Problem 107 - Physics for Scientists and Engineers with Modern Physics

Question

In a science-fiction novel, an intelligent ocean-dwelling creature's eye functions underwater with a near point of 25 \(\mathrm{cm} .\) This creature would like to create an underwater magnifier out of a thin plastic container filled with air. What shape should the air-filled plastic container have (i.e., determine radii of curvature of its surfaces) in order for it to be used by the creature as a \(3.0 \times\) magnifier? Assume the eye is focused at its near point.

Step-by-Step Solution

Answer

The magnifier should be a planoconvex lens with one surface radius of 2.08 cm.

Step 1: Understanding the magnification requirement

The creature wants a magnifier that provides a magnification of 3.0. The magnification (\(M\)) for simple magnifiers is defined as \(M = \dfrac{25}{F}\), where \(F\) is the focal length in centimeters. Therefore, to achieve a 3.0 magnification, we can set up the equation: \(3.0 = \dfrac{25}{F}\).

Step 2: Solve for the focal length

To determine the focal length \(F\) that provides the desired magnification, solve the equation \(3.0 = \dfrac{25}{F}\):\[F = \dfrac{25}{3.0} = 8.33\, \text{cm}\]

Step 3: Determine the lens maker's equation

We use the lens maker's formula to find the radii of curvature of the lens needed for this focal length. The lens maker's equation is:\[\dfrac{1}{F} = (n - 1) \left( \dfrac{1}{R_1} - \dfrac{1}{R_2} \right)\]where \(n\) is the refractive index of the lens material, and \(R_1\) and \(R_2\) are the radii of curvature of the lens surfaces.

Step 4: Analyze refraction conditions

The magnifier is underwater, so the refractive index of the lens \(n\) in water becomes: \(n_{air/water} = \dfrac{n_{air}}{n_{water}} \approx \dfrac{1.0}{1.33} \approx 0.75\). Substitute \(F = 8.33\, \text{cm}\) and \(n_{air/water} = 0.75\) into the lens maker's equation: \[\dfrac{1}{8.33} = (0.75 - 1) \left( \dfrac{1}{R_1} - \dfrac{1}{R_2} \right)\] This simplifies to:\[-0.12 = -0.25 \left( \dfrac{1}{R_1} - \dfrac{1}{R_2} \right)\] which further simplifies to: \[0.48 = \dfrac{1}{R_1} - \dfrac{1}{R_2}\]

Step 5: Decide radii of curvature for the magnifier

To achieve a positive focal length (convex lens), we can choose one surface with a positive radius and the other with a negative radius or zero if simpler. Possible options:1. Planoconvex lens: \(R_1 = 2.08\, \text{cm}, R_2 = \infty\) gives \(1/2.08\) fulfilling \(0.48\).2. Symmetrical Convex lens: take symmetrical radii like \(R_1 = 4.16\, \text{cm}, R_2 = -4.16\, \text{cm}\).Either configuration would work based on practicality in creating the lens.

Key Concepts

Magnification

Magnification is a key concept in optics and is used to describe how much larger or smaller an object appears in a given optical instrument, like a magnifying glass or a lens. It is often represented by the symbol

• M = \( \dfrac{25}{F} \)

Where \( M \) stands for magnification, and \( F \) represents the focal length of the lens in centimeters. In the context of simple magnifiers, they are typically defined to have a magnification based on a standard observation distance, often taken as 25 cm. This distance is roughly the closest point at which the average human eye can clearly focus on an object without strain. For the intelligent ocean-dwelling creature in the exercise, this means altering the focal length of their magnifying lens to achieve a specific magnification level. For their desired 3x magnification, one would set \( 3.0 = \dfrac{25}{F} \) and solve to find the focal length that pairs with this magnification, resulting in \( F = 8.33 \, \text{cm} \).

By understanding magnification, one effectively alters the perceived size of objects under observation through careful adjustment of the focal length.

Lens Maker's Equation

The lens maker's equation is an essential piece of optics that assists in determining the design of lenses. It's essentially a bridge between the lens material's properties and the construction parameters required for a particular optical performance.

• \[ \dfrac{1}{F} = (n - 1) \left( \dfrac{1}{R_1} - \dfrac{1}{R_2} \right) \]

In the equation:

• \(F\) is the focal length,
• \(n\) is the refractive index of the lens material,
• \(R_1\) and \(R_2\) are the radii of curvature of the lens surfaces.

For this underwater magnifier, since the lens was filled with air and submerged in water, the adjusted refractive index became \( n_{air/water} \approx 0.75 \).

Substituting values into the lens maker's equation helps derive the necessary radii of curvature to fulfill the optical design of the magnifier for the ocean creature. This meticulous adjustment is pivotal for the performance of optical instruments in varying environments, such as underwater conditions.

Focal Length

Focal length is an important characteristic of lenses, describing the distance between the lens and the image sensor when the subject is in focus. It determines not only magnification but also the field of view.A shorter focal length translates into a wider field of view, while a longer focal length gives a more narrow field of view. In the context of a magnifying lens

• The focal length \(F\) influences how much the image will be magnified.

As derived in the solution, we determined \(F = 8.33\, \text{cm}\) to achieve a magnification of 3.0. This was calculated using the dimension of the near point as a reference for practical reading or observation distance.The entire optical system's design will pivot around ensuring the indicated focal length aligns with the desired usage environment, in this case, underwater, braced for the creature’s unique vision characteristics.Focusing on focal length is crucial for formulating lenses for specific applications, much like tailoring a tool for precise tasks.