91Ó°ÊÓ

The lens formula is a central concept in optics and is used to determine the relationship between the focal length, the object distance, and the image distance for a lens. This formula can be expressed mathematically as:

• \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)

Here, \( f \) is the focal length of the lens, \( v \) is the image distance, and \( u \) is the object distance. In this formula, distances are considered positive if they are in the direction of the incoming light and negative if they are in the opposite direction. This is the sign convention used in optics.

To apply the lens formula, one simply plugs in the known values and solves for the unknown variable. In our exercise, the lens formula was used to find the image distance \( v \). Knowing that the focal length \( f = 30 \mathrm{cm} \) and the object distance \( u = -75 \mathrm{cm} \), we created the equation:

• \( \frac{1}{30} = \frac{1}{v} + \frac{1}{75} \)

Solving this equation gives the image distance \( v = -37.5 \mathrm{cm} \), indicating that the image is formed on the same side of the lens as the object.

Image formation in lenses is a fascinating aspect of optics. A lens can form real or virtual images, depending on the placement of the object relative to the lens.

• A real image is formed when the light rays physically converge to form the image on the same side as the incoming light.
• A virtual image is one where the light rays appear to diverge from the image after passing through the lens.

In the given exercise, the resulting image is formed at a distance of \(-37.5 \mathrm{cm}\) from the lens, on the same side as the object. The negative sign before the image distance indicates that we have a real image, which typically is inverted. The location and characteristics of the image depend on the two governing factors:

• Focal Length: Which determines how strongly the lens will converge or diverge the incoming light.
• Object Distance: Which affects the magnification and nature of the image formed.

Magnification in optics describes how much larger or smaller the image is compared to the object itself. It is denoted by the symbol \( M \) and is calculated using the following formula:

• \( M = \frac{h_i}{h_o} = -\frac{v}{u} \)

Where \( h_i \) is the height of the image and \( h_o \) is the height of the object. In the provided problem, this formula was used to determine the image height knowing that the image distance \( v = -37.5 \mathrm{cm} \) and the object distance \( u = -75 \mathrm{cm} \).

• The magnification \( M = \frac{-(-37.5)}{-75} = -0.5 \).

This calculation reveals that the image is inverted and half the size of the object, indicated by the negative sign and magnitude respectively. Magnification conveys crucial information regarding not only the size but also the orientation of the image concerning the original object.