Question

A myopic person's eye has a relaxed focal length of \(1.9 \mathrm{~cm}\). What is the maximum distance from the eye at which she can see an object clearly, and what is the magnification of the lens for an object at this point? A. \(0=1 \mathrm{~cm}, m=-\frac{1}{2}\) B. \(0=38 \mathrm{~cm}, m=-\frac{1}{38}\) C. \(0=38 \mathrm{~cm}, n=-\frac{1}{19}\) D. \(0=50 \mathrm{~cm}, \mathrm{mt}=-\frac{1}{50}\)

Short answer

The maximum object distance is \(1.9 \mathrm{~cm}\), and the magnification is \(-1\). None of the given options match this answer.

Step-by-step solution

Determine the maximum object distance

To find the maximum distance at which the person can see the object clearly, she will need to see it at the farthest point she can focus the light from the object onto her retina. This occurs when the image distance (i) is at its minimum. For a myopic person, this minimum image distance corresponds to the relaxed state of her eye. The relaxed focal length is given by 1.9 cm in this case. Therefore, i = 1.9 cm. Now, let's use the lens formula to find the maximum object distance (o): \(\frac{1}{f} = \frac{1}{o} + \frac{1}{i}\) Substitute f = 1.9 cm and i = 1.9 cm into the formula: \(\frac{1}{1.9} = \frac{1}{o} + \frac{1}{1.9}\)

Solve for the maximum object distance (o)

Simplify the equation and solve for o: \(\frac{1}{o} = \frac{1}{1.9} - \frac{1}{1.9}\) \(\frac{1}{o} = \frac{1}{1.9}\) Flip both sides: o = 1.9 cm So, the maximum object distance at which she can see an object clearly is 1.9 cm.

Calculate the lens magnification (m)

Now we will calculate the magnification using the formula: m = -\(\frac{i}{o}\) Substitute the values for i = 1.9 cm and o = 1.9 cm: m = -\(\frac{1.9}{1.9}\) m = -1 The lens magnification at this point is -1. The maximum object distance is 1.9 cm, and the magnification is -1. Comparing the answer to the available options, we can see that none of them match the correct solution. This means, there might be a typographical error in the given options, or the context of the question might have been incorrectly understood.

Key concepts

Optics

Optics is a branch of physics that deals with the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. This area of physics is crucial in understanding how lenses and mirrors form images by focusing light. The study of optics helps us explore how visual perception is affected by different focal lengths and lens designs.
By using the principles of optics, we can understand how corrective lenses, like those used for myopia, work to compensate for deficiencies in the eye's focusing ability. Optics is divided into two main parts: geometrical optics, which describes light propagation in terms of rays, and physical optics, which considers the wave nature of light. Both concepts are central to the study of vision and lens design.

Focal Length

The focal length is the distance between the center of a lens and its focal point, where light rays converge to form a clear image. It's a crucial element in understanding how lenses function, especially in devices like cameras, microscopes, and corrective eyewear.
In our exercise, the myopic person's eye has a focal length of 1.9 cm when relaxed. This means that when no accommodation occurs, the eye focuses light from objects at this specific distance directly onto the retina. If the focal length is too short, as in myopia, objects in the distance appear blurry because they focus in front of the retina.

• A positive focal length indicates a converging lens while a negative focal length implies a diverging lens.
• The focal length affects the field of view and depth of field of an optical system.

Corrective lenses are used to adjust the focal length so the light focuses correctly on the retina.

Magnification

Magnification describes the degree to which the size of an image is increased from its original size by an optical system. In simple terms, it's how much larger or smaller an object appears when viewed through a lens or other optical device. Magnification is an essential concept in optics, applicable in eyeglasses and devices like microscopes that require enlargement of small details.
The magnification formula used in the exercise is: \[ m = -\frac{i}{o} \]where \( i \) is the image distance and \( o \) is the object distance. The negative sign indicates that the image formed is inverted in relation to the object. In the context of the exercise, with both the image and object distance being 1.9 cm, the magnification is -1. This means the image is the same size as the object but inverted.

Lens Formula

The lens formula is a mathematical relationship that connects the focal length of a lens with the object distance and the image distance. It is expressed as: \[ \frac{1}{f} = \frac{1}{o} + \frac{1}{i} \]where \( f \) is the focal length, \( o \) is the object distance, and \( i \) is the image distance. This formula is vital because it provides a quantitative method to locate the image formed by a lens based on given object and focal distances.
In the problem, the formula helps determine the maximum distance at which the myopic person can see objects clearly by setting the image distance \(i\) equal to the focal length due to her relaxed eye state. Using the lens formula, you solve for the object distance \(o\) and find it to be the same as the focal length, 1.9 cm. This illustrates how lenses correct visual perception by shifting the convergence point of light on the retina.