Question

The typical length of a human eyeball is \(2.50 \mathrm{~cm} .\) a) What is the effective focal length of the two-lens system made from a normal person's cornea and lens when viewing objects far away? b) What is the effective focal length for viewing objects at the near point?

Short answer

Answer: The relationship between the effective focal length, individual focal lengths, and the distance between the lenses in the human eye for both distant and near point objects is given by \(F_{eff} = F_1 + F_2 - d\).

Step-by-step solution

Set up the equation

Since we are considering distant objects, we assume the object is at infinity. We can use the lens formula: \(\frac{1}{f} = \frac{1}{o} - \frac{1}{i}\), where \(f\) is the focal length, \(o\) is the object distance, and \(i\) is the image distance. When the object is at infinity, the image distance equals the focal length: \(i = f\).

Find the effective focal length of the two-lens system

We use the equation for effective focal length for the two-lens system: \(\frac{1}{F_{eff}} = \frac{1}{F_1} + \frac{1}{F_2} - \frac{d}{F_1 F_2}\). Since \(i = f\) and the image distance equals the eyeball length (2.5 cm) for distant objects, we have \(F_1 + F_2 - d = 2.5\) cm. As the actual values of \(F_1\) and \(F_2\) are not given, we cannot find an exact numerical value for the effective focal length. However, the relationship between the focal lengths and the eyeball length is given by \(F_{eff} = F_1 + F_2 - d\). b) Effective focal length for near point objects.

Calculate the object distance for the near point

For the near point, the object distance includes the near point distance \(D\) and the distance \(L\) between the near point and the eyeball, i.e., \(o = D + L\). The near point distance for normal vision is given as 25 cm.

Use the lens formula to find the image distance for the near point

We can use the lens formula again, \(\frac{1}{f} = \frac{1}{o} - \frac{1}{i}\) to find the image distance for the near point. Since \(o = D + L = 25 + L\) cm, and we do not have the exact value of the individual focal lengths, we cannot find the exact numerical value for the image distance in this case.

Find the effective focal length for near point objects

We can use the equation for the effective focal length for the two-lens system again: \(\frac{1}{F_{eff}} = \frac{1}{F_1} + \frac{1}{F_2} - \frac{d}{F_1 F_2}\). As the actual values of \(F_1\), \(F_2\), and \(d\) are not given, we cannot find an exact numerical value for the effective focal length for near point objects. However, the relationship between the focal lengths of the two lenses and their distance is given by \(F_{eff} = F_1 + F_2 - d\) for the near point as well.

Key concepts

Lens Formula

The lens formula is a fundamental principle in optics that describes the relationship between an object's position with respect to a lens, the image it produces, and the focal length of the lens. The formula is given as \( \frac{1}{f} = \frac{1}{o} - \frac{1}{i} \), where \( f \) denotes the lens's focal length, \( o \) is the distance of the object from the lens, and \( i \) is the distance of the image from the lens.

To understand how this applies to everyday life, consider a photography enthusiast using a camera. When they adjust the focal length of a camera lens, they're unknowingly using principles of the lens formula to create a sharp image. For objects that are very far away, effectively at infinity, the image distance \( i \) becomes equal to the focal length \( f \) because the rays arriving at the lens are parallel and converge at the focal point.

This concept is fundamental when studying the optics of the human eye or using systems of lenses, which often involve combining the optical powers of multiple lenses to achieve the desired magnification or image clarity.

Optics in the Human Eye

The human eye can be thought of as an advanced optical system, mainly consisting of the cornea and the lens. Like a camera lens, the eye's optics focus light to form an image on the retina.

The cornea, which is the eye's front surface, provides much of the eye's optical power, and the lens inside the eye fine-tunes this focus. Our eyes adjust the focal length depending on whether we're looking at distant mountains or reading the text on a screen, a process known as accommodation.

For instance, when looking at distant objects, the lens flattens, increasing its focal length, so the image is projected correctly onto the retina. On the contrary, while looking at close objects, the lens becomes rounder, reducing its focal length to maintain a sharp image on the retina. This illustrates why understanding optics is essential in fields like optometry, where custom lenses are prescribed to compensate for various vision impairments.

Two-Lens System

A two-lens system is an optical setup where two lenses are placed in sequence along the same axis. This is common in binoculars, microscopes, and telescopes. The effective focal length of the system, \( F_{eff} \), describes how the entire system converges or diverges light.

The equation to find the effective focal length in a two-lens system is \( \frac{1}{F_{eff}} = \frac{1}{F_1} + \frac{1}{F_2} - \frac{d}{F_1 F_2} \), where \( F_1 \) and \( F_2 \) are the focal lengths of the individual lenses, and \( d \) is the distance between them. The last term, \( \frac{d}{F_1 F_2} \), accounts for the fact that the two lenses aren't a single unit - there's a physical gap that adjusts the overall power of the system.

The adjustment of this effective focal length is key in practices like photography or in the design of optical instruments, where the goal is to manipulate the zoom, clarity, and field of view. A common everyday example would be using reading glasses in conjunction with the eyes to read fine print, combining the optical powers of both to see clearly.