Final answer:
To find the distance between the lenses when the microscope is adjusted for a relaxed eye, one must use the thin lens equations to first calculate the image distance from the objective lens, after which the focal length of the eyepiece is added.
Step-by-step explanation:
To calculate the distance between the lenses when a compound microscope is adjusted for a relaxed eye, we must determine the image position formed by the objective lens and then use this position as the object distance for the eyepiece. For a relaxed eye, we want the final image to be at infinity.
Firstly, we use the lens formula 1/f = 1/do + 1/di, where do is the object distance, di is the image distance, and f is the focal length of the lens. The object distance to the objective lens (do) is given as 0.800 cm, shorter than the focal length of the objective lens, so it will produce a real, inverted image on the other side of the lens.
For the objective lens with do = 0.800 cm and f = 0.740 cm, the image distance (di) can be calculated using the thin lens equation. Because we want the final image to be at infinity, the image distance for the eyepiece must be equal to its focal length, which is 2.80 cm. Thus, the total distance between the objective and eyepiece lenses is the sum of the image distance from the objective and the focal length of the eyepiece.
Using the lens formula for the objective lens:
1/f = 1/do + 1/di
1/0.740 cm = 1/0.800 cm + 1/di
Solving for di gives us the image distance from the objective lens.
Once we have di, we add the focal length of the eyepiece to find the total distance between the lenses.