Question

A person can see objects clearly when they lie between 40 cm and 400 cm from his eye. In order to increase the maximum distance of distant vision to infinity, the type of lens and power of correction lens required respectively are:

• A. Convex, \( 0.25 \) Diopter
• B. Concave, \( -0.25 \) Diopter
• C. Concave, \( -0.5 \) Diopter
• D. Convex, \( 0.5 \) Diopter

Hint:

Myopia (short-sightedness) is corrected using a concave lens. The power of the lens is given by \( P = \frac{1}{f} \), where \( f \) is in meters.

Answer 1

The Correct Option is B

Step 1: Identifying the Eye Defect - The person can see near objects clearly but distant objects appear blurred. - This is a case of myopia (short-sightedness). - A concave lens is required to correct this.
Step 2: Calculating the Lens Power The lens formula is: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u}. \] For distant vision correction: - Far point = \( 400 \) cm. - To correct myopia, the image should be formed at \( 400 \) cm when the object is at infinity (\( u = \infty \)). \[ \frac{1}{f} = \frac{1}{400} - \frac{1}{\infty} = \frac{1}{400}. \] \[ f = 400 \text{ cm} = 4 \text{ m}. \] \[ P = \frac{1}{f} = \frac{1}{-4} = -0.25 \text{ D}. \] Thus, the correct answer is: \[ \boxed{\text{Concave, } -0.25 \text{ D}}. \]

Answer 2

To solve this problem, we must first identify the issue with the person's vision. When an individual's far-point, which is the maximum distance at which they can see objects clearly, is closer than infinity, they are typically suffering from myopia (nearsightedness). In this case, the person's far-point is 400 cm. In order to correct this and increase the person's far-point to infinity, we require a lens that can diverge light, thus correcting the focus onto the retina. Such a lens is a concave lens, which has negative power.

Using the lens formula for myopia correction, we set the image distance \( v \) for a clear distant view as negative infinity, and the object distance \( u \) as the far-point of the person, which is 400 cm.

The lens formula is:

\(\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\)

Since \( v = -\infty \), we have:

\(\frac{1}{f}=0-\frac{1}{400}\)

This simplifies to:

\(f=-400 \, \text{cm} = -4 \, \text{m}\)

The power \( P \) of the lens in diopters (D) is given by the formula:

\(P=\frac{1}{f \, (\text{in meters})}\)

Substituting the value of \( f \):

\(P=-\frac{1}{4}=-0.25 \, \text{D}\)

Therefore, the person requires a concave lens with a power of -0.25 diopters to extend their maximum distance of vision to infinity. Hence, the correct choice is "Concave, \(-0.25 \, \text{Diopter}\)".