Problem 25 - Master Resource Book in JEE Main Physics

Question

The dispersive power of the material of lens of foeal length \(20 \mathrm{~cm}\) is \(0.08 .\) The longitudinal chromatic aberration in of the lens is [a) \(0.08 \mathrm{~cm}\) (b) \(1.6 \mathrm{~cm}\) (c) \(0.8 \mathrm{~cm}\) (d) \(0.16 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The longitudinal chromatic aberration of the lens is 1.6 cm.

Step 1: Understand the Given Data

We are given the focal length (f) of the lens as 20 cm and the dispersive power (\(\omega\)) of the lens material as 0.08. We need to find the longitudinal chromatic aberration.

Step 2: Recall the Formula for Longitudinal Chromatic Aberration

The formula for calculating the longitudinal chromatic aberration (LCA) is given by \( \text{LCA} = f \cdot \omega \), where \(f\) represents the focal length and \(\omega\) represents the dispersive power of the lens material.

Step 3: Substitute the Known Values

Substitute the known values into the formula: \( \text{LCA} = 20 \text{ cm} \times 0.08 \).

Step 4: Calculate the Result

Perform the multiplication: \( 20 \times 0.08 = 1.6 \text{ cm} \). Thus, the longitudinal chromatic aberration is \(1.6 \text{ cm}\).

Step 5: Verify with Options

Among the given options, the closest value to our calculated result is option (b) \(1.6 \text{ cm}\). Hence, this is the correct choice.

Key Concepts

Dispersive Power

The concept of dispersive power is crucial in understanding how different colors of light are separated by a lens. Dispersive power, symbolized as \(\omega\), represents the measure of a lens material's ability to separate different wavelengths of light due to dispersion. This occurs because different wavelengths of light have varying speeds when they pass through a material, thus bending light to different extents.
In optics, dispersive power is expressed as a dimensionless number. The higher the dispersive power, the more the material separates colors, which can lead to phenomena such as chromatic aberration. This property is significant in designing optical systems where precise color layering is essential, such as in cameras and telescopes. Being aware of dispersive power helps in selecting the right lens material for specific applications, particularly in minimizing unwanted fringing effects.

Focal Length

Focal length is a fundamental concept in optics that plays a major role in how lenses bend light. Represented usually by the letter \(f\), it refers to the distance between the lens and the point where light rays converge to form a clear image. Simply put, it defines the "power" of the lens to focus light.
Shorter focal lengths mean that the lens can converge light quickly, resulting in a stronger magnifying effect and a wider field of view. This is why short focal lengths are preferred in wide-angle photography. On the other hand, lenses with longer focal lengths tend to zoom in on distant objects, suitable for applications like telephoto photography. In terms of chromatic aberration, focal length affects how significantly different colors separate along the optical axis, impacting the clarity and color accuracy of an image.

Lens Material

The material of the lens is pivotal as it affects both the dispersive power and the focal length, directly influencing optical performance. Each type of lens material, like glass or plastic, has its own refractive index which determines how light is bent when passing through.
For instance, specialty optical glasses are engineered to have specific refractive properties that allow for precise control over focal length and minimize chromatic aberration. The material’s ability to transmit light effectively without adding color distortion means it is crucial in scientific instruments and high-quality photography lenses. Selecting the right lens material involves balancing factors like cost, weight, and desired optical outcomes, often making trade-offs based on the application's specific needs.

Optics Formula

Understanding and applying optics formulas is essential when dealing with lenses and light manipulation. In the context of this exercise, the formula for longitudinal chromatic aberration—\( \text{LCA} = f \cdot \omega \)—highlights the critical relationship between focal length \(f\) and dispersive power \(\omega\). This formula helps quantify how much colors will separate or "spread out" along the axis of a lens.
Optics formulas are tools used to predict how lenses behave under various conditions, facilitating the design of optical systems. From calculating image formation to analyzing the paths of light rays through different media, these formulas are indispensable in the field of optics. By understanding these relationships, one can make informed decisions about lens selection and configuration to achieve desired visual results.