Answer:
The diameter of the objective needed to resolve two objects only 350 nm apart using the microscope in air with a light wavelength of 550 nm is approximately 1.63 mm.
Step-by-step explanation:
To determine the diameter of the objective needed to resolve two objects that are only 350 nm apart using a microscope in air with light wavelength of 550 nm, we can use the formula for the minimum resolvable distance (the Rayleigh criterion):
Minimum resolvable distance (d) = 1.22 * (λ / NA)
Where:
λ = wavelength of light
NA = numerical aperture of the objective
The numerical aperture (NA) of an objective lens is defined as the measure of its ability to gather light and resolve fine details at a fixed object distance. It is related to the diameter of the objective (D) and the focal length of the objective (f) as follows:
NA = D / (2 * f)
Given that the focal length of the objective (f) is 1.5 mm (or 1.5 * 10^-3 meters) and the wavelength of light (λ) is 550 nm (or 550 * 10^-9 meters), we can calculate the numerical aperture (NA):
NA = D / (2 * f)
NA = D / (2 * 1.5 * 10^-3)
Now, we need to calculate the minimum resolvable distance (d) using the Rayleigh criterion:
d = 1.22 * (λ / NA)
d = 1.22 * (550 * 10^-9) / (D / (2 * 1.5 * 10^-3))
d = 1.22 * (550 * 10^-9) * (2 * 1.5 * 10^-3) / D
Now, let's plug in the value of the desired minimum resolvable distance (350 nm or 350 * 10^-9 meters):
350 * 10^-9 = 1.22 * (550 * 10^-9) * (2 * 1.5 * 10^-3) / D
Now, we can solve for D:
D = 1.22 * (550 * 10^-9) * (2 * 1.5 * 10^-3) / (350 * 10^-9)
D = 1.63 mm
So, the diameter of the objective needed to resolve two objects only 350 nm apart using the microscope in air with a light wavelength of 550 nm is approximately 1.63 mm.