Final answer:
To obtain eight maxima within the central maximum of the single-slit pattern, the slit width (a) should be one-sixteenth of the slit separation (d). The calculation is based on the single-slit diffraction formula and the phase difference for the maxima in the double-slit interference pattern.
Step-by-step explanation:
The question asks what should be the width (a) of each slit to obtain eight maxima of two double-slit patterns (slit separation d) within the central maximum of the single-slit pattern. To solve for the slit width, we utilize the single-slit diffraction formula for the first minimum which is given by a sin(θ) = λ where θ is the angle to the first minimum and λ is the wavelength of the light used.
For double-slit interference, the position of maxima is given by δm = m λ where δm is the phase difference and m is the order of the maximum. In the scenario described, we desire that the first minimum of the single-slit pattern coincides with the eighth maximum of the double-slit pattern. To achieve this, we must adjust the width a such that the formula δm = d sin(θ) is satisfied, where δm for the eighth double-slit maximum with m=8 would be 8λ.
Combining these two equations and knowing that eight maxima are observed within the same angle as the first minimum of the single-slit pattern, we find that a = d/16. Thus, for eight double-slit maxima to fit within the single-slit central maximum, the slit width should be one-sixteenth of the slit separation d.