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Zorluk: ZorThin Lenses, Optical Instruments, and Defects of Vision

A hypermetropic eye has its unassisted near point situated at 75 cm75\text{ cm} from the eye. Calculate the focal length, in cm\text{cm}, of the converging spectacle lens needed to enable the person to read a book comfortably at a distance of 25 cm25\text{ cm} from the eye.

Cevap: 37.5 cm

Cevap

The focal length of the required converging lens is 37.5 cm37.5\text{ cm}.
To correct hypermetropia, the spectacle lens must form a virtual image of an object located at the desired near point (u=+25 cmu = +25\text{ cm}) at the eye's actual, unassisted near point (v=−75 cmv = -75\text{ cm}). Substituting u=+25 cmu = +25\text{ cm} and v=−75 cmv = -75\text{ cm} into the thin lens formula 1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v} gives 1f=125−175=275 cm−1\frac{1}{f} = \frac{1}{25} - \frac{1}{75} = \frac{2}{75}\text{ cm}^{-1}. Taking the reciprocal yields f=37.5 cmf = 37.5\text{ cm}.

Adım Adım Çözüm

1
Determine the object distance and image distance with sign conventions
u=+25 cmu = +25\text{ cm} and v=−75 cmv = -75\text{ cm}.
The object is placed at the normal reading distance (25 cm25\text{ cm}), and the spectacle lens creates a virtual image on the same side of the lens at the person's near point (75 cm75\text{ cm}).
2
Set up the thin lens formula
1f=125−175.\frac{1}{f} = \frac{1}{25} - \frac{1}{75}.
The thin lens formula relates the focal length to object and image distances.
3
Calculate the focal length ff
f = 37.5\text{ cm}.
Simplifying 3−175=275\frac{3 - 1}{75} = \frac{2}{75} and taking the reciprocal yields f=752=37.5 cmf = \frac{75}{2} = 37.5\text{ cm}.

Anahtar Kavram

Correction of hypermetropia (farsightedness) using thin lens formula with virtual image sign convention
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Konular

Thin Lenses, Optical Instruments, and Defects of Vision

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