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The lens formula is a crucial tool in optics problems, especially when dealing with lenses and the formation of images. It helps us determine the relationship between the object distance \(d_o\), the image distance \(d_i\), and the focal length of the lens \(f\). The formula is given by:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}. \]This fundamental equation applies to both converging and diverging lenses. By rearranging the lens formula, you can solve for any of the values if the other two are known. It's important to remember:

• For converging lenses, the focal length \(f\) is positive.
• For diverging lenses, the focal length \(f\) is negative.
• When the image distance \(d_i\) is positive, the image is real and formed on the opposite side of the object.
• When \(d_i\) is negative, the image is virtual and appears on the same side as the object.

Image position is an essential concept in understanding how lenses create images. The position where the image is formed, either real or virtual, is determined using the lens formula. In our specific problem, for the first converging lens with a focal length of \(+5.0 \mathrm{~cm}\), the object position was \(15.0 \mathrm{~cm}\) away.
By using the lens formula, we calculated the image distance \(d_{i1} = 7.5 \mathrm{~cm}\), indicating that the image is formed behind the lens. For the second diverging lens, the object distance is derived from this image position, acting as a virtual object, resulting in a negative value.
The image position relative to a lens helps determine characteristics like size, orientation, and type (real or virtual). Adjusting the object or the type of lens will influence where this image forms.

Focal length is a vital characteristic that describes how strongly a lens converges or diverges light. It is denoted as \(f\) and plays a pivotal role in the lens formula. Positive focal lengths signify converging lenses, while negative values correspond to diverging lenses.
In the exercise, the first lens has a focal length of \(+5.0 \mathrm{~cm}\), indicating it is a converging lens. It brings light rays to a focus, creating a real image. On the other hand, the second lens has a focal length of \(-12.0 \mathrm{~cm}\), classifying it as a diverging lens. Such lenses spread light rays outward, usually forming virtual images unless accompanied by other lenses.
Understanding focal length is crucial in lens systems, as it helps predict how lenses will act on incoming light, and it aids in calculating image distances and sizes in optical setups.

Magnification in optics refers to the process of enlarging or reducing the apparent size of an object through a lens. It is defined by the magnification formula:
\[ m = -\frac{d_i}{d_o}, \]where \(m\) represents magnification, \(d_i\) is the image distance, and \(d_o\) is the object distance.
Magnification indicates not only the size of the image relative to the object but also its orientation. A negative magnification value indicates an inverted image, while a positive value signifies an upright image.

• The first lens gave a magnification of \(-0.5\), suggesting the image was half the size of the object and inverted.
• The second lens, a diverging lens, significantly enhanced the size with a magnification of \(6\), resulting in a much larger image.

The final magnification is the product of both lens magnifications, resulting in an overall change in size and orientation for the final image.

In multi-lens systems, a "virtual object" is created when the image from one lens serves as the object for the subsequent lens. This typically happens when two lenses are close together, allowing the image formed by the first lens to be used by the second lens to form another image.
In the exercise, the first lens creates an image that acts as a virtual object for the second lens. Virtual objects are considered to have negative distances, as they are not physically located on the same side as the camera or eye observing the second lens.
This concept allows lenses to manipulate light to achieve complex image formations, crucial in intricate optical designs like microscopes and telescopes.