Question

An object is located \(10.0 \mathrm{cm}\) in front of a converging lens with focal length \(12.0 \mathrm{cm} .\) To the right of the converging lens is a second converging lens, \(30.0 \mathrm{cm}\) from the first lens, of focal length \(10.0 \mathrm{cm} .\) Find the location of the final image by ray tracing and verify by using the lens equations.

Short answer

**Answer:** 1. Ray tracing for each lens individually 2. Lens equations for each lens

Step-by-step solution

Understand the given exercise and object location

Consider an object located 10.0 cm in front of the first converging lens (Lens 1) with a focal length of 12.0 cm and a second converging lens (Lens 2) 30.0 cm away from Lens 1 with a focal length of 10.0 cm.

Ray tracing for Lens 1

For Lens 1: 1. Draw a ray parallel to the principal axis which refracts through the focal point of Lens 1. 2. Draw a ray passing through the center of Lens 1, which remains undeviated. 3. Draw a ray passing through the focal point of Lens 1, which refracts parallel to the principal axis after passing Lens 1. These three rays will converge at a point defining an intermediate image formed by Lens 1.

Find the image distance for Lens 1

Use the image formed in step 2 as an object for Lens 2 and repeat the ray tracing process for Lens 2: 1. Draw a ray parallel to the principal axis which refracts through the focal point of Lens 2. 2. Draw a ray passing through the center of Lens 2, which remains undeviated. 3. Draw a ray passing through the focal point of Lens 2, which refracts parallel to the principal axis after passing Lens 2. These rays will converge at a common point where the final image is formed.

Lens equations for Lens 1

The lens equation for Lens 1 is given by \(\frac{1}{f_1}=\frac{1}{d_o}+\frac{1}{d_{i1}}\), where: \(f_1 = 12.0\,\mathrm{cm}\) is the focal length of Lens 1, \(d_o = 10.0\,\mathrm{cm}\) is the object distance from Lens 1, \(d_{i1}\) is the distance of the intermediate image formed by Lens 1. Solve for \(d_{i1}\) to get the image distance formed by Lens 1.

Lens equations for Lens 2

The lens equation for Lens 2 is given by \(\frac{1}{f_2}=\frac{1}{d_{o2}}+\frac{1}{d_{f}}\), where: \(f_2 = 10.0\,\mathrm{cm}\) is the focal length of Lens 2, \(d_{o2}\) is the object distance from Lens 2, which is equal to the image distance \(d_{i1}\) from Lens 1 plus the distance between the two lenses (30.0 cm), \(d_{f}\) is the final image distance formed by Lens 2. Solve for \(d_{f}\) to get the final image distance formed by the entire lens system.

Compare the results from ray tracing and lens equations

Compare image distances obtained through ray tracing and lens equations. If they are in agreement, the final image location obtained is accurate.