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In the field of optics, a real image is formed when light rays from an object converge at a point after passing through a lens. This type of image can be projected onto a screen, which is not the case for virtual images. Real images are always inverted. Meaning, they appear upside down relative to the original object. In the exercise, a real image is formed to the right of the lens, indicating that the lens is converging the light rays at a specific point. This is crucial because it enables us to analyze and calculate the properties related to the lens, such as magnification and focal length.

Magnification describes how much larger or smaller the image is compared to the actual object. Mathematically, it is the ratio of the image height to the object height, which also relates to the image and object distances. The formula for magnification is:

• \( m = \frac{-d_i}{d_o} \)

The negative sign indicates the image is inverted. In the problem, the image size is mentioned to be one-half the size of the object. Therefore, magnification is

• \( m = -\frac{1}{2} \)

This reflects the fact that the image is inverted and half the size of the object. By using this relationship, we can find distances related to the object and image, which are key for subsequent calculations like finding the focal length.

Focal length is the measure of how strongly a lens converges (or diverges) light. It is the distance from the center of the lens to the focal point, where parallel rays of light converge. A shorter focal length indicates a more powerful lens that bends light more sharply. To find the focal length, we utilize the distances from the lens to the object (\(d_o\)) and to the image (\(d_i\)). The exercise solves for these distances first, providing information to calculate focal length with the lens formula. Knowing the focal length helps understand a lens's properties and its application in various optics scenarios.

The lens formula is fundamental in optics, providing a mathematical relationship between object distance (\(d_o\)), image distance (\(d_i\)), and the focal length (\(f\)). The lens formula is:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]This equation is the key to solving many optical problems. In this exercise, once the distances were found, they were substituted into the lens formula to find the focal length.

• By substituting \(d_o = 60\) cm and \(d_i = 30\) cm, we calculated the focal length as 20 cm.

Understanding how to rearrange and solve the lens formula is critical, providing insight into how lenses affect light and form images. This formula is especially useful when working with converging and diverging lenses in various settings, from simple school experiments to complex optical devices.