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The focal length of a lens is a crucial concept in optics, as it determines how the lens bends light. Specifically, it is the distance from the lens where parallel rays of light either converge or diverge. For a thin lens, the focal length can be positive or negative. A positive focal length indicates a converging lens, like many magnifying glasses. On the other hand, a negative focal length, as given in the exercise (\(f = -100 \text{ cm}\)), signifies a diverging lens. This type of lens spreads light rays apart, giving it its characteristic trait of forming virtual images. When you use your thin lens formula, \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), understanding the sign convention is key to determining the nature of the focal length and the resulting image.

A virtual image is an image that appears to be located at a position from which light rays seem to diverge. In reality, the light does not converge there. This contrasts with a real image, where light actually converges at a point. Virtual images are typically formed by diverging lenses or mirrors.
In the context of the thin lens formula used in the exercise, the image distance \(d_i\) is negative, signifying a virtual image. This happens because the image forms on the same side as the object, which in this case is to the right of the lens (\(d_i = -50 \text{ cm}\)). Virtual images are always upright compared to the original object and, in this situation, they also appear smaller due to the diverging effect of the lens.

The object distance is essentially how far the original object is from the lens. In optical terms, this is denoted by \(d_o\), and in our scenario, it is \(100 \text{ cm}\). When evaluating the thin lens formula, knowing the object distance helps determine how light will behave when interacting with the lens.

• If an object is placed far from a lens, it might produce a clearer, real image on the opposite side.
• When close and if a converging lens is used, it can magnify the object into a virtual image.

The exercise has a unique setup with a diverging lens and an object on the same side. This peculiar configuration with \(d_o = 100 \text{ cm}\) beautifully illustrates how a virtual image forms with a negative lens.

Understanding image characteristics means analyzing how and where an image forms concerning its parent object. Key characteristics include whether the image is real or virtual, upright or inverted, and magnified or reduced.
For our exercise, the characteristics are defined as follows:

• **Nature:** The image is virtual because it forms on the same side as the object (\(d_i = -50 \text{ cm}\)). Virtual images can't be projected onto a screen given they are just apparent locations.
• **Orientation:** This image is upright, maintaining the same vertical orientation as the actual object.
• **Size:** It is smaller than the actual object, which is a consistent feature of diverging lenses as they spread light rays apart.

These characteristics are directly tied to the parameters given and showcase how applying the thin lens formula provides a deeper understanding of lens behavior.