In a Young's double slit experiment, the two slits act as coherent sources of waves of equal amplitude A and wavelength `lamda` in another experiment with the same arrangement the two slits are made to act as incoherent sources of waves of same amplitude and wavelength. if the intensity at the middle point of the screen in te first case is `I_(1)` and in te second case `I_(2)` then the ratio `(I_(1))/(I_(2))` is

Options

A. `2`
B. `1`
C. `0.5`
D. `4`

Explanation

To solve the problem, we need to analyze the two cases: when the slits are coherent sources and when they are incoherent sources.

### Step 1: Intensity in the Coherent Source Case
In the case of coherent sources (Young's double slit experiment), the intensity at a point on the screen can be calculated using the formula:
\[ I_1 = 4A^2 \]
where \( A \) is the amplitude of the waves coming from the two slits. This is derived from the principle of superposition, where the resultant amplitude is \( 2A \) (since both waves are in phase), leading to:
\[ I_1 = (2A)^2 = 4A^2 \]

### Step 2: Intensity in the Incoherent Source Case
For incoherent sources, the intensities simply add up. Therefore, if each slit has an intensity \( I \) (where \( I = A^2 \)), the total intensity \( I_2 \) at the same point on the screen is:
\[ I_2 = I + I = 2A^2 \]

### Step 3: Finding the Ratio of Intensities
Now, we can find the ratio of the intensities \( \frac{I_1}{I_2} \):
\[ \frac{I_1}{I_2} = \frac{4A^2}{2A^2} = \frac{4}{2} = 2 \]

### Conclusion
Thus, the ratio of the intensities at the middle point of the screen in the two cases is:
\[ \frac{I_1}{I_2} = 2 \]

### Final Answer
The ratio \( \frac{I_1}{I_2} = 2 \).

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