Problem 16 - Master Resource Book in JEE Main Physics

Question

A double convex lens made out of glass (refractive index, \(\mu=15\) ) has both radii of curvature of magnitudes \(20 \mathrm{~cm}\). Incident light rays parallel to the axis of this lens will converge at a distance, \(d\) such that (a) \(d=10 \mathrm{~cm}\) (b) \(d=\frac{20}{3} \mathrm{~cm}\) (c) \(d=40 \mathrm{~cm}\) (d) \(d=20 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The light rays converge at a distance \(d = 20 \mathrm{~cm}\), option (d).

Step 1: Understanding the Lens Maker's Formula

To find the focal length of a lens, we use the lens maker's formula: \(\frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\), where \(n\) is the refractive index of the lens material, and \(R_1\) and \(R_2\) are the radii of curvature of the lens surfaces.

Step 2: Identify Parameters

Given that the lens is double convex, both radii of curvature have magnitude \(20 \mathrm{~cm}\). For a double convex lens, \(R_1 = +20 \mathrm{~cm}\) and \(R_2 = -20 \mathrm{~cm}\). The refractive index \(\mu = 1.5\) for glass.

Step 3: Plug Values into Lens Maker's Formula

Plug the known values into the formula: \(\frac{1}{f} = (1.5-1) \left( \frac{1}{20} - \frac{1}{-20} \right)\). This simplifies to \(\frac{1}{f} = 0.5 \left( \frac{1}{20} + \frac{1}{20} \right)\).

Step 4: Calculate the Focal Length

Calculate the terms inside the parentheses first: \(\frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10}\). Therefore, \(\frac{1}{f} = 0.5 \times \frac{1}{10} = \frac{1}{20}\).

Step 5: Find the Focal Distance

Since \(\frac{1}{f} = \frac{1}{20}\), the focal length \(f = 20 \mathrm{~cm}\). Thus the distance \(d = f = 20 \mathrm{~cm}\).

Key Concepts

Double Convex Lens

A double convex lens is a type of lens that is curved outward on both sides, similar to the shape of a lentil seed. This unique curvature allows it to converge light beams that enter parallel to its principal axis.

The double convex lens works on the principle that light slows down when entering a denser medium, such as glass. As light enters the lens, it bends towards the normal due to refraction. Once it exits the lens, it bends again, away from the normal, converging at a focal point on the other side. This focal point is where all parallel rays of light meet after passing through the lens.

• Converges light rays, reducing their spread.
• Forms real and inverted images if the object is outside the focal length.
• Curved outward from both sides, resembling a convex shape.

With applications ranging from eyeglasses to magnifying glasses, double convex lenses are widely used in optics to focus light more effectively.

Refractive Index

The refractive index, represented by the symbol \( n \) or sometimes \( \mu \), is a measure of how much the speed of light reduces when passing through a medium. In essence, it quantifies the bending of light when entering a material.

Mathematically, it is the ratio of the speed of light in a vacuum to that in a given medium. For example, the refractive index of glass is often about 1.5, meaning light travels 1.5 times slower in glass than in a vacuum.

• Refractive Index \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum and \( v \) is the speed of light in the medium.
• A higher refractive index indicates greater bending ability.
• Different materials have different refractive indices influencing how they bend light.

In lens making, the refractive index helps in determining the focal length, leveraging the lens maker's formula. Understanding this property is crucial in designing lenses for precise optical instruments.

Radii of Curvature

The radii of curvature refer to the distances from the center of curvature to the lens' surfaces. For a double convex lens, this involves two radii, each corresponding to one of the convex surfaces.

In a simple visual: imagine slicing a sphere and examining the curve. The radius of curvature is like the radius of the circular slice you see. In lenses, it determines how pronounced these curves are; a smaller radius indicates a sharper curve.

• Labelled as \( R_1 \) and \( R_2 \) for the two surfaces.
• Positive value for surfaces bulging outwards (convex).
• Used in lens maker’s formula to calculate a lens’ focal length.

For this exercise's double convex lens, both radii are of equal magnitude but opposite signs: \( R_1 = +20 \text{ cm} \) and \( R_2 = -20 \text{ cm} \). These values influence how light converges through the lens, affecting focal point and image formation.