What is the ratio of slit widths when amplitudes of light waves from them have ratio `1 : sqrt(2)` ?

Options

A. `1 : sqrt(2)`
B. `sqrt(2) : 1`
C. `1 : 2`
D. `2 : 1`

Answer

To find the ratio of slit widths when the amplitudes of light waves from them have a ratio of \(1 : \sqrt{2}\), we can follow these steps:

### Step 1: Understand the relationship between amplitude and intensity
The intensity \(I\) of a wave is proportional to the square of its amplitude \(A\). This can be expressed as:
\[ I \propto A^2 \]

### Step 2: Define the amplitudes
Let the amplitudes of the light waves from the two slits be \(A_1\) and \(A_2\). According to the problem, the ratio of the amplitudes is given as:
\[ \frac{A_1}{A_2} = \frac{1}{\sqrt{2}} \]

### Step 3: Calculate the ratio of intensities
Using the relationship between intensity and amplitude, we can express the ratio of intensities \(I_1\) and \(I_2\) as:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]
Substituting the amplitude ratio:
\[ \frac{I_1}{I_2} = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 4: Relate intensity ratio to width ratio
The intensity ratio is also related to the ratio of the slit widths \(w_1\) and \(w_2\):
\[ \frac{w_1}{w_2} = \frac{I_1}{I_2} \]
Thus, we have:
\[ \frac{w_1}{w_2} = \frac{1}{2} \]

### Step 5: Conclusion
The ratio of the slit widths is:
\[ \frac{w_1}{w_2} = \frac{1}{2} \]

### Final Answer
The ratio of the slit widths is \(1 : 2\).

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