Problem 13 - Arihant AIEEE Physics

Question

The focal lengths of a thin convex lens for red and violet colours are \(44.6 \mathrm{~cm}\) and \(42.5 \mathrm{~cm}\). The focal length for the mean colour and dispersive power of the lens are respectively: (a) focal length \(=43.53 \mathrm{~cm}\) dispersive power \(=0.048\) (b) focal length \(=28.53\) dispersive power \(=0.048\) (c) focal length \(=63.53 \mathrm{~cm}\) dispersive power \(=8.48\) (d) focal length \(=30.43\) dispersive power \(=4.8\)

Step-by-Step Solution

Answer

(a) focal length \(=43.53 \, \text{cm}\), dispersive power \(=0.048\)

Step 1: Identify Given Data

We have the focal lengths for red and violet colors as \(f_r = 44.6 \, \text{cm}\) and \(f_v = 42.5 \, \text{cm}\). We need to find the focal length for the mean color and the dispersive power of the lens.

Step 2: Calculate the Focal Length for Mean Color

The formula for the focal length for the mean color \(f_m\) is the arithmetic mean of \(f_r\) and \(f_v\). Calculate it: \[f_m = \frac{f_r + f_v}{2} = \frac{44.6 + 42.5}{2} = \frac{87.1}{2} = 43.55 \, \text{cm}\]

Step 3: Calculate the Dispersive Power

The dispersive power, \(\omega\), is given by the formula: \[\omega = \frac{f_r - f_v}{f_m} = \frac{44.6 - 42.5}{43.55} = \frac{2.1}{43.55} \approx 0.0482\]Round to two decimal places, \(\omega \approx 0.048\).

Step 4: Choose the Correct Option

Comparing our results with the given options:- The calculated focal length is \(43.55 \, \text{cm}\) (very close to \(43.53 \, \text{cm}\))- The dispersive power is \(0.048\).Thus, option (a) matches our calculation.

Key Concepts

Convex Lens

A convex lens is a piece of transparent material, like glass, with at least one surface that curves outward. It's thicker in the middle than at the edges. This type of lens converges light rays that pass through it, bringing them to a focus. Convex lenses are used in various optical devices, such as cameras, glasses, and telescopes. When light travels through a convex lens, it bends toward the thicker center of the lens.

• It gathers light and focuses it to form images, which can be real or virtual.
• Convex lenses are essential for magnifying and projecting images.
• They have diverse applications in scientific and everyday life, allowing us to observe small details or distant objects.

Understanding how a convex lens works is fundamental to optics and is key in studying light behavior and image formation.

Focal Length

The focal length of a lens is the distance from the center of the lens to the point where converging light rays meet, called the focus. For a convex lens, this is where parallel rays of light come together. It determines how strongly the lens converges or diverges light. The smaller the focal length, the more the lens bends light to bring it into focus.

• Focal length is inversely related to the bending or convergence power of a lens.
• It is measured in centimeters (cm) or millimeters (mm) in optics.
• The focal length influences the field of view of optical devices, such as cameras or microscopes, determining how much of the scene is captured.

Calculating the average focal length for different colors can show how a lens might affect various wavelengths of light differently.

Dispersive Power

Dispersive power measures a lens's ability to separate light into its component colors, like a prism. It results from varying refractive indices for different wavelengths of light, meaning each color bends by a different amount. In a convex lens, dispersive power could lead to chromatic aberrations if the colors cannot focus at the same point.

• Calculating dispersive power helps in designing lenses that minimize these aberrations, improving image quality.
• A lens with high dispersive power makes red and violet colors focus at different points, but modern lenses aim to minimize this effect.
• By knowing the dispersive power, optical engineers can select materials that reduce unwanted color fringing.

Understanding dispersive power helps ensure that lenses provide clear and accurate images.

Mean Color

Mean color, in the context of lenses, refers to the average focal length derived from the focal lengths for different colors, usually red and violet. Different colors have different focal lengths due to their dispersion through lenses. The mean color represents the average focus of all colors and helps predict the general behavior of light passing through the lens.

• Calculating mean color helps in evaluating the general performance of a lens for mixed light.
• The mean color calculation is crucial for aligning the focus of optical devices using white light, which comprises all visible colors.
• This concept assists in correcting lenses to achieve sharper and more accurate imaging, especially in devices like cameras and microscopes.

Knowing how to find the mean color can refine the design and application of optical lenses for better performance.