Problem 80 - Fundamentals of Physics

Question

Two-lens systems. In Fig. \(34-45\), stick figure \(O\) (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to \(O\), which is at object distance \(p_{1}\). Lens 2 is mounted within the farther boxed region, at distance \(d .\) Each problem in Table \(34-9\) refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by \(\mathrm{C}\) for converging and D for diverging; the number after C or \(\mathrm{D}\) is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated). Find (a) the image distance \(i_{2}\) for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification \(M\) for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of lens 2 as object \(O\) or on the opposite side. \(\begin{array}{lllll}\mathbf{8 0} & +10 & \mathrm{C}, 15 & 10 & \mathrm{C}, 8.0\end{array}\)

Step-by-Step Solution

Answer

(a) Image distance \( i_2 = 13.33 \) cm. (b) Magnification \( M = 2 \). (c) Real, (d) Inverted, (e) Opposite side.

Step 1: Understand the Problem

We have a two-lens system consisting of two thin lenses, with lens 1 being a converging lens (focal length 15 cm) and lens 2, also a converging lens (focal length 8 cm). The object is placed 10 cm from lens 1, and the lenses are separated by 10 cm.

Step 2: Lens 1 Image Distance Calculation

First, calculate the image distance \( i_1 \) for lens 1 using the lens formula: \( \frac{1}{f_1} = \frac{1}{p_1} + \frac{1}{i_1} \), where \( f_1 = +15 \) cm and \( p_1 = 10 \) cm.\[\frac{1}{i_1} = \frac{1}{15} - \frac{1}{10}\]Solving this gives \( i_1 = 30 \) cm.

Step 3: Finding Intermediate Image Position

The intermediate image from lens 1 acts as the object for lens 2. Since \( i_1 = 30 \) cm and the lenses are 10 cm apart, the object distance for lens 2, \( p_2 \), is:\[ p_2 = 30 - 10 = 20 \text{ cm} \] This distance is positive since it's measured in the direction of the original object’s light path.

Step 4: Lens 2 Image Distance Calculation

Calculate the image distance \( i_2 \) for lens 2 using the lens formula: \( \frac{1}{f_2} = \frac{1}{p_2} + \frac{1}{i_2} \), where \( f_2 = +8 \) cm and \( p_2 = 20 \) cm.\[\frac{1}{i_2} = \frac{1}{8} - \frac{1}{20}\]Solving this gives \( i_2 = +13.33 \) cm.

Step 5: Calculate Overall Lateral Magnification

The overall magnification \( M \) is the product of the magnifications from each lens:\[ M = m_1 \times m_2 \]where \( m_1 = -\frac{i_1}{p_1} = -\frac{30}{10} = -3 \) and \( m_2 = -\frac{i_2}{p_2} = -\frac{13.33}{20} \approx -0.6665 \).\[M = -3 \times -0.6665 = 2 \]

Step 6: Determine the Nature of the Final Image

(c) The image is real (R) since \( i_2 \) is positive. (d) The image is inverted (I) because overall magnification \( M = 2 \) is positive, opposite the inversion of the first lens. (e) The final image is on the opposite side of lens 2 from the object since \( i_2 \) is positive.

Key Concepts

Converging lenses

Converging lenses are one of the most common types of lenses used in optics. They are also known as convex lenses. A converging lens bends incoming light rays towards a single point known as the focus or focal point. This property makes converging lenses ideal for focusing light to form clear and sharp images.

• These lenses are thicker at the center than at the edges.
• They have a positive focal length, which means they converge light rays to a point.

Understanding converging lenses is crucial when analyzing two-lens systems as they play a vital role in image formation.

Lens formula

The lens formula is a key mathematical tool used in optics to determine the relationship between the object distance (\( p \)), the image distance (\( i \)), and the focal length (\( f \)) of a lens. It is expressed as: \[\frac{1}{f} = \frac{1}{p} + \frac{1}{i}.\]This formula is applicable to both converging and diverging lenses. However, it's important to remember:

• A positive focal length indicates a converging lens, while a negative focal length stands for a diverging lens.
• The object distance (\( p \)) is usually taken positive if the object is on the same side as the incoming light.
• The image distance (\( i \)) is positive if the image forms on the side opposite the incoming light from the lens.

The lens formula is foundational for calculating image distances in lens systems.

Image distance calculation

Calculating the image distance is a critical step in understanding how lenses form images. For each lens in a system, you use the lens formula to find this distance. For instance, if you know the object distance and focal length for a particular lens, plug these values into the lens formula to solve for the image distance (\( i \)).
You can then determine many aspects of the image, such as size, orientation, and type (real or virtual).

• First, solve for image distance (\( i \)) to determine where the image will form.
• Use this image distance as the object distance for the next lens, if more than one lens is involved.

Understanding how to calculate the image distance helps in predicting the behavior of light in complex optical systems.

Lateral magnification

Lateral magnification is a measure of how much larger or smaller the image is compared to the object. It is calculated for a single lens using the formula:\[m = -\frac{i}{p},\]where (\( m \)) is the magnification, (\( i \)) the image distance, and (\( p \)) the object distance.

• If the magnification (\( m \)) is positive, the image is upright relative to the object.
• If (\( m \)) is negative, the image is inverted.

The overall magnification in a two-lens system is the product of the magnifications from each lens. Hence, understanding lateral magnification is vital for evaluating how the overall image will appear in terms of size and orientation.

Real and virtual images

Real and virtual images are two fundamental types of images formed by lenses. Understanding the difference between them is key in lens systems. A **real image** is formed when light rays converge at a point. These images can be projected onto a screen as they appear on the side of the lens opposite the origin of light. For converging lenses, real images are usually inverted. A **virtual image**, on the other hand, occurs when light rays only appear to diverge from a point behind the lens. These cannot be projected on a screen and appear upright. Virtual images are typically formed when the lens cannot bring the light rays to actually converge and appear on the same side as the object. In lens systems, discerning whether an image is real or virtual aids in interpreting the lens's effect on viewed objects.