Problem 93 - Fundamentals of Physics

Question

Someone with a near point \(P_{n}\) of \(25 \mathrm{~cm}\) views a thimble through a simple magnifying lens of focal length \(10 \mathrm{~cm}\) by placing the lens near his eye. What is the angular magnification of the thimble if it is positioned so that its image appears at (a) \(P_{n}\) and (b) infinity?

Step-by-Step Solution

Answer

(a) 3.5 (b) 2.5

Step 1: Understanding the Concept of Angular Magnification

Angular magnification is the factor by which a lens can make an object appear larger when the image is viewed from a given near point. It is defined as the ratio of the angle subtended by the image at the eye to the angle subtended by the object at the reference point.

Step 2: Calculate Angular Magnification for Image at Near Point

For an image at the near point, the angular magnification \( M \) can be calculated using the formula:\[M_n = 1 + \frac{d}{f}\]where \( d \) is the near point distance (\( P_n = 25 \text{ cm} \)) and \( f \) is the focal length of the lens (\( 10 \text{ cm} \)). Substitute the given values:\[M_n = 1 + \frac{25}{10} = 1 + 2.5 = 3.5\]

Step 3: Calculate Angular Magnification for Image at Infinity

When the image is at infinity, the angular magnification \( M \) of the lens is given by:\[M_\infty = \frac{d}{f}\]Substitute the given values for the near point \( d = 25 \text{ cm} \) and focal length \( f = 10 \text{ cm} \):\[M_\infty = \frac{25}{10} = 2.5\]

Step 4: Conclusion

The calculated angular magnifications for the given scenarios are 3.5 when the image is at the near point and 2.5 when the image is at infinity.

Key Concepts

Optical Physics

Optical physics is the field of study that deals with the behavior and properties of light, including its interactions with matter. A key aspect of this discipline is understanding how lenses and mirrors can focus or magnify light to form images. When light passes through a lens, it bends (or refracts) due to the change in medium, altering the path of the light rays.

In optical physics, lenses are used to magnify objects, which allows for a larger image to be seen by the observer's eye. The science behind this is based on fundamental principles like the refractive index, Snell's Law, and lens maker's equation. Lenses come in various shapes, such as convex and concave, and they are essential in many optical devices like glasses, cameras, and microscopes.

• Convex lenses converge light rays to a point, making them ideal for magnifying objects.
• Concave lenses diverge light rays, often used to correct for visual deficiencies like myopia.

By utilizing these optical elements, humans have been able to extend their range of vision, examine small objects in detail, and make significant advancements in fields such as medicine, astronomy, and photography.

Focal Length

The focal length of a lens is a critical parameter in optical physics. It represents the distance from the lens to the focal point, where light rays initially parallel to the optical axis converge.

The focal length (\( f \)) determines how strongly the lens converges or diverges light. For a convex lens, a shorter focal length implies stronger convergence and greater magnification of the object. When considering a simple magnifying lens, the focal length influences the angular magnification, determining how much larger the object appears when viewed through the lens.

The equations used to calculate angular magnification rely on the lens's focal length. These calculations help in figuring out the dimension by which an image is enlarged under different conditions:

• For an image at the near point: \( M_n = 1 + \frac{d}{f} \)
• For an image at infinity: \( M_\infty = \frac{d}{f} \)

Understanding the concept of focal length and its application helps optimize the design and usage of lenses in various optical instruments.

Near Point Distance

The near point distance is the closest an object can be to the eye while remaining in clear focus. For a person with normal vision, this distance is typically about 25 cm. The near point affects how we perceive magnification through a lens.

In optical physics, the near point is crucial when establishing the correct positioning of the object and lens to achieve the desired magnification. When the image of an object forms at the near point, the lens allows us to see the object’s details more clearly. This happens because the object is closer to the eye, enhancing focus and clarity. In the exercise:

• An object positioned at the near point distance provides a greater angular magnification of 3.5, as seen through a lens with a focal length of 10 cm.
• When the image forms at infinity where the eye is relaxed for far vision, the angular magnification reduces to 2.5.

Understanding the near point distance helps in tailoring the use of lenses for improved vision, ensuring clear and sharp viewing, even for people with varying degrees of eyesight quality.