Problem 47 - Physics for Scientists and Engineers with Modern Physics

Question

(II) One lens of a nearsighted person's eyeglasses has a focal length of \(-23.0 \mathrm{cm}\) and the lens is 1.8 \(\mathrm{cm}\) from the eye If the person switches to contact lenses placed directly on the eye, what should be the focal length of the corresponding contact lens?

Step-by-Step Solution

Answer

The focal length of the contact lens should be -24.8 cm.

Step 1: Understanding the Lens Formula

The lens formula is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. For glasses aiding nearsightedness, the glasses form a virtual image of distant objects very close to the eye.

Step 2: Calculate Image Distance with Glasses

When the object is very far away, \( d_o \approx \infty \), making \( \frac{1}{d_o} \approx 0 \). Hence, \( \frac{1}{f} = \frac{1}{d_i} \). For glasses, since \( f = -23.0 \) cm, \( d_i = f = -23.0 \) cm (the image distance relative to the glasses).

Step 3: Determine the Image Position Relative to the Eye

The glasses are 1.8 cm away from the eye. Thus, the image distance with respect to the eye is \( -23.0 - 1.8 = -24.8 \) cm.

Step 4: Use the Lens Formula for Contact Lenses

For contact lenses, the lens is directly on the eye, so the image distance \( d_i \) must be \(-24.8\) cm directly from the lens. We want to find the new focal length \( f' \) such that \( \frac{1}{f'} = \frac{1}{d_o} + \frac{1}{d_i} \), but \( \frac{1}{d_o} \approx 0 \) since the object is very far away.

Step 5: Calculate Focal Length for Contact Lenses

Using \( \frac{1}{f'} = \frac{1}{-24.8} \), we find that \( f' = -24.8 \) cm. This means the contact lens should have the same effective focal length as the distance from the eye with glasses adjustment.

Key Concepts

Lens Formula

The lens formula is an essential equation in optical physics, often applied when studying lenses and their effects. It is expressed as \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where:

• \( f \) represents the focal length of the lens.
• \( d_o \) is the distance from the object to the lens, termed the object distance.
• \( d_i \) denotes the distance from the lens to the image formed, known as the image distance.

This formula helps us understand how lenses focus light to form images. By adjusting any two of these variables, one can determine the third. This relationship is crucial when figuring out how glasses or contact lenses will affect a person's vision. For nearsightedness, the lens in glasses creates a virtual image that appears close to the eye, facilitating easier vision of distant objects. When the object distance is very large, the formula simplifies as \( \frac{1}{d_o} \approx 0 \), which is particularly useful for eyeglasses or contact lenses calculations.

Focal Length

The focal length of a lens is a measure of how strongly the lens converges or diverges light. In the context of nearsightedness, this concept is highly relevant because:

• It defines the distance at which parallel rays of light will converge after passing through the lens. A negative focal length, such as \(-23.0\, \text{cm}\) in the case of glasses for nearsightedness, indicates a diverging lens.
• These lenses spread light rays outward, which helps in altering the pathway of light to form a virtual image nearby.
• For contact lenses, the focal length needs to compensate for the additional distance typically accounted by glasses being in front of the eye.

Choosing the correct focal length for contact lenses is crucial, as the lens is directly on the eye. This specific task involves finding an equivalent to the setup created by glasses, adjusted for the position shift.

Nearsightedness

Nearsightedness, also known as myopia, is a common vision condition where distant objects appear blurry, while close objects are seen clearly. This happens because the eye focuses light more in front of the retina than on it. Correcting this requires altering the way light moves through the eye:

• Special lenses are used, which are typically diverging lenses with a negative focal length.
• These lenses spread light rays in a manner that helps the eye bring these rays into focus directly onto the retina, allowing for clear vision of distant objects.
• Glasses for nearsightedness sit some distance away from the eye, typically around 1 to 2 cm, while contact lenses sit directly on the eye, eliminating this distance.

Switching from glasses to contact lenses means the corrective lens needs to have a different focal length to account for its changed position. This adjustment ensures the light correctly reaches the retina without the intermediate distance glasses once provided.