \right)^2 \]
where \( \beta = \frac{\pi a \sin(\theta)}{\lambda} \), \( a \) is the slit width, \( \lambda \) is the wavelength of light, and \( \theta \) is the angle of diffraction.

### Step 3: Identify Consecutive Maxima
The positions of the maxima can be found at angles where \( \sin(\beta) \) is maximized. The first few maxima occur at angles corresponding to \( \beta = n\pi \) for \( n = 1, 2, 3, \ldots \).

### Step 4: Calculate Intensities of Consecutive Maxima
The intensity of the first maximum (at \( n=1 \)) can be denoted as \( I_1 \), and the second maximum (at \( n=2 \)) can be denoted as \( I_2 \). The ratio of the intensities of these maxima can be calculated as follows:
\[ \frac{I_1}{I_2} = \frac{I_0 \left( \frac{\sin(\beta_1)}{\beta_1} \right)^2}{I_0 \left( \frac{\sin(\beta_2)}{\beta_2} \right)^2} \]
where \( \beta_1 = \frac{\pi a \sin(\theta_1)}{\lambda} \) and \( \beta_2 = \frac{\pi a \sin(\theta_2)}{\lambda} \).

### Step 5: Simplify the Ratio
As \( n \) increases, the intensity of the maxima decreases. The ratio of intensities of consecutive maxima can be approximated as:
\[ \frac{I_n}{I_{n+1}} \approx \left( \frac{n}{n+1} \right)^2 \]
This indicates that the intensity of the maxima decreases as we move to higher orders.

### Final Step: Conclusion
Thus, the ratio of the intensities of consecutive maxima in the diffraction pattern due to a single slit is approximately:
\[ \frac{I_n}{I_{n+1}} \approx \left( \frac{n}{n+1} \right)^2 \]