Diffraction and the Resolution of Optical Instruments The Hubble Space Telescope (HST) has excellent resolving power because there is no atmospheric distortion of the light. Its 2.4 -m-diameter primary mirror can collect light from distant galaxies that formed early in the history of the universe. How far apart can two galaxies be from each other if they are 10 billion light-years away from Earth and are barely resolved by the HST using visible light with a wavelength of \(400 \mathrm{nm} ?\)

Answer: To find the distance between the galaxies, follow these steps: 1. Calculate the angular separation of the galaxies using Rayleigh's criterion: $$ θ = 1.22 \frac{(400 \times 10^{-9}\,\text{m})}{(2.4\,\text{m})} $$ Calculate the value of θ. 2. Convert the distance to the galaxies (10 billion light-years) to meters: $$ \text{Distance to galaxies} = (10 \times 10^9\,\text{light-years}) \times (9.461 \times 10^{15} \, \text{m/light-year}) $$ Calculate the value of the distance in meters. 3. Use the small-angle approximation and the formula relating angular separation, distance between galaxies, and distance to the galaxies from Earth to find the distance (d) between the galaxies: $$ θ = \frac{d}{\text{Distance to galaxies}} $$ Solve for d using the value of θ from Step 1 and the distance to the galaxies in meters from Step 2. By following these steps, you'll find the distance between the two galaxies barely resolved by the Hubble Space Telescope.