The near point of a patient's eye is 58.2 cm.
(a) What should be the refractive power ℛ of a corrective lens prescribed to enable the patient to clearly see an object at 22.0 cm?
ℛ = dpt
(b) When using the new corrective glasses, the patient can see an object clearly at 23.8 cm but not at 22.0 cm. By how many diopters did the lens grinder miss the prescription? (dpt)
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To calculate the refractive power of the corrective lens prescribed, we can use the lens formula:
1/ℛ = 1/f - 1/d
Where ℛ is the refractive power of the lens in diopters, f is the focal length of the lens in meters, and d is the object distance in meters.
(a) To find the refractive power ℛ of the corrective lens prescribed to enable the patient to clearly see an object at 22.0 cm (0.22 m), we need to calculate the focal length of the lens. The object distance d is given as 0.22 m:
1/ℛ = 1/f - 1/d
Since the near point of the patient's eye is 58.2 cm (or 0.582 m), the object distance d for the patient is:
d = 0.582 m
Substituting the values into the lens formula:
1/ℛ = 1/f - 1/d
1/ℛ = 1/f - 1/0.582
To solve for ℛ, we rearrange the equation:
1/ℛ = (0.582 - f) / (0.582f)
Now we can solve for ℛ:
ℛ = (0.582f) / (0.582 - f)
Substituting d = 0.22 m into the equation:
ℛ = (0.582f) / (0.582 - f) = (0.582 * 0.22) / (0.582 - 0.22) = 0.1286 / 0.362 = 0.355 diopters (approximately)
Therefore, the refractive power of the corrective lens prescribed is ℛ = 0.355 diopters.
(b) To find the prescription error by the lens grinder, we need to compare the prescribed refractive power (ℛ) with the actual refractive power of the lens used. The patient can see clearly at 23.8 cm (0.238 m) but not at 22.0 cm (0.22 m).
Using the lens formula, we can calculate the actual refractive power (ℛ_actual) of the lens needed to see clearly at 23.8 cm:
1/ℛ_actual = 1/f_actual - 1/d
With d = 0.238 m:
1/ℛ_actual = 1/f_actual - 1/0.238
Similarly to part (a), we can calculate the actual refractive power (ℛ_actual) using the lens formula:
ℛ_actual = (0.582f_actual) / (0.582 - f_actual)
Substituting d = 0.238 m into the equation:
ℛ_actual = (0.582f_actual) / (0.582 - f_actual) = (0.582 * 0.238) / (0.582 - 0.238) = 0.1386 / 0.344 = 0.403 diopters (approximately)
The difference between the prescribed refractive power (0.355 diopters) and the actual refractive power (0.403 diopters) is:
0.403 - 0.355 = 0.048 diopters (approximately)
Therefore, the lens grinder missed the prescription by 0.048 diopters.