Problem 70 - College Physics

Question

A plastic plano-concave lens has a radius of curvature of \(50 \mathrm{~cm}\) for its concave surface. If the index of refraction of the plastic is \(1.35,\) what is the power of the lens?

Step-by-Step Solution

Answer

The power of the lens is \(-0.7\) diopters.

Step 1: Understand the Lens Maker's Formula

The Lens Maker's formula is used to calculate the power of lenses, given by \( P = \left( n - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \) where \( P \) is the power of the lens, \( n \) is the index of refraction, and \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces. For a plano-concave lens, one side is flat (\( R_1 = \infty \)), and the concave side has a specific radius of curvature (\( R_2 = -50 \mathrm{~cm} \)).

Step 2: Apply Lens Maker’s Formula Parameters for Plano-Concave Lens

For a plano-concave lens, the formula simplifies as \( R_1 = \infty \) and \( R_2 = -50 \mathrm{~cm} \). Substitute these values into the formula:\[ P = \left( 1.35 - 1 \right) \left( \frac{1}{\infty} - \frac{1}{-50} \right) \]This simplifies to:\[ P = 0.35 \times \left( 0 + \frac{1}{50} \right) \]

Step 3: Compute the Power of the Lens

Now, calculate the power by evaluating the above expression:\[ P = 0.35 \times \frac{1}{50} \]\[ P = 0.35 \times 0.02 \]\[ P = 0.007 \text{ or } -0.7 \text{ diopters} \]Since the lens is concave, the focus is virtual and negative, thus the power is negative.

Key Concepts

Understanding Plano-Concave Lenses

A plano-concave lens is a special type of lens that has one flat surface and one inwardly curved surface. This makes it different from other types of lenses like convex or biconvex lenses. Because of its unique shape, a plano-concave lens can spread out light rays that pass through it, causing the light to diverge. This property is useful in applications where light needs to be spread out or where a wider field of view is preferable. Understanding this can help when dealing with optical devices, as these lenses are often used to manage and alter light paths.

The Role of the Index of Refraction

The index of refraction is a crucial concept in optics, as it measures how much a ray of light bends, or refracts, when it enters a different medium. A plano-concave lens made of plastic with an index of refraction of 1.35 means that light entering the lens will bend slightly more than it would in air (which has an index of refraction of approximately 1.00).

• A higher index of refraction indicates greater bending of light.
• The index value is determined by the composition of the material.
• It affects how efficiently the lens can direct and focus light.

Radius of Curvature in Lenses

The radius of curvature of a lens surface refers to the radius of the sphere from which the lens surface is a segment. For the plano-concave lens, this is only applicable to the curved side. The given radius of curvature is \(50 \, \text{cm}\), and being negative indicates that it is a concave surface. Here's why knowing the radius is essential:

• It determines how strongly the lens can converge or diverge light.
• The larger the radius, the more gradual the curve, leading to less bending of light.
• In calculations, it's crucial to note the sign, as it indicates the direction of curvature.

Lens Power Calculation

Calculating the power of a lens involves understanding its ability to converge or diverge light, which is measured in diopters. For a plano-concave lens, this power is determined via the lens maker’s formula. With one side flat and one curved, the formula simplifies. By implementing the given parameters:\[P = \left( n - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\]- Here, \(n\) is the index of refraction, given as 1.35.- \(R_2\) is \(-50 \, \text{cm}\)Substituting these:\[P = 0.35 \times \left( 0 + \frac{1}{50} \right) = -0.7 \text{ diopters}\]This negative value represents the divergent nature of the lens.