Problem 46 - College Physics

Question

A thin planoconvex lens has a radius of curvature of magnitude 22.5 \(\mathrm{cm}\) on the curved side. When a color chart is placed 48.0 \(\mathrm{cm}\) from the lens, green light of wavelength 550 \(\mathrm{nm}\) is focused 277 \(\mathrm{cm}\) from the lens and blue light of wavelength 450 \(\mathrm{nm}\) is focused 17 \(\mathrm{I} \mathrm{cm}\) from the lens. What are the indices of refraction for these two wavelengths of light?

Step-by-Step Solution

Answer

The refractive indices are approximately 1.541 for green light and 2.793 for blue light.

Step 1: Understand the Lensmaker's Equation

The lensmaker's formula is used to determine the focal length of a lens given its physical properties and the refractive index of the material:\[ \frac{1}{f} = (n - 1)\left( \frac{1}{R} - \frac{1}{R'} \right) \]For a planoconvex lens, one side is flat, thus \( R' = \infty \), and the equation simplifies to:\[ \frac{1}{f} = (n - 1)\frac{1}{R} \]

Step 2: Use Gaussian Lens Formula

The Gaussian lens formula relates the object distance (\(d_o\)), the image distance (\(d_i\)), and the focal length (\(f\)) of the lens:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]Use this equation to find the focal length \(f\) for each wavelength.

Step 3: Calculate Focal Length for Green Light

For green light with \( d_o = 48.0 \, \text{cm} \) and \( d_i = 277 \, \text{cm} \), use:\[ \frac{1}{f_g} = \frac{1}{48} + \frac{1}{277} \]Calculate \( f_g \):\[ f_g = \frac{48 \times 277}{48 + 277} \approx 41.60 \, \text{cm} \]

Step 4: Calculate Refractive Index for Green Light

Using the simplified lensmaker's equation for green light:\[ \frac{1}{41.60} = (n_g - 1) \frac{1}{22.5} \]Solving for \( n_g \):\[ n_g = \frac{22.5}{41.60} + 1 \approx 1.541 \]

Step 5: Calculate Focal Length for Blue Light

For blue light with \( d_o = 48.0 \, \text{cm} \) and \( d_i = 17 \, \text{cm} \), use:\[ \frac{1}{f_b} = \frac{1}{48} + \frac{1}{17} \]Calculate \( f_b \):\[ f_b = \frac{48 \times 17}{48 + 17} \approx 12.55 \, \text{cm} \]

Step 6: Calculate Refractive Index for Blue Light

Using the simplified lensmaker's equation for blue light:\[ \frac{1}{12.55} = (n_b - 1) \frac{1}{22.5} \]Solving for \( n_b \):\[ n_b = \frac{22.5}{12.55} + 1 \approx 2.793 \]

Key Concepts

Lensmaker's Equation

The Lensmaker's Equation is a powerful formula that helps us find the focal length of a lens by considering both its shape and the refractive index. It generally looks like this:\[ \frac{1}{f} = (n - 1)\left( \frac{1}{R} - \frac{1}{R'} \right) \]This formula is handy because it takes into account the different curvatures of the lens surfaces. For a planoconvex lens, which has one flat surface, the equation simplifies:\[ \frac{1}{f} = (n - 1)\frac{1}{R} \]Here, \(R'\) is infinite for the flat side, simplifying the equation. By knowing the curvature and how light bends in the material, you can calculate the lens's focal length, helping predict how it will focus light in practical applications.
It's crucial to understand this principle when designing lenses for cameras or glasses, ensuring optimal light convergence or divergence.

Gaussian Lens Formula

The Gaussian Lens Formula links the object distance, image distance, and focal length of a lens. It's widely used in optics to determine where an image will form based on the object's position relative to the lens:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]Here:

• \(f\) is the lens's focal length.
• \(d_o\) is the distance from the object to the lens.
• \(d_i\) is the distance from the image to the lens.

By rearranging and using this equation, you can solve for the unknown focal length if the object and image positions are known. This formula is fundamentally important when determining how a lens will transform light paths, allowing you to design optical systems like cameras precisely or predict how they perform in different conditions.
Understanding this relation helps students and professionals in analyzing and constructing optical tools.

Refractive Index

The refractive index of a material is a measure of how much it slows down light passing through it compared to the speed of light in a vacuum. It's denoted by \(n\) and is a crucial concept in optics as it affects light bending when transitioning between materials. When light enters a denser medium, it bends towards the surface normal, and if it enters a less dense medium, it bends away from the normal. This behavior is why lenses can focus or disperse light.In the textbook exercise, refractive indices for different wavelengths (colors) of light are calculated using derived focal lengths. For green light, the refractive index was approximately 1.541, while blue light had a significantly higher index of around 2.793. These differences occur due to wavelength-dependent variations in how light refracts, explaining phenomena like dispersion in prisms.
Knowing the refractive index helps in selecting materials for lenses based on specific applications, ensuring functionality such as corrective lenses' accuracy or reducing chromatic aberrations in optical systems.

Planoconvex Lens

A planoconvex lens has one flat surface and one convex surface. It's commonly used in optical devices where converging light rays to a focal point is essential.
The shape helps in focusing light efficiently with minimal spherical aberration, making it ideal for tasks like laser focusing or as a component in imaging systems. For a planoconvex lens, the Lensmaker's Equation simplifies significantly because one of the surface curvatures is infinite. This simplification lets us focus on the properties of the convex side more closely. Such lenses are crafted from materials like glass or plastic, considering their refractive index to achieve the desired focal length. They are also chosen based on the application needs, whether for focusing light strongly or weakly. The study and use of planoconvex lenses underscore the importance of geometry and material properties in optical design.
Understanding their function broadens your ability to apply lens optics in real-world scenarios, ensuring effective design and use of optical systems.