Question

A person wearing bifocal glasses is reading a newspaper a distance of \(25 \mathrm{~cm}\). The lower part of the lens is converging for reading and has a focal length of \(70 . \mathrm{cm} .\) The upper part of the lens is diverging for seeing at distances far away and has a focal length of \(50 . \mathrm{cm} .\) What are the uncorrected near and far points for the person?

Short answer

Answer: The uncorrected near point is 38.89 cm behind the lower lens, and the uncorrected far point is 50 cm behind the upper lens.

Step-by-step solution

Analyze the converging (lower) lens

The lower part of the bifocal glasses is a converging lens for reading. Its focal length is \(f = 70cm\). The image distance, \(d_i\), is the distance at which the person is reading the newspaper, which is \(25cm\). Now, we can use the lensmaker's equation to solve for the object distance, \(d_o\), for the lower lens.

Use the lensmaker's equation for the lower lens

Apply the lensmaker's equation to the converging lens with \(f = 70cm\) and \(d_i = 25cm\). We get \(\frac{1}{70} = \frac{1}{d_o} + \frac{1}{25}\). Now, solve for \(d_o\).

Calculate the object distance for the lower lens

We can solve for \(d_o\) by first subtracting \(\frac{1}{25}\) from both sides of the equation: \(\frac{1}{d_o} = \frac{1}{70} - \frac{1}{25}\) Next, find a common denominator for both fractions, which is 350: \(\frac{1}{d_o} = \frac{5-14}{350}\) Now, simplify the fraction: \(\frac{1}{d_o} = -\frac{9}{350}\) Finally, invert both sides of the equation to find \(d_o\): \(d_o = -\frac{350}{9}\mathrm{~cm}\) Since the object distance is negative, this means that the person's uncorrected near point is a virtual object, located \(38.89 \mathrm{~cm}\) behind the lens.

Analyze the diverging (upper) lens

The upper part of the bifocal glasses is a diverging lens for seeing at larger distances. Its focal length is \(f = -50cm\). For a fully corrected vision, the image needs to be at infinity. So, the image distance \(d_i\) is \(\infty\). Now, we can use the lensmaker's equation to solve for the object distance, \(d_o\), for the upper lens.

Use the lensmaker's equation for the upper lens

Apply the lensmaker's equation to the diverging lens with \(f = -50cm\) and \(d_i = \infty\). We get \(\frac{1}{-50} = \frac{1}{d_o} + \frac{1}{\infty}\). Note that \(\frac{1}{\infty}\) is 0, so now solve for \(d_o\).

Calculate the object distance for the upper lens

Since \(d_i = \infty\), we get \(\frac{1}{-50} = \frac{1}{d_o}\). Invert both sides of the equation to find \(d_o\): \(d_o = -50\mathrm{~cm}\) Once again, since the object distance is negative, this means that the person's uncorrected far point is a virtual object, located \(50 \mathrm{~cm}\) behind the lens.

Determine the uncorrected near and far points

The uncorrected near point for the person wearing the bifocal glasses is \(38.89 \mathrm{~cm}\) behind the lower lens. The uncorrected far point is \(50 \mathrm{~cm}\) behind the upper lens.

Key concepts

Lensmaker's Equation

The lensmaker's equation is a fundamental tool used in optics to relate the physical properties of a lens to their optical effects. It's particularly helpful when designing lenses to correct vision, such as those found in bifocal glasses. The equation itself is given by \[ \frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \[ where \(f\) is the focal length of the lens, \(d_o\) is the object distance, \(d_i\) is the image distance, and \(n\) is the refractive index of the lens material. \(R_1\) and \(R_2\) are the radii of curvature of the lens surfaces.
By rearranging the terms, it gives us a clear method to calculate where an image will form for an object at a certain distance from the lens—vital for creating the correct prescription in bifocal glasses.

Converging and Diverging Lenses

In optics, lenses are categorized based on their ability to converge or diverge light rays.

Converging lenses

• Thicker at the center than at the edges.
• Bring parallel rays of light to a focus at the focal point.
• Used for correction of hyperopia or farsightedness.

Diverging lenses

• Thinner in the middle and thicker at the edges.
• Spread out light rays, making them diverge away from the focal point.
• Often used to correct myopia or nearsightedness.

In the context of bifocal glasses, the lower part of the lens typically converges light for reading, while the upper part diverges light to better focus on distant objects.

Object and Image Distance

Object and image distance are key concepts when understanding how lenses form images of various objects.

Object Distance (\(d_o\))

This is the distance between the object and the lens. If the object is real, the distance is positive, and if virtual, it's negative.

Image Distance (\(d_i\))

This is the distance between the image formed by the lens and the lens itself. A positive image distance indicates a real image on the same side of the lens as the object, while a negative image distance indicates a virtual image on the opposite side. Understanding these distances is critical in determining how to correct an individual's vision with bifocal lenses.

Uncorrected Near and Far Points

The uncorrected near and far points refer to the closest and furthest distances at which a person can see clearly without corrective lenses.

Near Point

This is the closest point at which one can see an object clearly. As people age or if they have certain vision conditions, such as presbyopia, this point may move further away, necessitating reading glasses.

Far Point

Conversely, this is the furthest point at which one can focus on an object. For those with nearsightedness, this point is closer than normal. Identifying these points helps determine the necessary lens power for bifocals to correct vision at various distances.