Problem 44 - Physics for Scientists and Engineers with Modern Physics

Question

(II) An eye is corrected by a \(-4.50-\mathrm{D}\) lens, \(2.0 \mathrm{~cm}\) from the eye. \((a)\) Is this eye near- or farsighted? \((b)\) What is this eye's far point without glasses?

Step-by-Step Solution

Answer

(a) Nearsighted. (b) Far point is -24.2 cm.

Step 1: Understanding Lens Power

The lens power is given as \(-4.50\) diopters (D). Lens power in diopters indicates whether the lens is converging or diverging. A negative diopter value indicates that the lens is a diverging lens, which is used to correct nearsightedness, also known as myopia.

Step 2: Determine Eye Condition

Since the lens power is negative, the lens is diverging, which corrects near-sightedness (myopia). Thus, the eye in question is near-sighted.

Step 3: Far Point Calculation Formula

For a near-sighted person, the correction lens brings light into focus on the retina. The far point (the furthest point at which an object is clearly visible) without glasses can be calculated using the formula for lens power: \[ P = \frac{1}{f} \] where \( P = -4.50 \) diopters and \( f \) is the focal length in meters.

Step 4: Calculating the Focal Length

Convert the lens power into the focal length: \(-4.50\) D means \[ f = \frac{1}{-4.50} = -0.222 \text{ meters} \], or equivalently, \(-22.2\) cm.

Step 5: Considering Lens Position Relative to Eye

The lens is positioned \(2.0\) cm in front of the eye. Therefore, the actual far point is equal to the focal length of the correction lens plus the distance from the lens to the eye: \[ ext{Far Point} = -22.2 ~ ext{cm} - 2.0 ~ ext{cm} = -24.2 ~ ext{cm}. \] Thus, the far point of the eye without glasses is \(-24.2\) cm.

Key Concepts

Near-sightedness (Myopia)

Near-sightedness, or myopia, is a common vision condition where distant objects appear blurry while those that are near can be seen clearly. This happens because the light entering the eye focuses in front of the retina instead of directly on it.
A myopic eye has either an elongated shape or excessive curvature of the cornea, both leading to the incorrect focus of incoming light. To correct this, lenses known as diverging (or concave) lenses are used.
These lenses help diverge light rays slightly before they enter the eye, allowing the light to be correctly focused onto the retina.

• This makes far away objects appear clearer.
• The need for diverging lenses is indicated by a negative diopter value, as found in the corrective lenses for myopia.

Lens Power (Diopters)

Lens power measures how much a lens converges or diverges light and is denoted in diopters (D). The diopter value is crucial for understanding a lens's capability to correct vision.
A positive diopter number indicates a converging (or convex) lens, usually used to correct farsightedness. Conversely, a negative diopter number signifies a diverging (or concave) lens, necessary for correcting near-sightedness.
The power of a lens in diopters is derived from the formula:
\[ P = \frac{1}{f} \]
where P represents lens power in diopters and f is the focal length in meters.

• For a lens with power \(-4.50 D\), the focus is on diverging light to treat near-sightedness.
• This means helping the image form at the correct distance on the retina for improved vision clarity.

Focal Length Calculation

Calculating the focal length is essential in determining how a lens will correct vision issues. Focal length is the distance over which diverted light is brought back to a point of focus.
This is particularly important for corrective lenses, where the focal length needs to be adjusted just enough to position the image perfectly onto the eye's retina.
Using the lens power formula
\[ f = \frac{1}{P} \]
we can find the focal length. For a lens power of \(-4.50 D\), the calculation becomes:

• \[ f = \frac{1}{-4.50} = -0.222 \text{ meters}\], or \(-22.2\) cm.

This negative focal length indicates that the lens is indeed diverging the light. It is not just the lens's focal length that is crucial, but the distance between the lens and the eye also plays a significant role in its corrective capacity.

• In the given exercise, the focal length calculation helps determine the incorrect far point of the eye prior to correction.
• If the lens is placed \(2.0\) cm in front of the eye, the effective focal length calculation results in a far point determination of \(-24.2\) cm for the eye without glasses.

Understanding focal length adjustment is key to tailoring lenses that perfectly compensate for myopia.