## Thin Lenses

14.4 Thin Lenses

### Learning Objectives

By the end of this section, you will be able to:

- Use ray diagrams to locate and describe the image formed by a lens
- Employ the thin-lens equation to describe and locate the image formed by a lens

### Ray Diagrams

Here are the steps to drawing ray diagrams:

- Draw the lens (a side view like I’ve been doing) and a long, horizontal line through the center. This line is called the central axis, or the optical axis.
- Draw both focal points of the lens (remember they are symmetric, so the distance
*f*will be the same on both sides). Just draw them as dots. - Then draw the object at the appropriate distance from the lens (
*d*). We usually draw the object as a vertical arrow pointing upward. This is to allow us to figure out if the image will be inverted (upside down) or not. An arrow has a definite top and bottom to it, so it makes it easy._{o} - Draw exactly 3 special rays, and trace them through the lens. These special rays are called
*principle light rays*and they follow very specific paths. Each ray starts at the top of the object.- Draw a ray that starts at the top of the object and runs parallel to the optical axis. When this light ray gets to the lens, it will bend inward and go through the focal point.
- Draw a ray that starts at the top of the object and goes through the center of the lens. This will continue in a straight line through the lens and won’t bend.
- Draw a ray that starts at the top of the object and goes through the focal point on the object-side of the lens. When this reaches the lens, it will bend so that it exits the lens parallel to the optical axis. (Remember how lenses are symmetrical.)

- Where the three principle light rays converge is where the image forms. Since the rays started at the top of the object, you will see an image of the top of the object at the point where the light rays converge.

### Converging (Convex) Lenses

**Practice!**

Practice 14.4.1 |
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A convex lens has a focal length f. An object is placed at a distance between f and 2f on a line perpendicular to the center of the lens. The image formed is located at what distance from the lens? |

Practice 14.4.2 |
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A convex lens has a focal length f. An object is placed between infinity and 2f from the lens along a line perpendicular to the center of the lens. The image is located at what distance from the lens? |

### Diverging Lenses

### The Lens Equation and Magnification

The equations we use for spherical lenses are the same as those for spherical mirrors. The lens equation is

and the lateral magnification is

#### Sign Conventions

We have to be careful about the signs (+ or -) that we use when we describe the distances and the magnification. Here is a table to help with the signs:

Converging (Convex) Lens | Diverging (Concave) Lens | |

Real Object | + d_{o} | + d_{o} |

Real Image | + d_{i} | N/A (concave lenses cannot form real images) |

Virtual Image | – d_{i} | – d_{i} |

Focal Length | + f | – f |

Pause & Predict 14.4.1 |
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What is the focal length of your eye’s lens when you look at the apple up close, at 10 cm from your eyes? |

Pause & Predict 14.4.2 |
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What focal length should a diverging corrective lens have for this nearsighted person to clearly see an object 5.45 m away? |

**Practice!**

Practice 14.4.3 |
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An object is placed at 29.0 cm to the right of a diverging lens whose focal length is -25 cm. What is the location of the image relative to the lens? |

Practice 14.4.4 |
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A thin diverging lens has focal length f = -12 cm. If an object 9 cm tall is placed 24 cm from the lens, what is the height of the image? |

Practice 14.4.5 |
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A convex, or converging, lens with a focal length of 34 cm is used to observe an image of an apple than is 80 cm away from the lens. At what distance from the lens does the image form and what type of image is it? |

Practice 14.4.6 |
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A person’s eye has a diameter of 2.5 cm and a converging lens that produces a clear, real image on the retina. When viewing an apple, the eye’s lens has a focal length of 2.4 cm. What is the distance between the apple and the person’s eyes? |

Practice 14.4.7 |
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A nearsighted person wants to see an apple that is 7 meters away but can only clearly see objects that are at most 62 cm away from her eyes. Eyeglasses made of diverging corrective lenses can help her to see the apple clearly. If her glasses are 2.5 cm in front of her eyes, what should be the power of these corrective lenses to allow her to see the apple? |

The power of the corrective lenses is equal to the inverse of the focal length measured in meters. Use the thin lens equation to determine the inverse of the focal length, or the power of the corrective lenses:

making sure to use object and image distances that are measured from the corrective lens and not the eye’s lens.

**Discuss!**

Given the object distance and focal length of the converging lens in the figure, determine the image distance.

Suppose the image of an object is upright and magnified 1.95 times when the object is placed 17.5 cm from a particular converging lens.

(a) Find the location of the image.

(b) Find the focal length of the lens.