The ratio of the intensities at minima to maxima in the interference pattern is 9 : 25. What will be the ratio of the widths of the two slits in the young's double slit experiment ?

Options

A. `8 : 1`
B. `16 : 1`
C. `4 : 1`
D. `9 : 1`

Answer

To solve the problem, we need to find the ratio of the widths of the two slits in Young's double slit experiment based on the given ratio of intensities at minima to maxima.

### Step-by-Step Solution:

1. **Understand the Given Ratio**:
The problem states that the ratio of the intensities at minima to maxima is given as 9:25. This means:
\[ I_{min} : I_{max} = 9 : 25 \]

2. **Express Intensities in Terms of Slit Widths**:
In Young's double slit experiment, the intensity at maxima \(I_{max}\) and minima \(I_{min}\) can be expressed in terms of the amplitudes (or widths) of the two slits \(a_1\) and \(a_2\):
\[ I_{max} = (a_1 + a_2)^2 \]
\[ I_{min} = (a_1 - a_2)^2 \]

3. **Set Up the Ratio of Intensities**:
From the given ratio, we can write:
\[ \frac{I_{min}}{I_{max}} = \frac{9}{25} \]
Substituting the expressions for \(I_{min}\) and \(I_{max}\):
\[ \frac{(a_1 - a_2)^2}{(a_1 + a_2)^2} = \frac{9}{25} \]

4. **Cross Multiply to Eliminate the Fraction**:
Cross multiplying gives us:
\[ 25(a_1 - a_2)^2 = 9(a_1 + a_2)^2 \]

5. **Expand Both Sides**:
Expanding both sides:
\[ 25(a_1^2 - 2a_1a_2 + a_2^2) = 9(a_1^2 + 2a_1a_2 + a_2^2) \]
This simplifies to:
\[ 25a_1^2 - 50a_1a_2 + 25a_2^2 = 9a_1^2 + 18a_1a_2 + 9a_2^2 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 25a_1^2 - 9a_1^2 - 50a_1a_2 - 18a_1a_2 + 25a_2^2 - 9a_2^2 = 0 \]
Which simplifies to:
\[ 16a_1^2 - 68a_1a_2 + 16a_2^2 = 0 \]

7. **Factoring the Quadratic**:
Dividing through by 16:
\[ a_1^2 - 4.25a_1a_2 + a_2^2 = 0 \]
This can be factored or solved using the quadratic formula.

8. **Finding the Ratio of Slit Widths**:
By solving the quadratic equation, we find:
\[ a_1 = 4a_2 \]
Therefore, the ratio of the widths of the two slits is:
\[ \frac{a_1}{a_2} = 4:1 \]

### Final Answer:
The ratio of the widths of the two slits is \(4:1\).

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