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 spectacles, what will be the range over which you will be able to see objects distinctly?

Short answer

Answer: The range of distinct vision for a person wearing -0.20 diopter spectacles is from \(2.5\,\mathrm{m}\) to an undefined far point.

Step-by-step solution

Calculate the focal length of the spectacles

Using the power (P) of the spectacles and the equation: \(P = \frac{1}{f}\), where f is the focal length, we can find the focal length of the -0.20 diopter spectacles: $$ f = \frac{1}{P} = \frac{1}{-0.20} = -5 \,\mathrm{m} $$

Calculate the new near point with the spectacles

To find the new near point, we will use the lens formula: $$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$ where \(f\) is the focal length of the spectacles, \(d_o\) is the object distance (the person's original near point distance) and \(d_i\) is the image distance (the new near point distance with the spectacles). We'll first convert the person's original near point distance to meters. $$ d_o = 2.0\cdot10^{1}\,\mathrm{cm} * \frac{1 \,\mathrm{m}}{100\,\mathrm{cm}} = 0.2\,\mathrm{m} $$ Now we can use the lens formula to find the new near point distance \(d_i\): $$ \frac{1}{-5} = \frac{1}{0.2} - \frac{1}{d_i} $$ Rearranging to solve for \(d_i\), we get: $$ d_i = \frac{1}{\frac{1}{0.2} - \frac{1}{-5}} = \frac{1}{0.4} = 2.5\,\mathrm{m} $$

Calculate the new far point with the spectacles

The far point is when the image distance (\(d_i\)) is at infinity. We can set \(d_i\) to infinity in the lens formula to find the corresponding object distance (\(d_o\)): $$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{\infty} $$ Since \(\frac{1}{\infty}\) approaches zero, we get: $$ \frac{1}{f} = \frac{1}{d_o} $$ Substitute the value of f from step 1 to get: $$ \frac{1}{-5} = \frac{1}{d_o} $$ Solving for \(d_o\) gives the new far point: $$ d_o = -5\,\mathrm{m} $$ However, considering that negative distances aren't physically meaningful in this context, we can conclude that a person wearing -0.20 diopter spectacles will not have a new far point, meaning that they won't be able to see objects at infinity clearly.

Determine the range of distinct vision with the spectacles

From steps 2 and 3, we have found the new near point to be \(2.5\,\mathrm{m}\), and we concluded that there will be no new far point for the person wearing -0.20 diopter spectacles. Therefore, the range over which the person will be able to see objects distinctly with the spectacles will be from \(2.5\,\mathrm{m}\) to an undefined far point.

Key concepts

Focal Length Calculation

Understanding focal length is crucial when it comes to spectacles and vision adjustment. The focal length of a lens determines how it converges or diverges light. This concept is particularly important when figuring out how diopter spectacles will affect one's vision.

In the given exercise, the spectacles have a power of -0.20 diopters. The calculation of focal length is straightforward using the formula:
\[ f = \frac{1}{P} \]
where \( f \) is the focal length in meters and \( P \) is the power of the spectacles in diopters. For a lens with negative power, such as -0.20 diopters, the focal length is also negative, indicating a diverging lens. In this case, the focal length calculation yields \( f = -5 \) meters, meaning the lens will cause parallel rays of light to appear to diverge from a virtual source 5 meters behind the lens.

Being able to calculate the focal length is essential for further calculating the range over which objects can be seen distinctly with the spectacles.

Lens Formula

The lens formula is a fundamental equation in optics that relates an object's distance from a lens to the image's distance and the lens's focal length. It is represented as:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
where \( f \) is the focal length, \( d_o \) is the object distance (distance from the lens to the object), and \( d_i \) is the image distance (distance from the lens to the image formed).

In our exercise, we used this formula to find the new near point when spectacles with a power of -0.20 diopters are worn. By plugging in the provided near point and the calculated focal length into the lens formula, we can solve for the new image distance, giving us the distance at which someone can see clearly with the spectacles.

The exact application of the lens formula in different scenarios, like finding the new far point, is essential for understanding how corrective lenses alter our vision, aiming to bring objects into sharp focus for individuals with impaired sight.

Near and Far Point of the Eye

The terms 'near point' and 'far point' are often used when discussing vision. The near point is the closest distance at which the eye can focus on an object, while the far point is the farthest. For a person with normal vision, the near point is approximately 25 centimeters, and the far point is at infinity.

However, those with vision difficulties may have a near point that is farther than normal or a far point that is closer than infinity. The use of corrective lenses aims to reset these points to more normal ranges. For instance, in our exercise, we determined that with -0.20 diopter spectacles, a person's near point shifted to 2.5 meters, and due to the divergent nature of plano-concave lenses indicated by negative diopters, it is impossible to have a definite far point.

This explains why individuals might need different strengths for reading glasses versus glasses for distance vision, addressing their unique near and far points. It's key for students to understand these concepts to comprehend how lenses correct vision and what realistic improvements can be expected with specific diopter values.