Problem 69 - College Physics

Question

A plastic convex meniscus (Fig. 23.14) contact lens is made of plastic with an index of refraction of \(1.55 .\) The lens has a front radius of \(2.50 \mathrm{~cm}\) and a back radius of \(3.00 \mathrm{~cm} .\) (a) The signs of \(R_{1}\) and \(R_{2}\) are \((1)+,+,(2)+,-\) (3) \(-,+,(4)-,-.\) Explain. (b) What is the focal length of the lens?

Step-by-Step Solution

Answer

(a) +, + (b) Focal length is approx. 2.48 cm.

Step 1: Understanding radius signs

The radii of curvature for lenses are signs that guide in determining the direction of curvature. A radius is positive if the corresponding surface is convex as seen from the outside of the lens. For a convex meniscus lens, the front radius \( R_1 = 2.50 \) cm is of a convex surface; thus, it is positive. The back radius \( R_2 = 3.00 \) cm is also convex from the outside, which makes it positive as well. Hence, the correct sign combination is (+,+).

Step 2: Selecting option for part (a)

Based on the reasoning that both \( R_1 \) and \( R_2 \) are positive, we select option (1) which is the combination of \(+,+\).

Step 3: Applying Lensmaker's formula

To find the focal length \( f \) of the lens, use the Lensmaker's formula:\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]where \( n = 1.55 \) is the index of refraction, \( R_1 = 2.50 \) cm, and \( R_2 = -3.00 \) cm. Notice, conventionally the back radius \( R_2 \) is negative when using the Lensmaker's formula.

Step 4: Calculating with given values

Substitute the given values into the Lensmaker's formula:\[ \frac{1}{f} = (1.55 - 1) \left( \frac{1}{2.50} - \frac{1}{-3.00} \right) \]\[ \frac{1}{f} = 0.55 \left( 0.4 + 0.3333 \right) \]Calculate the above expression to find \( f \).

Step 5: Solving for focal length

Continuing from the calculation in the previous step:\[ \frac{1}{f} = 0.55 \times 0.7333 \]\[ \frac{1}{f} = 0.4033 \]Therefore, \( f = \frac{1}{0.4033} \approx 2.48 \) cm. The focal length of the lens is approximately 2.48 cm.

Key Concepts

Index of Refraction

The index of refraction is a key concept when dealing with lenses. This number describes how much a material can bend light.
It is a measure of the speed of light in a given medium compared to the speed of light in a vacuum.

• A higher index means the light slows down more and bends more as it enters the material.
• In the exercise, the plastic lens has an index of refraction of 1.55, meaning light travels 1.55 times slower in the lens than in open air.

This property is important because it affects how lenses focus light.
By bending the light rays, lenses can form images and magnify objects.

Radius of Curvature

The radius of curvature refers to the radius of a sphere from which a lens surface has been sliced. For lenses, it determines whether the lens is convex or concave.

• A surface with a positive radius of curvature is convex as it bulges outwards.
• If the radius is negative, the surface is concave, curving inwards.

In the context of the given problem, it is essential to correctly identify these radii.
The lens has a front radius of 2.50 cm and a back radius of 3.00 cm; both are positive, indicating convex surfaces.
Understanding these radiations helps in applying the Lensmaker's formula to find out how the lens will focus light.

Focal Length

The focal length of a lens is the distance between the lens and the focal point where parallel light rays converge. It tells you how strongly the lens bends incoming light. Brighter or larger images are formed when the lens has a short focal length, which means strong bending for a given index of refraction and radius. The Lensmaker's formula is a vital tool in determining this property:\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]By substituting the values, namely the index of refraction 1.55 and the radii 2.50 cm and \(-3.00\) cm, we calculate the focal length as approximately 2.48 cm, which helps in finding out how the lens will work in practical scenarios.

Convex Lens

A convex lens is a type of lens where the surfaces bulge outward. This shape causes light rays to converge or focus, making it ideal for applications like magnifying glasses or corrective lenses.

• Convex lenses have positive focal lengths, indicating that they converge light.
• They are commonly used in devices that need to magnify or focus light onto a specific point.

The plastic convex meniscus discussed in the exercise serves this purpose.
It has both radii as convex, leading to a converging lens which sharply focuses the light to assist with vision or magnification tasks.