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Understanding the thin lens equation is essential when dealing with the behavior of light as it passes through lenses. The fundamental relationship governing the optics in a thin lens is summarized with the equation \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length of the lens, \(d_o\) is the object distance, and \(d_i\) is the image distance.

When handling problems related to lenses, this equation allows us to determine one of the three variables if the other two are known. For example, if a magnifier with a known focal length is used by an observer to focus on an object, and we know where the image is formed (like at the observer's near point), we can rearrange the equation to find the required object distance for a clear image. This is the backbone of creating magnified images in optics.

The object distance, \(d_o\), in optics, refers to the space between the object being viewed and the lens. Its value is crucial in determining how the lens will form the image—a key point for students studying magnification and lens equations. The object distance directly affects the properties of the formed image, including whether it's real or virtual, upright or inverted, and magnified or reduced.

When using the thin lens equation, if the object distance is less than the focal length of the lens, the resulting image is virtual and magnified. As in our example with the magnifier, setting the object at the precise distance allows the viewer to see a magnified image at their near point, enabling a clear view of small details.

The concept of image height, \(h_i\), is integral when understanding magnified images through lenses. This height represents the size of the image produced by the lens as compared to the actual size of the object, \(h_o\). The relation between the object and the image size is given by the magnification \(\text{M}\), where \(\text{M} = \frac{h_i}{h_o}\) or the negative ratio of image distance to object distance \(\text{M} = - \frac{d_i}{d_o}\).

The sign in the magnification formula indicates the image orientation; if it is negative, the image is inverted. By knowing the object's height and the magnification, we can calculate the height of the image. In optics exercises where precision is vital, like our textbook problem, understanding how to compute image height is key to grasp how a magnifier amplifies small objects, allowing for detailed examination.