Problem 71 - College Physics

Question

An optometrist prescribes a corrective lens with a power of \(+1.5 \mathrm{D}\). The lens maker starts with a glass blank that has an index of refraction of 1.6 and a convex front surface whose radius of curvature is \(20 \mathrm{~cm}\). To what radius of curvature should the other surface be ground? Is the surface convex or concave?

Step-by-Step Solution

Answer

The second surface should be ground with a radius of curvature of approximately \\(-0.41 \, \mathrm{cm} \\) and it is concave.

Step 1: Understanding Lensmaker's Formula

The lensmaker's formula relates the power of a lens to the radii of curvature of its two surfaces and the index of refraction of the material. It is given by: \[ P = \left( (n-1) \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] where \(P\) is the lens power, \(n\) is the index of refraction, \(R_1\) is the radius of curvature of the first surface (convex), and \(R_2\) is the radius of curvature of the second surface. We need to solve for \(R_2\).

Step 2: Substitute Known Values

Substitute the given values into the lensmaker's formula: the lens power \(P = +1.5 \mathrm{D}\), the index of refraction \(n = 1.6\), and the radius of curvature for the convex surface \(R_1 = +20 \mathrm{~cm}\) (note that the sign is positive for a convex surface). The equation becomes: \[ +1.5 = \left(1.6 - 1\right) \left(\frac{1}{20} - \frac{1}{R_2}\right) \] Simplify the equation to find \(R_2\).

Step 3: Solve for \\(R_2\\)

First, compute the difference in indices: \[ 1.6 - 1 = 0.6 \] Now, multiply both sides of the equation by 20:\[ 1.5 \times \frac{1}{0.6} = \frac{1}{20} - \frac{1}{R_2} \] Solving for \(\frac{1}{R_2}\), we get: \[ 2.5 = \frac{1}{20} - \frac{1}{R_2} \] \( R_2 \) can then be calculated from: \[ \frac{1}{R_2} = \frac{1}{20} - 2.5 \] Calculate this value.

Step 4: Calculate Final Result

Perform the calculation from the earlier step:\[ \frac{1}{R_2} = \frac{1}{20} - 2.5 = \frac{1 - 50}{20} = \frac{-49}{20} \]Therefore, \(R_2 = -\frac{20}{49}\ ext{cm}\).Because \(R_2\) is negative, the surface is concave.

Key Concepts

Index of Refraction

The index of refraction is a fundamental property of materials that relates to how light travels through them. It's a dimensionless number that describes how much the speed of light is reduced inside the material compared to in a vacuum.
This property is crucial in lens design because it influences how much light bends, or refracts, when passing through a lens.

• A higher index of refraction means light will bend more, allowing for lenses with greater focusing power.
• This property is symbolized by the letter \( n \).
• In our lens problem, the index of refraction is 1.6 for the glass.

The choice of the index affects the lensmaker's formula, which is used to calculate the necessary shape and surface curvature of lenses to achieve a desired optical power. Using materials with a higher index of refraction enables thinner lenses for the same optical effect, which can be beneficial both for aesthetics and practicality.

Radius of Curvature

When dealing with lenses, the radius of curvature of their surfaces plays a pivotal role. It's the radius of the sphere from which the lens' surface could be imagined as a segment.
The radius of curvature determines how strongly the lens surfaces will refract or bend light. A smaller radius means a stronger curvature, leading to greater optical power.

• Lenses can have two radii of curvature: one for each side.
• The first radius, \( R_1 \), is given as 20 cm for the convex surface.
• The challenge in the solution involves finding \( R_2 \), the radius for the other surface, to achieve the needed lens power.

The lensmaker's formula incorporates these radii, along with the index of refraction, to calculate the lens's power. Precise modification of these curvatures helps in designing lenses with specific optical properties that can correct vision impairments or focus light as needed.

Concave and Convex Surfaces

Surfaces of lenses can be either convex or concave, affecting how light is directed. Convex surfaces bulge outwards, while concave surfaces curve inwards.

Understanding the difference is essential for applications like corrective lenses in eyeglasses.

• Convex lenses converge light rays, making them ideal for correcting farsightedness.
• In optical terms, a convex surface has a positive radius of curvature.
• Concave lenses diverge light rays, commonly used for nearsightedness.
• Here, concave surfaces feature negative radii of curvature.

In our example, the front surface is convex with a \( R_1 \) of 20 cm. Calculating \( R_2 \) results in a negative value, indicating the required surface is concave. This interplay between concave and convex shapes allows lenses to adjust or correct vision by precisely steering light rays to focus accurately on the retina.