Final answer:
To find the angle for the first dark fringe in single slit diffraction, the equation asin(θ) = mλ is used, where 'a' is the slit width and 'm' is the fringe order. Without the wavelength, the numerical angle cannot be given, hence just the method is provided.
Step-by-step explanation:
The question posed requires the application of the principle of single slit diffraction in the field of Physics. The phenomenon occurs when a wave encounters an obstacle which causes a spread of the wave into regions where it would not have reached by straight-line travel. The resulting diffraction pattern on a screen will have bright and dark fringes due to constructive and destructive interference.
To find the angle that locates the first dark fringe, you use the equation for single slit diffraction minima: asin(θ) = mλ, where 'a' is the slit width, 'θ' is the diffraction angle, 'm' is the order number of the dark fringe, and 'λ' is the wavelength of the light.
In this question, we are looking for the first dark fringe, so m=1. However, the wavelength of light has not been provided which is necessary for the calculation. Assuming we had the wavelength, for part (a) with a slit width of 1.8 x 10⁻⁴ m, and for part (b) with a slit width of 1.8 x 10⁻⁶ m, we would solve the equation θ = arcsin(mλ/a) for each part to get the respective angles for the dark fringes.