Problem 73 - Physics

Question

A nearsighted person wears contacts to correct for a far point that is only \(3.62 \mathrm{~m}\) from his eyes. The near point of his unaided eyes is \(25.0 \mathrm{~cm}\) from his eyes. If he does not remove the lenses when reading, how close can he hold a book and see it clearly?

Step-by-Step Solution

Answer

The book can be held approximately 19 cm from the eyes.

Step 1: Understanding the Problem

A person is wearing contact lenses that correct his far point to infinity. The unaided far point is at `3.62 m`, and he wants to read a book. We need to find how close the book can be to the eyes while wearing the lenses. This is determined by his near point distance with the lenses.

Step 2: Determine the Power of the Lens

To determine how the lenses correct the vision, calculate the power needed to shift the far point from `3.62 m` to infinity. The lens formula is: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]For correction to infinity, \(d_i \) (image distance) should be infinite, making \[ \frac{1}{f} = \frac{1}{d_o} + 0 = \frac{1}{-3.62} \] (negative as it's a real object distance for correction)Thus, the power \(P\) of the lens is \[ P = \frac{1}{f} = -\frac{1}{3.62} \approx -0.276 \text{ diopters} \].

Step 3: Calculate the New Near Point with Lenses

Now, apply the power of the lens to find the modified near point distance. With the lenses, the effective object distance should have the same image distance of `25 cm` as the unaided near point: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]Given the lens power \(P = -0.276\), \[ -0.276 = \frac{1}{d_o} - \frac{1}{0.25} \]Solving for \(d_o\): \[ d_o = \left( \frac{1}{-0.276 + \frac{1}{0.25}} \right)^{\!-1} \approx 0.19 \text{ m} \] or `19 cm`.

Key Concepts

Nearsightedness

Nearsightedness, also known as myopia, is a common vision condition where people can see objects near to them clearly, but objects far away are blurry. This occurs because light entering the eye is focused in front of the retina instead of directly on it. The retina is the thin layer of tissue that lines the back of the eye and is vital for visual sensation.
To understand this better, think of a camera focusing mechanism that doesn’t quite bring a distant subject into sharp focus. People with myopia see distant objects in a similar way — they appear blurry. Unfortunately, the condition usually becomes evident during childhood and may progress until adulthood. While the exact cause isn’t fully understood, it’s believed to be a result of genetics or environmental factors such as excessive near work or lack of outdoor activities. Common signs include squinting, eye strain, headaches, and difficulty seeing distant objects like road signs.
Myopia can be corrected with glasses, contact lenses, or refractive surgery. The goal of corrective lenses is to refocus light onto the retina to produce a clear image.

Lens Power

Lens power is a crucial concept in optics, especially when dealing with corrective lenses for myopia or other refractive errors. The power of a lens is essentially its ability to bend light. It's measured in diopters, which is the reciprocal of the focal length in meters.Let's break it down:

• A positive diopter lens is a converging lens, often used to correct farsightedness (hyperopia).
• A negative diopter lens is a diverging lens, which is what is used to correct nearsightedness (myopia). For example, a lens power of \(-0.276\) indicates a diverging lens.

When a myopic person uses a lens with a negative power, it diverges light rays before they reach the eye, allowing them to be focused precisely on the retina. This changes the point of focus of images and makes distant objects come into clear view.
Calculating lens power involves understanding the lens formula: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]. Here, \(f\) is the focal length, \(d_o\) is the object distance (negative for lens correction), and \(d_i\) is the image distance. By using this formula, one can determine the necessary lens power required for correcting nearsightedness.

Near Point Distance

The near point distance is the closest distance at which the eye can focus on an object. For most young people, this distance is typically around 25 cm. However, this can change with age or visual conditions. To comprehend how near point distance works, consider it as the eye's own natural focusing limit.

• For nearsighted individuals, their unaided near point can sometimes be closer than 25 cm because their lenses are better at focusing nearby.
• When corrected with lenses, the near point distance might change, enabling the person to focus on objects even closer.

In our example, after calculating the lens power required to correct the far point, we used the lens formula to find the new near point distance. The corrected near point distance was found to be 19 cm, thanks to the negative power of the lenses that allowed the eyes to focus on objects closer than the initial 25 cm unaided distance.
This change means that when wearing the correcting lenses, he can comfortably read a book held as close as 19 cm away.