A

Step 1: Recall that the optical path length (OPL) change is given by:
$$ OPL = n imes d $$
where n is the refractive index and d is the thickness of the plate.

Step 2: Given that we want the change in optical path to be equal to one fourth of the wavelength, we can set up the equation:
$$ ext{Change in OPL} = rac{1}{4} imes ext{wavelength} = rac{1}{4} \lambda $$

Step 3: Since the light passes through the plate normally, we can equate the two expressions for OPL:
$$ rac{1}{4} \lambda = \mu \times d $$

Step 4: Solving for d (thickness of the plate), we get:
$$ d = \frac{\frac{1}{4} \lambda}{\mu} = \frac{\lambda}{4\mu} $$

Therefore: The thickness of the plate required to produce the desired change in optical path is given by:
$$ d = \frac{\lambda}{4\mu} $$