Final answer:
The position y1 of the first dark band from the center of the central maximum using the small angle approximation is y1 = (λd)/a, where λ is the wavelength of light, d is the distance from the screen to the slit, and a is the width of the slit.
Step-by-step explanation:
The student is asking about the small angle approximation in the context of an interference pattern created by light passing through a single slit resulting in a diffraction pattern on a screen.
In this context, the position y1 of the first dark band (also known as the first minimum) from the center of the central maximum can be calculated using the equation y1 = (λd)/a, where λ is the wavelength of light, d is the distance from the screen to the slit, and a is the width of the slit.
In the small angle approximation, the equation for the position of the minima in the diffraction pattern is derived assuming that the sin of the angle θ is approximately equal to the angle in radians, since θ is small. As a result, sin θ ≈ θ. From the single-slit diffraction equation d sin θ = mλ (where m is the order of the minimum), and using the small angle approximation, we can determine that y1 for the first minimum (where m = 1) is given by y1 = λD/a, misunderstanding this with the distance d from the slit to the screen, which is part of the correct formula for the distance between fringes in a double-slit experiment (Δy = xλ/d) can lead to confusion between the phenomena of single-slit diffraction and double-slit interference.