Question

For a person whose near point is \(115 \mathrm{~cm}\), so that he can read a computer monitor at \(55 \mathrm{~cm}\), what power of reading glasses should his optician prescribe, keeping the lens-eye distance of \(2.0 \mathrm{~cm}\) for his spectacles?

Short answer

Answer: The optician should prescribe reading glasses with a power of approximately 2.76 diopters.

Step-by-step solution

Find the distance at which the person can read without glasses

If the person's near point is at a distance of 115 cm, that means they can comfortably read without glasses at a distance of 115 cm.

Determine the distance the person needs to read a computer monitor comfortably

According to the problem, the person wants to read the computer monitor at a distance of 55 cm.

Identify the lens-eye distance

The lens-eye distance is given as 2.0 cm.

Calculate the effective distance from the lens to the computer monitor

Since the lens-eye distance is 2.0 cm, and the person will read the computer monitor at a distance of 55 cm, the effective distance from the lens to the computer monitor will be 55 cm - 2.0 cm = 53 cm.

Apply the lens equation to find the required focal length

The lens equation is given by the formula: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length, \(d_o\) is the object distance (distance from the lens to the computer monitor) and \(d_i\) is the image distance (near point). So the equation becomes \(\frac{1}{f} = \frac{1}{53} + \frac{1}{115}\).

Solve the lens equation for the focal length

To find the focal length \(f\), we first find the common denominator of the fractions which will be 53 * 115. Then we sum them up and calculate the reciprocal to determine the focal length: \(f = \frac{1}{\frac{1}{53} + \frac{1}{115}} = \frac{53 * 115}{53 + 115} = \frac{6095}{168} \approx 36.28 \mathrm{~cm}\).

Calculate the power of the reading glasses

The power of the reading glasses is the inverse of the focal length, expressed in diopters: \(P = \frac{1}{f}\), where \(f\) is in meters. To find the power, first convert the focal length to meters by dividing by 100: \(f = 36.28 \mathrm{~cm} = 0.3628 \mathrm{~m}\). Then calculate the power: \(P = \frac{1}{0.3628} \approx 2.76 \mathrm{~diopters}\). Therefore, the optician should prescribe reading glasses with a power of approximately 2.76 diopters.