Question

The radius of curvature of convex surface of a thin plano-convex lens is \(15 \mathrm{~cm}\) and refractive index of its material is \(1.6 .\) The power of the lens will be (A) \(6 \mathrm{D}\) (B) \(5 \mathrm{D}\) (C) \(4 \mathrm{D}\) (D) \(3 \mathrm{D}\)

Short answer

The power of the lens can be found using the lens maker's formula for a plano-convex lens and the given values of radius of curvature (15 cm) and refractive index (1.6). By calculating the focal length (f = 25 cm) and then the power using the formula Power = 1/focal length, we find that the power of the lens is \(4 \mathrm{D}\). Therefore, the answer is (C) 4D.

Step-by-step solution

Identify the given values

The radius of curvature of the convex surface (R) is 15 cm, and the refractive index (n) is 1.6.

Apply the lens maker's formula

The lens maker's formula for a plano-convex lens can be represented as follows:\[ \dfrac{1}{f} = (n - 1) \left[ \dfrac{1}{R_1} - \dfrac{1}{R_2}\right] \]Since the lens is plano-convex, one surface is flat, which means one radius of curvature is infinite (R₂ = ∞). The radius of curvature of the convex side is R₁ = 15 cm. Now, we can insert the given values into the lens maker's formula: \[ \dfrac{1}{f} = (1.6 - 1) \left[ \dfrac{1}{15} - \dfrac{1}{\infty}\right] \]

Calculate the focal length

To calculate the focal length from the given values, first simplify the equation: \[ \dfrac{1}{f} = 0.6 \times \dfrac{1}{15} \] \[ \dfrac{1}{f} = \dfrac{1}{25} \] Now, reciprocate both sides to find the focal length (f): \[ f = 25 \mathrm{~cm} \]

Calculate the power of the lens

The power of a lens is calculated using the formula Power = 1/focal length, with the focal length expressed in meters: \[ \text{Power} = \dfrac{1}{f} \] \[ \text{Power} = \dfrac{1}{0.25 \mathrm{~m}} \] \[ \text{Power} = \dfrac{1 \times 100}{25} \mathrm{D} \] \[ \text{Power} = 4 \mathrm{D} \]

Matching the answer

We have found that the power of the lens is 4D. Thus, the correct answer is (C) 4D.

Key concepts

Plano-Convex Lens

A plano-convex lens is a type of lens with one flat surface and one convex surface. This means that the lens is flat on one side and curves outward on the other side. The shape of a plano-convex lens directs light to converge to a single focal point, making it effectively a part of a sphere. These lenses are widely used in various optical applications due to their ability to focus light efficiently.

Some key characteristics of plano-convex lenses include:

• They are used to focus light, such as in magnifying glasses or telescopes.
• The concave shape on one end allows for efficient focusing with minimal aberrations.
• Their design makes them suitable for applications where light needs to traverse through the lens in one direction without significant scatter.

Overall, the unique shape of a plano-convex lens, combining both a curved and flat surface, makes it versatile in optical applications where precise light direction is needed.

Refractive Index

The refractive index is a fundamental property of materials that affects how they bend light. Represented by the symbol \( n \), it describes how much light slows down when it passes through a particular medium compared to its speed in a vacuum.

The refractive index is calculated using the ratio:\[ n = \dfrac{c}{v} \]where \( c \) is the speed of light in a vacuum and \( v \) is the speed of light in the material.

• A higher refractive index means the material bends light more sharply.
• It is crucial in designing lenses for cameras, glasses, or other optics.
• Materials with a high refractive index are often denser and heavier.

In the context of a lens, the refractive index determines how effectively the lens can focus light. A higher refractive index allows a lens to bend light more, facilitating greater optical power without significantly increasing the size of the lens.

Radius of Curvature

The radius of curvature pertains to the curved surface of a lens, especially in lenses like the plano-convex lens. It measures the distance from the center of the sphere from which the lens curvature is derived to the lens surface itself.

This property defines how much the lens surface bends and directly influences how well the lens focuses light.

• A lens with a smaller radius of curvature will bend light more steeply.
• The larger the radius, the gentler the curve of the lens.
• In convex lenses, the radius of curvature is typically positive, indicating that the center of curvature is on the same side as the source of light.

Understanding the radius of curvature is important when applying the lens maker's formula, as it helps determine the overall power and effectiveness of the lens in bending light accurately.

Optical Power

Optical power of a lens is a measure of its ability to converge or diverge light. It is quantified in diopters (D) and is mathematically the inverse of the focal length of the lens in meters.

Optical power is calculated as:\[ \text{Power} = \dfrac{1}{f} \]where \( f \) is the focal length.

Key points about optical power include:

• Positive power indicates a converging lens, like the plano-convex lens, which brings light to a point.
• Negative power indicates a diverging lens which spreads light apart.
• The higher the power, the stronger the lens's ability to focus or spread light.

In practical applications, knowing the optical power is essential for selecting the right lens for specific tasks, such as focusing cameras, eyeglasses prescriptions, or designing optical instruments.