Question

The distance from the lens (actually a combination of the cornea and the crystalline lens) to the retina at the back of the eye is \(2.0 \mathrm{~cm}\). If light is to focus on the retina, a) what is the focal length of the lens when viewing a distant object? b) what is the focal length of the lens when viewing an object \(25 \mathrm{~cm}\) away from the front of the eye?

Short answer

Answer: The focal length for a distant object is 2.0 cm, and for an object 25 cm away, the focal length is approximately 1.85 cm.

Step-by-step solution

Find the focal length when viewing a distant object

When viewing a distant object, the object distance is considered to be infinite. Therefore, the lens formula becomes: \(\frac{1}{f} = \frac{1}{\infty} + \frac{1}{2.0}\). The term \(\frac{1}{\infty}\) becomes 0, so: \(\frac{1}{f} = \frac{1}{2.0}\). Now, we need to find f. To find f, take the reciprocal of both sides: \(f = \frac{1}{\frac{1}{2.0}}\). The focal length, when viewing a distant object, is \(f = 2.0 \mathrm{~cm}\).

Find the focal length when viewing an object 25 cm away from the front of the eye

When viewing an object \(25 \mathrm{~cm}\) away from the front of the eye, the object distance, \(d_o\), is \(25.0 \mathrm{~cm}\). Now, we can use the lens formula: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\). Substitute the given values into the formula: \(\frac{1}{f} = \frac{1}{25.0} + \frac{1}{2.0}\). To solve for the focal length, f, we first need to find a common denominator for the fractions on the right side of the equation. In this case, the common denominator is 50: \(\frac{1}{f} = \frac{2}{50} + \frac{25}{50}\). Now, combine the fractions on the right side of the equation: \(\frac{1}{f} = \frac{27}{50}\). To find f, take the reciprocal of both sides: \(f = \frac{1}{\frac{27}{50}}\). The focal length, when viewing an object \(25 \mathrm{~cm}\) away from the front of the eye, is \(f \approx 1.85 \mathrm{~cm}\).

Key concepts

Lens Formula

The lens formula is an essential equation in optics, used to relate the distances and focal length of a lens. It is given by:

• \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)

where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. This formula allows us to calculate how lenses focus light to form clear images.
When viewing distant objects, the object distance is considered infinite, and so \( \frac{1}{d_o} \) is 0, simplifying calculations as seen in the original solution. This formula helps in adjusting the lens of your eye to view objects at varying distances.

Focal Length

Focal length is a critical concept in optics that measures how strongly a lens converges or diverges light. It is the distance between the lens and the point where parallel rays of light converge to a focus.

• A shorter focal length indicates a stronger converging power.
• The focal length varies when different objects are viewed, as seen when switching focus from a distant object to a closer one.

Your eye adjusts the lens's focal length to maintain a clear image on the retina regardless of the object's distance. This ability of the eye, similar to how a camera lens focuses, is key to good vision.

Object Distance

In optics, the object distance \( d_o \) is the space from the object being viewed to the lens. It's crucial when using the lens formula to determine the focal length.

• For distant objects, \( d_o \) is approximated as infinity, which simplifies the calculations, as animals naturally do when observing far-off items, like clouds or the horizon.
• For objects nearer, like a book, \( d_o \) is a finite value that affects image formation.

Understanding object distance is vital for practical applications in both everyday activities and complex optical devices, ensuring the lens correctly focuses on the target.

Retina

The retina is the light-sensitive tissue at the back of the eye, functioning similarly to a film in a camera. It plays a crucial role in vision.

• It captures light and converts it into neural signals for the brain to interpret.
• An image is formed when the eyes' lenses focus light onto the retina, providing the sense of sight.

Maintaining focus on the retina is essential for clear vision. This is why the lens of the eye adjusts its focal length depending on how far the object is. If the light doesn't meet precisely on the retina, vision may blur, leading to conditions like myopia or hyperopia.

Eye Anatomy

Understanding the anatomy of the eye aids in grasping how optics apply to vision.

• The eye contains two main refractive elements: the cornea and the crystalline lens. Both bend light to achieve focus.
• Light enters through the cornea, passes through the aqueous humor, the lens, and finally the vitreous humor before reaching the retina.

This structure resembles a complex optical device designed to deliver clear and accurate visual information. Each component, from the cornea's initial focus to the lens's adjustments, is vital to forming sharp images that our brains can quickly interpret, allowing us to see the world around us dynamically and remotely.