Problem 31 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

The refractive index is approximately 1.67.

Step 1: Understand Lens Maker's Equation

The Lens Maker's Equation is crucial in solving lens-related problems. For a thin lens, the formula is given by \(\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\), where \(f\) is the focal length, \(n\) is the index of refraction, and \(R_1\) and \(R_2\) are the radii of curvature of the lens surfaces. For a double-convex lens with equal radii, \(R_1 = -R_2 = 2.50\, \mathrm{cm}\).

Step 2: Relate Image Distance to Focal Length

Since the lens forms an image of a very distant object, the distance of the image is equal to the focal length of the lens. Thus, \(f = 1.87\, \mathrm{cm}\).

Step 3: Substitute Values into Lens Maker's Equation

Plug the known values into the equation: \(\frac{1}{1.87} = (n - 1) \left( \frac{1}{2.50} + \frac{1}{2.50} \right)\). The terms concerning the radii add up because the lens is symmetric.

Step 4: Simplify and Solve for the Refractive Index

Simplify the equation: \(\frac{1}{1.87} = (n - 1) \left( \frac{2}{2.50} \right)\). Calculate \(\frac{1}{1.87} \approx 0.535\) and \(\frac{2}{2.50} = 0.8\). Thus, \(0.535 = 0.8(n - 1)\). Solve for \(n\): \(n - 1 = \frac{0.535}{0.8}\), which simplifies to \(n \approx 1.67\).

Step 5: Conclusion and Verify Calculations

The calculated refractive index is \(n \approx 1.67\). Ensure calculations were done correctly by verifying each arithmetic step; the correct significant figures should be maintained based on given data.

Key Concepts

Double-Convex Lens

A double-convex lens is a lens that is curved outward on both sides, resembling the shape of a ready-to-pop bubble. This shape allows it to converge light, making it ideal for applications where light focusing is necessary.
Such lenses are commonly used in glasses, cameras, and optical instruments because they help in gathering and focusing light from an object.
This focusing ability is important when creating images of objects, as seen in today’s understanding of how your lens can form an image of a distant tree.

• In a double-convex lens, both sides have equal radii of curvature when symmetric, which means the bending of light on both sides is balanced.
• This symmetry simplifies calculations, especially in the context of the Lens Maker's Equation.
• The degree of curvature helps determine how much the light rays converge, impacting the lens’s focal length.

Understanding this construction allows us to calculate how light will behave as it passes through these types of lenses.

Index of Refraction

The index of refraction, denoted as ‘n’, is a measure of how much a substance can bend or refract light.
It is a crucial value that dictates how rays of light travel through different materials, such as glass or water.
The higher the index of refraction, the more the light will be bent.

• In the exercise, calculating the index of refraction was essential to understand how the lens forms an image.
• For the double-convex lens, we determined the lens had an index of refraction of approximately 1.67, meaning the lens material bends light significantly, making it efficient for focusing light.
• Using the Lens Maker's Equation, the index was solved considering known focal length and radii of curvature.

Understanding the index is key when determining the lens quality and what material to use for constructing various optical devices.

Focal Length

The focal length of a lens is the distance over which parallel rays of light either converge or appear to diverge after passing through the lens.
This distance is a crucial factor in determining the lens's ability to focus light and, consequently, produce clear images.
The focal length is denoted by ‘f’.

• In the given exercise, the focal length was directly related to the image formed by the lens at 1.87 cm.
• This means any distant object, like a tree, would be imaged at this distance from the lens, an essential concept in optics.
• A short focal length indicates a strong lens, which can converge light rays quickly. Longer focal lengths mean the lens converges light more gently.

The ability to determine focal length is central to designing lenses for specific applications, such as in cameras or scientific equipment.