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Chapter 24: Problem 62

(a) If Harry has a near point of \(1.5 \mathrm{m},\) what focal length contact lenses does he require? (b) What is the power of these lenses in diopters?

Short Answer

Expert verified
In this exercise, we are given the near point distance of Harry's eyes (1.5 meters) and asked to calculate the focal length and power of the contact lenses he requires. Using the lens formula, we determine that Harry needs contact lenses with a focal length of 1.2 meters. The power of these lenses is approximately 0.8333 diopters.

Step by step solution

01

Understand the lens formula

The lens formula relates the object distance (u), image distance (v) and the focal length (f) of a lens. The formula is given as: \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \) In this exercise, we are given the near point distance (u) and asked to find the focal length (f) while keeping the image distance (v) at the standard near point distance of 25 cm (0.25 meters).
02

Apply the lens formula

To find the focal length of the contact lenses, we need to substitute the given values of u and v into the lens formula: \( \frac{1}{f} = \frac{1}{1.5} + \frac{1}{0.25} \)
03

Calculate the focal length

Solve the lens formula for f: \( \frac{1}{f} = \frac{1}{1.5} + \frac{1}{0.25} = \frac{1}{1.5} + \frac{4}{1} = \frac{5}{6} \) \( f = \frac{6}{5} = 1.2 \ \mathrm{m} \) Thus, Harry requires contact lenses with a focal length of 1.2 meters.
04

Calculate the power of the lenses in diopters

The power (P) of a lens in diopters is defined as the reciprocal of its focal length in meters: \( P = \frac{1}{f} \) Substitute the value of f obtained in step 3: \( P = \frac{1}{1.2} = 0.8333 \) (rounded to 4 decimal places) Thus, the power of the contact lenses Harry requires is approximately 0.8333 diopters.

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