Question

Suppose the near point of your eye is $2.0\cdot {10}^1\mathrm{cm}$ and the far point is infinity. If you put on -0.20 diopter spec tacles, what will be the range over which you will be able to see objects distinctly?

Short answer

Answer: The new range of vision for the person wearing -0.20 diopter spectacles is 1 meter, between 4 meters and 5 meters from their eye.

Step-by-step solution

Calculate the focal length of the spectacles

We are given the power of the spectacles as P = -0.20 diopters. Using the definition of power, we can find the focal length f: $$f=\frac{1}{P}=\frac{1}{-0.20\mathrm{D}}=-5\mathrm{m}$$ The negative focal length indicates that the spectacles are concave lenses.

Find the new near point

To find the new near point, we need to consider the object distance (u) as the original near point, which is given as $2.0\cdot {10}^1\mathrm{cm}$. The new image distance (v) will be formed closer to the eye, which means the person can now see more distinctly objects closer than before. Using the lens formula: $$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$$ Plug in the values of f and u: $$\frac{1}{-5\mathrm{m}}=\frac{1}{2.0\cdot {10}^1\mathrm{cm}}+\frac{1}{v}$$ Convert the object distance to meters: $$\frac{1}{-5\mathrm{m}}=\frac{1}{0.2\mathrm{m}}+\frac{1}{v}$$ Solve for v: $$\frac{1}{v}=\frac{1}{-5\mathrm{m}}-\frac{1}{0.2\mathrm{m}}=-\frac{1}{4\mathrm{m}}$$ $$v=-4\mathrm{m}$$ The negative sign indicates that the new image is formed on the same side as the object. Since it's the minimum distance for distinct vision, we can ignore the negative sign. Therefore, the new near point is $4\mathrm{m}$.

Find the new far point

To find the new far point, we need to consider the object distance (u) as infinity, as the original far point is infinity. In this case, the new image distance (v) will be equal to the focal length of the lens. For the spectacles, we have already calculated the focal length as f = -5m. Since the new image distance (v) should be equal to the focal length, we have: $$v=-5\mathrm{m}$$ As the new far point should be a positive distance, we can use the concept of the negative sign indicating an object and image on the same side of the lens. Therefore, the new far point is 5 meters.

Determine the range of vision

Now that we have calculated the new near point and the new far point, we can find the range of vision over which a person can see objects distinctly with the given spectacles. The range of vision can be found by simply subtracting the new near point from the new far point: $$\text{range}=\text{new far point}-\text{new near point}=5\mathrm{m}-4\mathrm{m}=1\mathrm{m}$$ #Conclusion# With the -0.20 diopter spectacles, the person will be able to see objects distinctly within the range of 1 meter, between 4 meters and 5 meters from their eye.

Key concepts

Lens Formula

The lens formula is essential in optics for understanding how lenses form images. It relates the focal length ($f$), the object distance ($u$), and the image distance ($v$) and is mathematically expressed as:$$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$$This formula applies to diverging (concave) and converging (convex) lenses.
- **Focal Length ($f$)**: The distance from the center of the lens to the focal point. It determines the lens's power.- **Object Distance ($u$)**: The distance between the object and the lens.- **Image Distance ($v$)**: The distance between the image and the lens.You can use the formula to figure out either the image or object distance if you have the other two values. In scenarios involving glasses or lenses for vision correction, this formula helps predict where an image will form relative to the eye, impacting visual clarity.

Diopters

Diopters are the unit that measures the optical power of a lens. This power is calculated as the inverse of the focal length in meters:$$P=\frac{1}{f}$$- If a lens has a focal length of -5 meters, it has a power of -0.20 diopters.- Positive diopters correspond to convex lenses, which focus light.- Negative diopters, like those in the example, correlate with concave lenses that spread light.In vision correction, diopters indicate how much correction a lens provides. A prescription of -0.20 diopters implies that the lens is designed to assist someone with mild myopia (nearsightedness). For such lenses, the negative sign signifies a diverging lens, which helps the eyes adjust for distant vision by altering the focal length.

Near Point and Far Point

The near point and far point define a person's range of clear vision.
- **Near Point**: This is the closest distance at which one can see objects clearly. It varies with age and visual health. - **Far Point**: The furthest distance at which vision remains clear, typically extending to infinity for those with good vision. When corrective lenses are introduced, they shift these points. For example, introducing spectacles with -0.20 diopters modifies both the near and far points for someone whose natural far point is infinity but needs help with near vision. The lenses help them see clearly over a new range by moving the near point further from the eyes and adjusting the far point as per the lens's specifications.

Concave Lens

A concave lens, often used in eyeglasses, is distinguished by its thinner center compared to its edges. It diverges incoming light rays, making it an ideal corrective tool for myopia. Characteristics:

• Image Formation: Concave lenses produce virtual images that appear reduced and are upright.
• Optical Correction: Helps individuals focus distant objects by adjusting divergent rays before they hit the retina.
• Negative Focal Length: As seen in the example, the focal length is negative, implying that the light rays meet before the lens, not converging beyond it.

In practical application, such as spectacles, concave lenses adjust the eyes' focus range, altering how and where the images are formed and improving overall visual clarity. This adjustment is crucial for bringing distant objects into focus.