Problem 41 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A person struggles to read by holding a book at arm's length, a distance of \(55 \mathrm{~cm}\) away. What power of reading glasses should be prescribed for her, assuming they will be placed \(2.0 \mathrm{~cm}\) from the eye and she wants to read at the "normal" near point of \(25 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

The power of reading glasses needed is approximately -2.46 diopters.

Step 1: Understand the Problem

The person cannot read clearly at the normal reading distance of 25 cm and holds the book at 55 cm. We need to find the power of the lenses needed so that they can read at 25 cm. The lenses are placed 2.0 cm away from the eye.

Step 2: Calculate the Effective Distance

The glasses will be 2.0 cm in front of the eyes. Therefore, the image formed by the glasses must be at a distance of 25 cm - 2 cm = 23 cm from the eyes to appear at the near point.

Step 3: Use Lens Formula

The lens formula is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( d_o \) is the object distance (55 cm - 2 cm = 53 cm) and \( d_i \) is the image distance (23 cm). Substitute these values: \[ \frac{1}{f} = \frac{1}{53} + \frac{1}{-23} \] (the image distance is negative because it is virtual).

Step 4: Solve for Focal Length

Calculate \( \frac{1}{f} = \frac{1}{53} - \frac{1}{23} \). Use a common denominator for the fractions (\( 53 imes 23 \)):\[ \frac{1}{f} = \frac{23 - 53}{1219} = \frac{-30}{1219} \]Thus, the focal length \( f \) is: \[ f = -40.63 \text{ cm} \]

Step 5: Calculate the Power of the Lens

The power of the lens \( P \) in diopters (D) is given by \( P = \frac{100}{f} \) with \( f \) in meters. Convert the focal length to meters (\( -0.4063 \text{ m} \)):\[ P = \frac{100}{-0.4063} \approx -2.46 \text{ D} \]

Step 6: Interpret the Result

The negative power indicates that converging lenses (reading glasses) are needed with a power of approximately \(-2.46\) diopters.

Key Concepts

Lens Formula

The lens formula is an essential equation in optics used to calculate the focal length of lenses. It is expressed as \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where:

• \( f \) is the focal length of the lens.
• \( d_o \) represents the object distance from the lens.
• \( d_i \) indicates the image distance formed by the lens.

The formula helps in understanding how lenses focus light to form images. For virtual images, which occur when the image appears to be on the same side as the object, \( d_i \) is taken to be negative. This distinction is crucial when determining whether an image is real or virtual.
Applying this formula allows you to predict how lenses will behave in various situations, like adjusting for vision impairments. In practice, it aids in calculating the required focal length when you need corrective lenses, such as reading glasses. Knowing the focal length, you can prescribe lenses to help people read at the desired "normal" distance.

Focal Length

The focal length \( f \) of a lens is a measure of how strongly the lens converges or diverges light. In the context of corrective glasses, the focal length determines how the lens will adjust vision. To find the focal length in practical situations, use the lens formula. Here, the formula has been applied with specific object and image distances:

• Object distance \( d_o = 53 \text{ cm} \).
• Image distance \( d_i = -23 \text{ cm} \), indicating a virtual image.

The resulting calculation gives \( f = -40.63 \text{ cm} \).
A negative focal length signifies a diverging lens; however, for reading glasses, we refer to this effect as "converging to the eye." This means that the virtual image is formed closer to the viewer, simulating convergence and assisting in focusing on nearby objects. Understanding focal length in this context is vital for determining the power of lenses, allowing opticians to provide appropriate eyewear.

Reading Glasses

Reading glasses are specialized lenses designed to aid in seeing objects at a closer range, typically where near point vision shifts further due to age or individual eyesight conditions. These glasses are essential for individuals who struggle to read at standard distances, around 25 cm. By utilizing concepts like the lens formula and focal length, opticians can calculate the correct lens prescription.
In the provided exercise, reading glasses are used to adjust a near point distance to the normal reading distance of 25 cm. The calculation reveals a necessary lens power of approximately \(-2.46\) diopters. This power refers to how much the lenses need to diverge light rays before reaching the eyes to focus correctly.
The prescription of reading glasses is critical in enhancing everyday visual experiences, reducing the strain of focusing and allowing clearer and more natural reading habits. Corrective lenses like these bring significant visual comfort and improved quality of life to those with presbyopia or naturally limited near vision.