Problem 72 - Master Resource Book in JEE Main Physics

Question

Two points, separated by a distance of \(0.1 \mathrm{~mm}\), can just be inspeeted on a microscope when light of wavelength \(6000 \AA{A}\) is used. If the light of wavelength \(4800 \dot{A}\) is used, the limit of resolution is (a) \(0.8 \mathrm{~mm}\) (b) \(0.08 \mathrm{~mm}\) (c) \(0.1 \overline{\mathrm{mm}}\) (d) \(0.04 \mathrm{~mm}\)

Step-by-Step Solution

Answer

The limit of resolution is 0.08 mm (option b).

Step 1: Understand the Problem

We are given two points that can just be resolved using a microscope when the wavelength of light is \(6000 \text{ Å}\). We need to calculate the new limit of resolution (smallest distance) when the wavelength is changed to \(4800 \text{ Å}\).

Step 2: Recall the Resolving Power Formula

The resolving power of an optical instrument like a microscope is inversely proportional to the wavelength of light used. Mathematically, this relationship is described by Rayleigh's criterion: \[ d = \frac{1.22 \lambda}{D} \] where \(d\) is the limit of resolution, \(\lambda\) is the wavelength of light, and \(D\) is the diameter of the objective lens.

Step 3: Set Up Proportion for New and Old Conditions

Since the diameter \(D\) remains constant, the ratio of the old limit of resolution to the new one equals the ratio of their respective wavelengths: \[ \frac{d_\text{new}}{d_\text{old}} = \frac{\lambda_\text{new}}{\lambda_\text{old}} \] Given: \(d_\text{old} = 0.1\, \text{mm}\), \(\lambda_\text{old} = 6000 \text{ Å}\), and \(\lambda_\text{new} = 4800 \text{ Å}\).

Step 4: Calculate the New Limit of Resolution

By substituting the given values into the proportion, we get: \[ \frac{d_\text{new}}{0.1} = \frac{4800}{6000} \] Simplifying the fraction, \(\frac{4800}{6000} = \frac{4}{5}\). Therefore, \(d_\text{new} = 0.1 \times \frac{4}{5} = 0.08 \text{ mm}\).

Step 5: Verify the Answer

The result calculated from the proportion \(0.08 \text{ mm}\) matches option (b). Thus, the new limit of resolution when using \(4800 \dot{A}\) is \(0.08 \text{ mm}\).

Key Concepts

Wavelength Dependency

The concept of wavelength dependency is crucial in understanding how different wavelengths of light affect the resolving power of optical instruments like microscopes. Light travels in waves, and the distance between two consecutive peaks in a wave is known as the wavelength. When it comes to microscopes, the resolving power or the ability to distinguish two closely spaced points is influenced by the wavelength of light used.

• Shorter wavelengths (e.g., blue light) can offer better resolving power, helping you distinguish finer details in the specimen.
• Longer wavelengths (e.g., red light) tend to have lower resolving power, making it harder to separate closely spaced details.

This dependency arises because shorter wavelengths interact with smaller details more effectively. By switching to a light with a shorter wavelength, such as from 6000 Å to 4800 Å, the observable limit of resolution decreases, allowing us to view smaller gaps between points.

Rayleigh's Criterion

Rayleigh's criterion is a fundamental principle that defines the resolving power of optical systems. It explains under what conditions two points of light are perceived as separate. According to this criterion, two points are considered just visible separately if the central maximum of the diffraction pattern of one image coincides with the first minimum of the diffraction pattern of the other.Mathematically, the criterion is expressed as:\[ d = \frac{1.22 \lambda}{D} \]where:

• \(d\): the minimum distance at which two points can be resolved.
• \(\lambda\): the wavelength of the light.
• \(D\): the diameter of the lens or aperture.

This formula shows that the limit of resolution, \(d\), is directly proportional to the wavelength, meaning changes in wavelength directly affect how small of a detail you can observe with your microscope.Rayleigh's criterion helps us understand why changing from a 6000 Å wavelength to a 4800 Å wavelength allows for improved resolution, specifically reducing the limit of resolution from 0.1 mm to 0.08 mm.

Microscope Resolving Power

The resolving power of a microscope is a critical measure of its ability to distinctly observe two separate points that are close together. The effectively shorter the distance of these points, the greater the resolving power of the microscope. The resolving power is affected by several factors, including:

• The wavelength of light used, as dictated by wavelength dependency.
• The numerical aperture of the microscope lens, which is related to the size of the lens and its ability to gather light.

Higher resolving power leads to greater detail visible in the specimens being studied, which is especially important in biological and material sciences. Achieving high resolving power is often a balance between the light wavelength used and optimizing the physical properties of the microscope components. By applying principles like Rayleigh's criterion and considering wavelength effects, microscopes can be finely tuned to reach their optimal resolving power, which is essential for advanced and precise scientific research.