A farsighted person has a near point located 210 cm from the eyes. Find the focal length of the converging lenses in a pair of contacts that can be used to read a book held 25 cm from the eyes.
The correct answer and explanation is :
To find the focal length of the converging lens needed for a farsighted person to read a book at 25 cm, we use the lens formula:
[
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
]
where:
- ( f ) is the focal length of the lens,
- ( d_o ) is the object distance (where the book is held, 25 cm),
- ( d_i ) is the image distance (the person’s near point, 210 cm).
Step 1: Convert to meters
Since distances should be in meters for standard SI units:
[
d_o = 25 \text{ cm} = 0.25 \text{ m}, \quad d_i = 210 \text{ cm} = 2.10 \text{ m}
]
Step 2: Apply the Lens Formula
[
\frac{1}{f} = \frac{1}{0.25} + \frac{1}{-2.10}
]
[
\frac{1}{f} = 4 – 0.476
]
[
\frac{1}{f} = 3.524
]
Step 3: Solve for ( f )
[
f = \frac{1}{3.524} \approx 0.284 \text{ m} = 28.4 \text{ cm}
]
Explanation
A farsighted person has difficulty seeing nearby objects because their near point is farther than normal (typically 25 cm for a young person). This means their eye’s lens cannot converge light strongly enough to focus images of close objects onto the retina. A converging (convex) lens helps by pre-converging the light rays before they enter the eye, allowing the image to form at the person’s near point (210 cm) instead of at 25 cm.
The focal length of the lens is +28.4 cm, meaning it is a convex lens with a positive focal length. This corrective lens shifts the focus, enabling the person to read a book at 25 cm comfortably.