Question

A double-convex thin lens has surfaces with equal radii of curvature of magnitude 2.50 cm. Using this lens, you observe that it forms an image of a very distant tree at a distance of 1.87 cm from the lens. What is the index of refraction of the lens?

Short answer

The index of refraction of the lens is approximately 1.668.

Step-by-step solution

Understanding the Lensmaker's Formula

To solve this problem, we start by recognizing that it's based on the Lensmaker's Formula for thin lenses, which is given by: \[ \frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] where \( f \) is the focal length of the lens, \( n \) is the index of refraction, and \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces.

Identifying Known Values

From the problem, we know that the lens forms an image at 1.87 cm, which is the focal length \( f = 1.87 \) cm. The lens is double-convex with equal radii, so \( R_1 = 2.50 \) cm and \( R_2 = -2.50 \) cm (the negative sign indicates that one radius points in the opposite direction).

Substituting Known Values into the Formula

Substitute the known values into the lensmaker's formula: \[ \frac{1}{1.87} = (n - 1) \left( \frac{1}{2.50} - \frac{1}{-2.50} \right) \]. This equation simplifies further because \( \frac{1}{2.50} - \frac{1}{-2.50} = \frac{1}{2.50} + \frac{1}{2.50} = \frac{2}{2.50} \).

Simplifying the Equation

Simplify the equation: \[ \frac{1}{1.87} = (n - 1) \left( \frac{2}{2.50} \right) \]. Solving for \( n \), gives us: \[ n - 1 = \frac{2.50}{2 imes 1.87} \].

Calculating the Index of Refraction

Calculate the value for \( n \) using the simplified form: \[ n - 1 = \frac{2.50}{3.74} \] which simplifies to \[ n - 1 \approx 0.6684 \] so \[ n \approx 1.6684 \].

Conclusion

The index of refraction of the lens is approximately 1.668.

Key concepts

Index of Refraction

The index of refraction, often represented by the symbol \( n \), is a critical property of optical materials, like lenses. It describes how light bends or refracts when moving from one medium to another. A higher index means light bends more when entering the material.

In formulaic terms, the index is calculated by the ratio of the speed of light in a vacuum to the speed of light in the material. For example, the formula given,

• \( n = \frac{c}{v} \) where \( c \) is the speed of light in a vacuum and \( v \) is the speed of light in the material.

In the context of a lens, this refraction is what allows the lens to focus light and form images. The index of refraction varies between materials, allowing us to predict how they will interact with light in applications like glasses, microscopes, and cameras.

Focal Length

The focal length of a lens, denoted as \( f \), is the distance between the lens and the point where it converges light rays to a focus. It is a fundamental measurement that determines a lens's capacity to magnify objects or bring distant objects into focus.

For a thin lens, like the one in this exercise, the focal length is directly related to its optical power. The shorter the focal length, the more powerful the lens is, meaning it can bend light more sharply.

• It is given by \( rac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \), depending on the lens's index of refraction \( n \), and its radii of curvature \( R_1 \) and \( R_2 \).

Understanding focal length is essential for designing lenses for specific imaging applications where the sharpness and clarity of the image are paramount.

Radii of Curvature

The radii of curvature of a lens, represented as \( R_1 \) and \( R_2 \) for the two surfaces, tell us about the curvature of each surface. It’s how "bowed" or curved the lens is. The radius is considered positive if the center of curvature is on the same side as the incoming light, and negative if on the opposite side.

In our context, a double-convex lens will have one convex surface facing the light source and the other facing away, usually giving one positive and one negative radius. In the equation

• \( \frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \),

these values help determine the lens's focal length by affecting how much the lens can focus light into a single point.

Double-Convex Lens

A double-convex lens, sometimes just called a biconvex lens, is a type of lens where both surface curves are outward. It's the classic "magnifying glass" shape that aids in converging light to a point. This shape is effective for focusing light to create clear images.

Double-convex lenses are often used in applications requiring image magnification or correction of visual aberrations. Compared to other lenses, they typically provide a well-balanced image quality because both surfaces contribute to light focusing.

• Common in optometry and photography, their symmetric shape offers versatility and effectiveness in bringing distant objects into sharp view.

The focal length, calculated via the lensmaker's formula, is an essential attribute, affected by both the radii of curvature and the material's index of refraction.