A biology student prepares a slide with two bacteria that lie very close together, and she wants to…

A biology student prepares a slide with two bacteria that lie very close together, and she wants to see them as two separate dots. She uses green light of wavelength $520\ \text{nm}$ in vacuum, from a lamp under the stage. Her teacher reminds her that the best microscope is limited by the wave nature of light, whatever its magnification. Her microscope objective collects light from a cone with half-angle $\beta$, where $\sin\beta = 0.88$. In the first test, air with a refractive index of $1.00$ fills the space between the slide and the objective. In the second test, immersion oil with a refractive index of $1.52$ fills it. Use $d_{\min} = \dfrac{1.22\,\lambda}{2n\sin\beta}$. Find the smallest separation that the objective can resolve with the oil, and with the air. Then assume that the eye can just separate two points $0.10\ \text{mm}$ apart at $25\ \text{cm}$. Find the total magnification that the microscope needs so that two bacteria at the oil limit look as far apart as that. Say why a higher magnification than this does not show more detail.