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The numerical aperture (NA) of a microscope objective lens is a critical factor in determining the resolution of the microscope. The NA is essentially a measure of the lens's ability to gather light and resolve fine specimen detail at a fixed distance. It is defined as:\[NA = n \cdot \sin(\theta)\]where:

• \(n\) is the refractive index of the medium between the lens and the specimen, typically air or oil.
• \(\theta\) is the half-angle of the maximum cone of light that can enter or exit the lens.

A higher NA indicates better resolving power of the microscope. It allows more light to be captured from the specimen, resulting in a clearer and more detailed image. For instance, an NA of 1.0 as used in the exercise means the objective has a good capacity to resolve fine details. High NA lenses are essential for seeing very small structures with great clarity.

In microscopes, the wavelength of light used for illumination is crucial in determining the resolution. Resolution refers to the ability of a microscope to distinguish two points as separate entities, and it is inversely proportional to the wavelength of the light used. Shorter wavelengths provide better resolution.
Typical visible light wavelengths range from about 400 nm (nanometers) to 700 nm. Blue light, around 450 nm, and green light, around 500 nm (as in the exercise), are often used in microscopy because they offer a balance between high resolution and confortable viewing.
The use of light with a wavelength of 500 nm, as given in the exercise, is quite common because the green region of the spectrum is highly visible to the human eye and provides a decent resolution for most microscopy applications. When using a microscope, adjusting the wavelength can affect how detailed the image appears.

The resolution limit of a microscope is the smallest distance between two points that the microscope can distinguish as separate. It defines the level of detail visible in the microscope image and depends on both the numerical aperture and the wavelength of light used.
The formula for calculating the resolution limit \(d\) is:\[d = \frac{\lambda}{2 \cdot NA}\]where \(\lambda\) is the wavelength of light and \(NA\) is the numerical aperture. In the exercise, substituting \(\lambda = 500 \times 10^{-9} m\) and \(NA = 1.0\) gives:\[d = \frac{500 \times 10^{-9} m}{2 \times 1.0} = 250 \times 10^{-9} m\]This calculation shows the resolution limit is 250 nm, meaning objects smaller than this cannot be resolved or seen clearly under the microscope. It's important for microscope users to choose the right combination of light wavelength and numerical aperture to achieve the resolution needed for their observations.