B

Step 1: In a Young's double slit experiment, the fringe width (β) is given by the formula:

β = \frac{\lambda D}{d}
where
λ = wavelength of light,
D = distance between the slits and the screen,
d = distance between the slits.

Step 2: When the apparatus is immersed in a medium like water, the wavelength of light changes due to the refractive index. The new wavelength (λ') in the medium is:

λ' = \frac{\lambda}{n}
where
n = refractive index of the medium (in this case, n = \frac{4}{3}).

Therefore, λ' = \frac{\lambda}{\frac{4}{3}} = \frac{3\lambda}{4}.

Step 3: The new fringe width (β') can be calculated using the new wavelength:

β' = \frac{\lambda' D}{d} = \frac{(\frac{3\lambda}{4}) D}{d}

β' = \frac{3}{4} \cdot \frac{\lambda D}{d} = \frac{3}{4} β
where β is the original fringe width.

Step 4: Given the original fringe width β = 0.4 mm, we find:

β' = \frac{3}{4} \cdot 0.4 mm = 0.3 mm.

Conclusion: The new fringe width when the apparatus is immersed in water is 0.3 mm.