91Ó°ÊÓ

The lens formula is an essential tool in optics, helping us relate the focal length of a lens, the distance from the object to the lens, and the distance from the image to the lens. This formula is given by the equation \( \frac{1}{f} = \frac{1}{o} + \frac{1}{i} \), where:

• \( f \) is the focal length.
• \( o \) is the object distance.
• \( i \) is the image distance.

For a diverging lens, always remember that the focal length \( f \) is taken as negative. This negative value arises because a diverging lens spreads out light rays, rather than converging them like a converging lens.
In the context of the exercise, assigning the correct sign conventions is crucial. We take the object distance \( o = -20 \text{ cm} \) (negative because the object is on the same side as the incoming light) and the focal length \( f = -30 \text{ cm} \). These values are then substituted into the lens formula, allowing us to find the image distance \( i \). Always ensure the signs are consistent with the nature of the lens and the position of the object.

To determine the image distance \( i \), we input the known values into the lens formula: \( \frac{1}{-30} = \frac{1}{-20} + \frac{1}{i} \). Solving for \( \frac{1}{i} \) gives us \( \frac{1}{i} = 0.0167 \), which means the image distance \(i \) is calculated as \(60 \text{ cm} \).
The positive value of \(i\) indicates that the image forms on the opposite side of the lens from the object, which is characteristic of virtual images formed by diverging lenses. These images cannot be projected on a screen as they appear to originate from a different location than where light physically converges.
Understanding that a positive \(i\) for a diverging lens means the image is virtual and upright is key. The image will appear smaller than the object, a common attribute of images formed by this type of lens.

A ray diagram visually represents how rays interact with a lens to form an image. In this context, it helps track and verify the steps we took analytically to find the image distance.
For a diverging lens, there are three critical rays to consider:

• A ray parallel to the principal axis refracts as if it came from the focal point on the side from which light approaches.
• A ray traveling towards the lens center passes through without bending.
• A ray directed toward the focal point on the object side exits parallel to the principal axis.

When you trace these rays on the diagram, they appear to diverge and never meet. However, by extending them backwards, they converge at a point on the side opposite the object, defining where the virtual image forms.
The ray diagram not only enhances visualization but also confirms the virtual nature and position of the image, providing insight into how diverging lenses shape images.