Bill is farsighted and has a near point located 125 cm from...

Bill is farsighted and has a near point located $125 \mathrm{~cm}$ from his eyes. Anne is also farsighted, but her near point is $75.0 \mathrm{~cm}$ from her eyes. Both have glasses that correct their vision to a normal near point $(25.0 \mathrm{~cm}$ from the eyes), and both wear the glasses $2.0 \mathrm{~cm}$ from the eyes. Relative to the eyes, what is the closest object that can be seen clearly (a) by Anne when she wears Bill's glasses and (b) by Bill when he wears Anne's glasses?

Bill is farsighted and has a near point located $125 \mathrm{~cm}$ from his eyes. Anne is also farsighted, but her near point is $75.0 \mathrm{~cm}$ from her eyes. Both have glasses that correct their vision to a normal near point $(25.0 \mathrm{~cm}$ from the eyes), and both wear the glasses $2.0 \mathrm{~cm}$ from the eyes. Relative to the eyes, what is the closest object that can be seen clearly
(a) by Anne when she wears Bill's glasses and
(b) by Bill when he wears Anne's glasses?

Key Concepts

Farsightedness

Farsightedness, or hyperopia, is a condition in which the eye's lens does not adequately focus close objects onto the retina. This results in a near point that is farther away than what is considered normal, causing difficulty in seeing nearby objects clearly without corrective lenses.

Near Point

The near point is the minimum distance at which the eye can focus on an object clearly. In a normal eye, this distance is typically around 25 cm, but in individuals with farsightedness, the near point is further away, necessitating optical correction to see close objects clearly.

Corrective Lenses

Corrective lenses are used to adjust the effective focusing distance of the eye. For farsighted individuals, these lenses help by creating a virtual image of the close object at a distance where the eye can comfortably focus, effectively moving the near point closer to the normal value.

Thin Lens Equation

The thin lens equation, which relates the object distance, image distance, and focal length of a lens, is crucial in designing and analyzing corrective lenses. This relationship is used to calculate the required focal length so that the lens forms the correct image distance for clear vision.

Lens Power

Lens power, measured in diopters, is defined as the reciprocal of the focal length in meters. It quantifies how strongly a lens converges or diverges light and is a key parameter in determining the corrective capability of a lens for adjusting the near point.

Vertex Distance

Vertex distance refers to the space between the corrective lens and the eye. This distance can affect the effective power of the lens and the overall correction provided, making it an important factor in the design and use of corrective eyewear.

Transcript

00:01 Bill is far -sighted and has a near point located 125 centimeters from its eyes.

00:07 Anne is also far -sighted, but her near point is 75 centimeters from her eyes.

00:11 Both have glasses that correct their vision to a normal near point, which is 25 centimeters, and both wear glasses two centimeters from their eyes.

00:20 Relative to the eyes, what is the closest object that can be seen clearly by anne who wears bill's glasses and by bill who wears anne's glasses.

00:32 So the formula that we need here is 1 over f is equal to 1 over do plus 1 over d .i.

00:42 So let's look at bill.

00:46 So the object's distance of bill is 25 minus 2 centimeters, which is 23 centimeters.

00:56 The image distance for bill is negative, 125 minus 2 centimeters, so negative, 1923 centimeters.

01:11 The object distance for ann is, again, 25 minus 2, which is 23 centimeters, and the image distance for ann is negative 75 minus 2, which is negative 73 centimeters.

01:30 Okay, so all of the distance, the image distance are negative because all of these images are going to be virtual images based on the type of lenses.

01:42 So we can rewrite this equation in terms of the focal lengths, which is one over d .i plus one over d .o.

01:54 The negative one power, that makes it a little bit easier to see what's happening.

01:58 So the focal point, in this case for bill, is equal to when we plug in values, 28 .3 centimeters.

02:10 Same thing for ann, same equation, but using ann's numbers.

02:16 The focal length is 33 .6 centimeters...

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