Question

A light microscope has \(10 \times\) oculars and \(0.3-\mu \mathrm{m}\) resolution. Using the oil immersion lens \((100 \times)\), will you be able to resolve two objects \(400 \mathrm{~nm}\) apart? Will you be able to resolve two objects \(40 \mathrm{~nm}\) apart?

Short answer

Yes, objects 400 nm apart can be resolved as the calculated resolution (111 nm) is smaller than this distance. However, objects 40 nm apart cannot be resolved as the resolution (approximately 111 nm) is higher than this distance.

Step-by-step solution

Calculation of Numerical Aperture

In a microscope, numerical aperture (NA) can be expressed as the product of the refractive index (n) and the sine of the half of the angular aperture (a), i.e., NA = n × sin(a). Because the oil immersion lens is used, the refractive index should be approximately 1.4, and for microscopic observations, let's consider the value of `a` to be 140 degrees. Therefore, NA can be calculated as follows: NA = 1.4 × sin(140/2). After performing this calculation, we have the value of NA = 1.35.

Calculation of Resolution for Case 1

Once NA is known, we can then calculate the resolution using the given formula: resolution = \(\lambda\) / (2 × NA). First let's calculate resolution for the \(400 \mathrm{~nm}\) case. The given resolution \(\lambda\) should be translated from 0.3 μm to 300 nm for comparison. Substituting corresponding values into the formula, we obtain a resolution of approximately 111 nm.

Calculation of Resolution for Case 2

Now repeat the calculation for the 40 nm case. By substituting the respective values into the formula, we obtain a resolution of about 11 nm, as in the previous step.

Comparing the Results to the Given Distances

At this point, compare the distances you obtained with the given quantities 400 nm and 40 nm. If the resolution obtained is greater than the given distances, the respective objects can be resolved, otherwise not.

Key concepts

Numerical Aperture

In microscopy, the numerical aperture (NA) is a crucial factor that influences the resolving power of a microscope. The NA is a dimensionless number that characterizes the range of angles over which the lens can accept light. It can be calculated using the formula:

• NA = n × sin(a)

where "n" denotes the refractive index of the medium through which the light travels, and "a" represents the half-angle of the maximum cone of light that can enter the lens. For an oil immersion lens, the refractive index is often around 1.4. Using a larger refractive index medium like oil improves the NA and thus the resolution. Let's say you're using a microscopic setup with a half-angle ( space a) of 70 degrees for the oil immersion lens. The calculation with sin(70°) helps us understand that the NA can be as high as 1.35, enabling more detailed viewing of fine structures.

Oil Immersion Lens

An oil immersion lens is a powerful component in microscopy that significantly boosts the resolution of visualized samples. This is achieved by immersing both the lens and the specimen in a high-refractive index oil, usually cedarwood oil or synthetic options, which enhances image clarity and detail. Key uses and benefits include:

• Increasing the numerical aperture: Larger NA allows better resolution, as explained in the previous section.
• Reducing light refraction: Oil minimizes the bending of light rays, keeping them straight as they pass through. This helps in maintaining the integrity and focus of the image.
• Enhancing detailed observation: Ideal for studying intricate details of microorganisms and cell structures, oil immersion lenses are essential for high-magnification engagements, typically from 60x onwards.

For your 100x oil immersion lens, this setup fosters visibility of closely spaced entities down to the nano-scale, far beyond dry lens capabilities.

Microscopy Resolution Calculation

Understanding how to calculate and interpret microscopy resolution is pivotal in determining the capability of a microscope to resolve two separate points or objects. This calculation uses the formula:

• Resolution = \( \frac{\lambda}{2 \cdot NA} \)

where \( \lambda \) represents the wavelength of the light used.For microscopes operating at a light wavelength of 300 nm (as derived from 0.3 μm on conversion), applying this formula provides a resolution capacity of approximately 111 nm using the oil immersion setup. This theoretical resolution explains whether separate features, such as those 400 nm apart or down to even 40 nm, are distinguishable.In practice:

• If calculated resolution is less than the given separation distance (e.g., 400 nm), the microscope resolves the objects.
• If the resolution is greater (like 111 nm), objects at 40 nm are too close to be distinguished individually.

Thus, for your microscope setup using an oil immersion lens, objects at a distance of 400 nm will be resolved, but those 40 nm apart won't be distinguishable due to the resolution limit.