Repeat Exercise 34.41 using the same lenses except for the following changes: The second lens is a diverging lens having a focal length of magnitude 60.0 cm.

Verified step by step guidance

1. Identify the given data: The first lens is a converging lens with a focal length \( f_1 \), and the second lens is a diverging lens with a focal length \( f_2 = -60.0 \ \text{cm} \). The object distance \( d_o \) for the first lens is the same as in Exercise 34.41. Assume the lenses are separated by a distance \( d \).
2. Calculate the image distance \( d_{i1} \) for the first lens using the lens equation: \( \frac{1}{f_1} = \frac{1}{d_o} + \frac{1}{d_{i1}} \). Rearrange to solve for \( d_{i1} \): \( d_{i1} = \left( \frac{1}{f_1} - \frac{1}{d_o} \right)^{-1} \).
3. Determine the object distance for the second lens, \( d_{o2} \), by considering the separation \( d \) between the lenses. If the image from the first lens is real, \( d_{o2} = d - d_{i1} \). If the image is virtual, \( d_{o2} = d + |d_{i1}| \).
4. Use the lens equation for the second lens to find the final image distance \( d_{i2} \): \( \frac{1}{f_2} = \frac{1}{d_{o2}} + \frac{1}{d_{i2}} \). Rearrange to solve for \( d_{i2} \): \( d_{i2} = \left( \frac{1}{f_2} - \frac{1}{d_{o2}} \right)^{-1} \).
5. Analyze the final image properties (real or virtual, upright or inverted, magnified or reduced) by calculating the magnification for each lens. The total magnification is the product of the magnifications of the two lenses: \( M = M_1 \cdot M_2 \), where \( M_1 = -\frac{d_{i1}}{d_o} \) and \( M_2 = -\frac{d_{i2}}{d_{o2}} \).

Key Concepts

Lens Types

Lenses are optical devices that refract light to form images. There are two main types: converging (convex) lenses, which focus light to a point, and diverging (concave) lenses, which spread light rays apart. Understanding the type of lens used is crucial for predicting how it will affect the path of light and the characteristics of the resulting image.

Focal Length

The focal length of a lens is the distance from the lens to the focal point, where parallel rays of light converge (for converging lenses) or appear to diverge from (for diverging lenses). It is a key parameter that determines the lens's power and the nature of the images formed. A shorter focal length indicates a stronger lens, while a longer focal length indicates a weaker lens.

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens, expressed as 1/f = 1/v + 1/u. This equation is essential for analyzing lens systems, allowing us to calculate the position and nature of the image formed by the combination of lenses. It is particularly important when dealing with multiple lenses, as their combined effects must be considered.