91Ó°ÊÓ

A converging lens, also known as a convex lens, is designed to bring light rays to a point. This is achieved through its curved surface. The main feature is that it can focus light, making objects appear larger and sharp.
Converging lenses are defined by their focal length, which is the distance from the lens's center to the focal point where light rays meet. This focal length is positive due to the nature of the lens

• Positive focal length
• Focal point brings light together
• Used in a variety of optical devices like cameras and glasses

In the problem, the converging lens has a focal length of \(30.0 \; \mathrm{cm}\). Understanding how this lens works is crucial for determining how an image is formed. The object placed at \(40.0 \; \mathrm{cm}\) to the left of the lens indicates the distance from where light begins its course through the lens.

Unlike converging lenses, diverging lenses (or concave lenses) spread light rays apart. This means they cause parallel light to diverge as if it originated from a focal point behind the lens.
Diverging lenses have a negative focal length, representing this outward spread of light.

• Negative focal length
• Light is spread out
• Used in applications like correcting short-sightedness

In the textbook problem, the diverging lens has a focal length of \(-20.0 \; \mathrm{cm}\), situated \(110 \; \mathrm{cm}\) to the right of the converging lens. This divergence will affect the journey of the light rays coming from the image formed by the first lens, changing its position and size.

The concept of image magnification is central to understanding how lenses work. It refers to the factor by which an image is enlarged compared to the object. This ratio is a result of the distances and focal lengths involved.
To calculate magnification, the formula \(m = -\frac{v}{u}\) is used, where \(v\) is the image distance from the lens and \(u\) is the object distance from the lens.

• Magnification less than 1 means the image is smaller than the object
• Greater than 1 means the image is larger
• Positive magnification indicates an upright image
• Negative magnification indicates an inverted image

In the problem's solution steps, initial magnification is calculated after the converging lens. Then, in combination with the diverging lens, the final magnification is determined, and the orientation can be inferred through its sign.

The lens formula is a mathematical representation that relates the object distance, image distance, and the focal length of a lens. It is crucial for finding the precise location of an image formed by the lens.
The formula is written as: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]Where:

• \(f\) is the focal length of the lens (positive for converging, negative for diverging)
• \(v\) is the image distance
• \(u\) is the object distance

This formula allows you to solve for one variable if the others are known. Applying the lens formula in step-by-step solutions enables finding the positions of intermediate and final images in optics problems.