25.6: Image Formation by Lenses (2024)

Learning Objectives

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

• List the rules for ray tracking for thin lenses.
• Illustrate the formation of images using the technique of ray tracking.
• Determine power of a lens given the focal length.

Definition: CONVERGING OR CONVEX LENS

The lens in which light rays that enter it parallel to its axis cross one another at a single point on the opposite side with a converging effect is called converging lens.

Definition: FOCAL POINT F

The point at which the light rays cross is called the focal point F of the lens.

Definition: FOCAL LENGTH \(f\)

The distance from the center of the lens to its focal point is called focal length \(f\).

Definition: POWER \(P\)

The power \(P\) of a lens is defined to be the inverse of its focal length. In equation form, this is

\[P = \frac{1}{f}.\]

where \(f\) is the focal length of the lens, which must be given in meters (and not cm or mm). The power of a lens \(P\) has the unit diopters (D), provided that the focal length is given in meters. That is \(1 D = 1/m\) or \(1 m^{-1}\). (Note that this power (optical power, actually) is not the same as power in watts defined in "Work, Energy, and Energy Resources." It is a concept related to the effect of optical devices on light.) Optometrists prescribe common spectacles and contact lenses in units of diopters.

Example \(\PageIndex{1}\): What is the Power of a Common Magnifying Glass?

Suppose you take a magnifying glass out on a sunny day and you find that it concentrates sunlight to a small spot 8.00 cm away from the lens. What are the focal length and power of the lens?

Strategy:

The situation here is the same as those shown in Figure 1 and Figure 2. The Sun is so far away that the Sun’s rays are nearly parallel when they reach Earth. The magnifying glass is a convex (or converging) lens, focusing the nearly parallel rays of sunlight. Thus the focal length of the lens is the distance from the lens to the spot, and its power is the inverse of this distance (in m).

Solution

The focal length of the lens is the distance from the center of the lens to the spot, given to be 8.00 cm. Thus, \[f = 8.00 cm.\] To find the power of the lens, we must first convert the focal length to meters; then, we substitute this value into the equation for power. This gives \[P = \frac{1}{f} = \frac{1}{0.0800 m} = 12.5 D.\]

Discussion:

This is a relatively powerful lens. The power of a lens in diopters should not be confused with the familiar concept of power in watts. It is an unfortunate fact that the word “power” is used for two completely different concepts. If you examine a prescription for eyeglasses, you will note lens powers given in diopters. If you examine the label on a motor, you will note energy consumption rate given as a power in watts.

Definition: DIVERGING LENS

A lens that causes the light rays to bend away from its axis is called a diverging lens.

Ray Tracing and Thin Lenses

Ray tracing is the technique of determining or following (tracing) the paths that light rays take. For rays passing through matter, the law of refraction is used to trace the paths. Here we use ray tracing to help us understand the action of lenses in situations ranging from forming images on film to magnifying small print to correcting nearsightedness. While ray tracing for complicated lenses, such as those found in sophisticated cameras, may require computer techniques, there is a set of simple rules for tracing rays through thin lenses. A thin lens is defined to be one whose thickness allows rays to refract, as illustrated in Figure \(\PageIndex{1}\), but does not allow properties such as dispersion and aberrations. An ideal thin lens has two refracting surfaces but the lens is thin enough to assume that light rays bend only once. A thin symmetrical lens has two focal points, one on either side and both at the same distance from the lens (Figure \(\PageIndex{5}\)). Another important characteristic of a thin lens is that light rays through its center are deflected by a negligible amount, as seen in Figure \(\PageIndex{6}\).

Using paper, pencil, and a straight edge, ray tracing can accurately describe the operation of a lens. The Ruls for Ray Tracing for thin lenses are based on the illustrations already discussed:

Image Formation by Thin Lenses

In some circ*mstances, a lens forms an obvious image, such as when a movie projector casts an image onto a screen. In other cases, the image is less obvious. Where, for example, is the image formed by eyeglasses? We will use ray tracing for thin lenses to illustrate how they form images, and we will develop equations to describe the image formation quantitatively.

Consider an object some distance away from a converging lens, as shown in Figure \(\PageIndex{7}\). To find the location and size of the image formed, we trace the paths of selected light rays originating from one point on the object, in this case the top of the person’s head. The figure shows three rays from the top of the object that can be traced using the ray tracing rules given above. (Rays leave this point going in many directions, but we concentrate on only a few with paths that are easy to trace.) The first ray is one that enters the lens parallel to its axis and passes through the focal point on the other side (rule 1). The second ray passes through the center of the lens without changing direction (rule 3). The third ray passes through the nearer focal point on its way into the lens and leaves the lens parallel to its axis (rule 4). The three rays cross at the same point on the other side of the lens. The image of the top of the person’s head is located at this point. All rays that come from the same point on the top of the person’s head are refracted in such a way as to cross at the point shown. Rays from another point on the object, such as her belt buckle, will also cross at another common point, forming a complete image, as shown. Although three rays are traced in Figure \(\PageIndex{7}\), only two are necessary to locate the image. It is best to trace rays for which there are simple ray tracing rules. Before applying ray tracing to other situations, let us consider the example shown in Figure 7 in more detail.

The image formed in Figure \(\PageIndex{7}\) is a real image, meaning that it can be projected. That is, light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye, for example. Figure \(\PageIndex{8}\) shows how such an image would be projected onto film by a camera lens. This figure also shows how a real image is projected onto the retina by the lens of an eye. Note that the image is there whether it is projected onto a screen or not.

Several important distances appear in Figure 7. We define \(d_{0}\) to be the object of distance, the distance of an object from the center of a lens. Image distance \(d_{i}\) is defined to be the distance of the image from the center of a lens. The height of the object and height of the image are given the symbols \(h_{0}\) and \(h_{i}\), respectively. Images that appear upright relative to the object have heights that are positive and those that are inverted have negative heights. Using the rules of ray tracing and making a scale drawing with paper and pencil, like that in Figure 7, we can accurately describe the location and size of an image. But the real benefit of ray tracing is in visualizing how images are formed in a variety of situations. To obtain numerical information, we use a pair of equations that can be derived from a geometric analysis of ray tracing for thin lenses. The thin lens equations are

\[\frac{1}{d_{0}} + \frac{1}{d_{i}} = \frac{1}{f}\label{25.7.1}\]

and

\[\frac{h_{i}}{h_{0}} = - \frac{d_{i}}{d_{0}} = m.\label{25.7.2}\]

We define the ratio of image height to object height (\(h_{i}/h_{0}\)) to be the magnification \(m\). (The minus sign in the equation above will be discussed shortly.) The thin lens equations are broadly applicable to all situations involving thin lenses (and “thin” mirrors, as we will see later). We will explore many features of image formation in the following worked examples.

Example \(\PageIndex{2}\): Finding the Image of a Light Bulb Filament by Ray Tracing and by the Thin Lens Equations

A clear glass light bulb is placed 0.750 m from a convex lens having a 0.500 m focal length, as shown in the figure. Use ray tracing to get an approximate location for the image. Then use the thin lens equations to calculate

See Also

Is the human eye concave or convex?

1. the location of the image and
2. its magnification.

Verify that ray tracing and the thin lens equations produce consistent results.

Strategy and Concept

Since the object is placed farther away from a converging lens than the focal length of the lens, this situation is analogous to those illustrated in Figure 7 and Figure 8. Ray tracing to scale should produce similar results for \(d_{i}\). Numerical solutions for \(d_{i}\) and \(m\) can be obtained using the thin lens equations, noting that \(d_{o} = 0.750 m\) and \(f = 0.500 m\).

Solutions (Ray Tracing)

The ray tracing to scale in Figure 9 shows two rays from a point on the bulb’s filament crossing about 1.50 m on the far side of the lens. Thus the image distance \(d_{i}\) is about 1.50 m. Similarly, the image height based on ray tracing is greater than the object height by about a factor of 2, and the image is inverted. Thus \(m\) is about –2. The minus sign indicates that the image is inverted.

The thin lens equations can be used to find \(d_{i}\) from the given information: \[\frac{1}{d_{o}} + \frac{1}{d_{i}} = \frac{1}{f} . \] Rearranging to isolate \(d_{i}\) gives \[\frac{1}{d_{i}} = \frac{1}{f} - \frac{1}{d_{o}}.\] Entering known quantities gives a value for \(1/d_{i}\): \[\frac{1}{d_{i}} = \frac{1}{0.500 m} - \frac{1}{0.750m} = \frac{0.667}{m}.\] This must be inverted to find \(d_{i}\): \[d_{i} = \frac{m}{0.667} = 1.50m .\] Note that another way to find \(d_{i}\) is to rearrange equation: \[\frac{1}{d_{i}} = \frac{1}{f} - \frac{1}{d_{o}}.\] This yields the equation for the image distance as: \[d_{i} = \frac{fd_{o}}{d_{o} - f}.\label{25.7.3}\] Note that there is no inverting here.

The thin lens equations can be used to find the magnification \(m\), since both \(d_{i}\) and \(d_{o}\) are known. Entering their values gives

\[m = -\frac{d_{i}}{d_{o}} = -\frac{1.50 m}{0.750 m} = -2.00. \label{25.7.4}\]

Discussion

Note that the minus sign causes the magnification to be negative when the image is inverted. Ray tracing and the use of the thin lens equations produce consistent results. The thin lens equations give the most precise results, being limited only by the accuracy of the given information. Ray tracing is limited by the accuracy with which you can draw, but it is highly useful both conceptually and visually.

Real images, such as the one considered in the previous example, are formed by converging lenses whenever an object is farther from the lens than its focal length. This is true for movie projectors, cameras, and the eye. We shall refer to these as case 1 images. A case 1 image is formed when \(d_{o} \gt f\) and \(f\) is positive, as in Figure \(\PageIndex{10a}\). (A summary of the three cases or types of image formation appears at the end of this section.)

A different type of image is formed when an object, such as a person's face, is held close to a convex lens. The image is upright and larger than the object, as seen in Figure \(\PageIndex{10b}\), and so the lens is called a magnifier. If you slowly pull the magnifier away from the face, you will see that the magnification steadily increases until the image begins to blur. Pulling the magnifier even farther away produces an inverted image as seen in Figure \(\PageIndex{10a}\). The distance at which the image blurs, and beyond which it inverts, is the focal length of the lens. To use a convex lens as a magnifier, the object must be closer to the converging lens than its focal length. This is called a case 2 image. A case 2 image is formed when \(d_{o} \lt f\) and \(f\) is positive.

Figure \(\PageIndex{11}\) uses ray tracing to show how an image is formed when an object is held closer to a converging lens than its focal length. Rays coming from a common point on the object continue to diverge after passing through the lens, but all appear to originate from a point at the location of the image. The image is on the same side of the lens as the object and is farther away from the lens than the object. This image, like all case 2 images, cannot be projected and, hence, is called a virtual image. Light rays only appear to originate at a virtual image; they do not actually pass through that location in space. A screen placed at the location of a virtual image will receive only diffuse light from the object, not focused rays from the lens. Additionally, a screen placed on the opposite side of the lens will receive rays that are still diverging, and so no image will be projected on it. We can see the magnified image with our eyes, because the lens of the eye converges the rays into a real image projected on our retina. Finally, we note that a virtual image is upright and larger than the object, meaning that the magnification is positive and greater than 1.

A third type of image is formed by a diverging or concave lens. Try looking through eyeglasses meant to correct nearsightedness. (Figure \(\PageIndex{12}\)). You will see an image that is upright but smaller than the object. This means that the magnification is positive but less than 1. The ray diagram in Figure 13 shows that the image is on the same side of the lens as the object and, hence, cannot be projected—it is a virtual image. Note that the image is closer to the lens than the object. This is a case 3 image, formed for any object by a negative focal length or diverging lens.

The table summarizes the three types of images formed by single thin lenses. These are referred to as case 1, 2, and 3 images. Convex (converging) lenses can form either real or virtual images (cases 1 and 2, respectively), whereas concave (diverging) lenses can form only virtual images (always case 3). Real images are always inverted, but they can be either larger or smaller than the object. For example, a slide projector forms an image larger than the slide, whereas a camera makes an image smaller than the object being photographed. Virtual images are always upright and cannot be projected. Virtual images are larger than the object only in case 2, where a convex lens is used. The virtual image produced by a concave lens is always smaller than the object -- a case 3 image. We can see and photograph virtual images only by using an additional lens to form a real image.

In "Image Formation by Mirrors," we shall see that mirrors can form exactly the same types of images as lenses.

Summary

• Light rays entering a converging lens parallel to its axis cross one another at a single point on the opposite side.
• For a converging lens, the focal point is the point at which converging light rays cross; for a diverging lens, the focal point is the point from which diverging light rays appear to originate.
• The distance from the center of the lens to its focal point is called the focal length \(f\).
• Power \(P\) of a lens is defined to be the inverse of its focal length, \(P = \frac{1}{f}\).
• A lens that causes the light rays to bend away from its axis is called a diverging lens.
• Ray tracing is the technique of graphically determining the paths that light rays take.
• The image in which light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye is called a real image.
• Thin lens equations are \(\frac{1}{d_{o}} + \frac{1}{d_{1}} = \frac{1}{f}\) and \(\frac{h_{1}}{h_{o}} = m\) (magnification).
• The distance of the image from the center of the lens is called image distance.
• An image that is on the same side of the lens as the object and cannot be projected on a screen is called a virtual image.