White coherent light ( 400 nm – 700 nm ) is passed through the...

Given parameters are:
Slit separation, $d = 0.5 \text{ mm} = 0.5 \times 10^{-3} \text{ m}$
Distance to screen, $D = 50 \text{ cm} = 0.5 \text{ m}$
Position of the hole, $y = 1.0 \text{ mm} = 1.0 \times 10^{-3} \text{ m}$

The path difference $\Delta x$ at a distance $y$ from the central maximum is given by:
$\Delta x = \dfrac{y d}{D}$

Substituting the given values:
$\Delta x = \dfrac{(1.0 \times 10^{-3}) \times (0.5 \times 10^{-3})}{0.5} \text{ m}$

$\Delta x = 1.0 \times 10^{-6} \text{ m} = 1000 \text{ nm}$

For strong intensity (constructive interference), the path difference must be an integer multiple of the wavelength $\lambda$:
$\Delta x = n \lambda$
where $n = 1, 2, 3, \dots$

Therefore, the possible wavelengths are:
$\lambda = \dfrac{\Delta x}{n} = \dfrac{1000}{n} \text{ nm}$

Checking for different integer values of $n$:
For $n = 1$, $\lambda = 1000 \text{ nm}$
For $n = 2$, $\lambda = 500 \text{ nm}$
For $n = 3$, $\lambda \approx 333.3 \text{ nm}$

The only wavelength that falls within the given visible range of $400 \text{ nm}$ to $700 \text{ nm}$ is $500 \text{ nm}$, which corresponds to $n = 2$.