Problem 60 - Physics

Question

Two converging lenses are separated by \(24.00 \mathrm{~cm} .\) The focal length of each lens is 12.00 \(\mathrm{cm}\). An object is placed \(36.00 \mathrm{~cm}\) to the left of the lens that is on the left. Determine the final image distance relative to the lens on the right.

Step-by-Step Solution

Answer

The final image is 12 cm to the left of the lens on the right.

Step 1: Identify the Given Values

We are given two converging lenses with focal length \( f_1 = f_2 = 12.00 \) cm. The lenses are separated by \( d = 24.00 \) cm. The object distance from the left lens (Lens 1) is \( o_1 = 36.00 \) cm.

Step 2: Use the Lens Formula for Lens 1

For Lens 1, apply the lens formula \( \frac{1}{f_1} = \frac{1}{o_1} + \frac{1}{i_1} \). Substitute \( f_1 = 12 \) cm and \( o_1 = 36 \) cm into the equation to find the image distance \( i_1 \) for Lens 1. After solving, we get: \[ \frac{1}{12} = \frac{1}{36} + \frac{1}{i_1} \] \[ \frac{1}{i_1} = \frac{1}{12} - \frac{1}{36} = \frac{3 - 1}{36} = \frac{2}{36} \] \[ i_1 = 18 \text{ cm} \].

Step 3: Determine Object Distance for Lens 2

The image formed by Lens 1 acts as the object for Lens 2. Because the lenses are 24 cm apart, and the image from Lens 1 is 18 cm to the right of Lens 1, its position relative to Lens 2 is given by \( o_2 = d - i_1 = 24 - 18 = 6 \) cm.

Step 4: Use the Lens Formula for Lens 2

For Lens 2, apply the lens formula \( \frac{1}{f_2} = \frac{1}{o_2} + \frac{1}{i_2} \). Substitute \( f_2 = 12 \) cm and \( o_2 = 6 \) cm into the equation to find the image distance \( i_2 \) for Lens 2. Then, \[ \frac{1}{12} = \frac{1}{6} + \frac{1}{i_2} \] \[ \frac{1}{i_2} = \frac{1}{12} - \frac{1}{6} = \frac{1 - 2}{12} = -\frac{1}{12} \] \[ i_2 = -12 \text{ cm} \].

Step 5: Interpret the Final Image Distance

Since \( i_2 = -12 \) cm, it indicates that the final image is located 12 cm to the left of Lens 2.

Key Concepts

Converging Lenses

Converging lenses are central to many optical devices. They are designed to bend light rays to meet at a single point, called the focus. This is why they are sometimes called convex lenses. These lenses are thicker in the center than at the edges.

• They're commonly used in cameras, glasses, microscopes, and telescopes.
• Their purpose is to focus light to form clear images of objects.

Understanding how converging lenses work is key to solving problems involving image formation. By using the lens formula, we can determine where an image will form in relation to the lens.

Focal Length

The focal length is a crucial property of a lens, affecting how it bends light. Represented by the symbol \( f \), the focal length is the distance from the lens to its focus where parallel light rays converge.

• Shorter focal lengths indicate lenses that are more powerful at bending light.
• In the exercise, each lens has a focal length of 12 cm, meaning they are identical in their light-bending power.

Knowing the focal length helps in using the lens formula to find out where images might appear.

Image Distance

Image distance refers to how far the image is from the lens. In calculations, it is represented by \( i \). A positive value means the image is formed on the side opposite the object, while a negative value implies it's on the same side as the object.

• The exercise showed the first lens creating an image 18 cm away, indicating the location where the light rays converge into a clear image.
• The second lens forms an image at -12 cm, showing that the final image is actually to the left of this lens.

Understanding image distance helps in visualizing where the images form in optical setups.

Object Distance

Object distance is the space from the object to the lens, denoted by \( o \). It is an essential part of the lens formula, which helps determine where an image will appear based on where an object is located.

• In our example, the initial object is 36 cm from the first lens, which serves as the input for calculating the first image distance.
• The image created by the first lens becomes the object for the second lens, with the object distance calculated using the separation between lenses and the initial image position.

Determining object distance accurately is vital for precise image location predictions.