Problem 64 - College Physics

Question

Two converging lenses \(L_{1}\) and \(L_{2}\) have focal lengths of \(30 \mathrm{~cm}\) and \(20 \mathrm{~cm}\), respectively. The lenses are placed \(60 \mathrm{~cm}\) apart along the same axis, and an object is placed \(50 \mathrm{~cm}\) from \(L_{1}\left(110 \mathrm{~cm}\right.\) from \(L_{2}\) ). Where is the image formed relative to \(L_{2}\), and what are its characteristics?

Step-by-Step Solution

Answer

The image is 60 cm to the right of \( L_2 \) and is real and inverted.

Step 1: Calculate Image Distance for Lens L1

First, we need to find the image distance for the lens \( L_1 \), using the lens formula: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). The focal length \( f_1 \) is 30 cm and the object distance \( d_{o1} \) is 50 cm. Plugging in the values, the equation becomes \( \frac{1}{30} = \frac{1}{50} + \frac{1}{d_{i1}} \). Solving for \( d_{i1} \) gives us \( d_{i1} = 75 \text{ cm} \) (image distance from \( L_1 \)).

Step 2: Determine Object Distance for Lens L2

The image formed by \( L_1 \) serves as the object for \( L_2 \). The distance between \( L_1 \) and \( L_2 \) is 60 cm, so the object distance for \( L_2 \), \( d_{o2} \), is \( 75 \text{ cm} - 60 \text{ cm} = 15 \text{ cm} \).

Step 3: Calculate Image Distance for Lens L2

Now, use the lens formula for \( L_2 \), where the focal length \( f_2 \) is 20 cm, and the object distance \( d_{o2} \) is 15 cm. Applying the lens formula: \( \frac{1}{f_2} = \frac{1}{d_{o2}} + \frac{1}{d_{i2}} \). Substitute the values, \( \frac{1}{20} = \frac{1}{15} + \frac{1}{d_{i2}} \), and solve for \( d_{i2} \). This gives \( d_{i2} = 60 \text{ cm} \).

Step 4: Determine the Position Relative to Lens L2

The image distance \( d_{i2} \) for lens \( L_2 \) is 60 cm, meaning the image is 60 cm on the opposite side of \( L_2 \) (since it is positive, the image is real and formed on the opposite side). The position relative to \( L_2 \) is thus 60 cm to the right of \( L_2 \).

Step 5: Identify Image Characteristics

For a real, inverted image, characteristics can be derived from magnification, but that's unnecessary for position. The image formed by \( L_2 \) will be real, inverted and may be magnified or reduced based on prior distances.

Key Concepts

Lens Formula

When working with lenses, the lens formula is a critical tool for understanding and predicting how light behaves when it passes through a lens. This formula is expressed as:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
where:

• \( f \) is the focal length of the lens, which is the distance from the lens at which parallel rays of light converge.
• \( d_o \) is the object distance, the distance from the object to the lens.
• \( d_i \) is the image distance, the distance from the image to the lens.

This equation helps to relate the focal length with the object and image distances, allowing us to calculate one if the others are known. In the context of a converging lens, this formula aids in determining where an image will form relative to the lens, which is essential in optical systems like cameras and eyeglasses.

Image Distance

Image distance is a crucial concept in optics and one that you frequently calculate using the lens formula. In the given problem, we have two converging lenses and need to determine where the image forms after passing through both lenses.
For each lens, you use the lens formula to calculate the image distance:

• For lens \( L_1 \), placing the object 50 cm away, the resulting image distance \( d_{i1} \) ends up being 75 cm.
• This image becomes the object for lens \( L_2 \), with its position dictating a new image distance \( d_{i2} \) calculated as 60 cm in this example.

Positive image distance indicates the image is real and formed on the opposite side of the lens. This understanding is pivotal in determining characteristics like whether an image is inverted or upright.

Focal Length

Focal length is a defining property of lenses, particularly converging ones where it denotes the point at which light rays parallel to the principal axis converge. Each lens has its distinct focal length determining how it interacts with light.
In optics problems, such as the provided exercise, knowing the focal lengths is essential:

• Lens \( L_1 \) has a focal length of 30 cm, meaning that's where rays converge after passing through it.
• Lens \( L_2 \) with a shorter focal length of 20 cm, signifies a quicker convergence of light rays.

The focal length directly influences both the image distance and the overall behavior of the lens system. In converging lenses, shorter focal lengths imply stronger bending of light, which affects how lenses need to be spaced for optimal image formation.

Object Distance

Understanding object distance is vital when working with lenses as it sets up the entire optical configuration.
The object distance \( d_o \) is the separation between the object and the lens. It's the starting point to determine where the image will form:

• For lens \( L_1 \), an object placed 50 cm away establishes this configuration, leading to an image formed at 75 cm distance.
• For lens \( L_2 \), this image becomes an object itself at 15 cm (75 cm from \( L_1 \) minus the separation of 60 cm), forming a new image at 60 cm distance.

This step-by-step understanding of object distances ensures correct application of the lens formula, ultimately determining the kind and position of the image produced. It's fundamental to mastering optical systems and solving complex lens combinations.