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# Lens

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Students will learn how light behaves when passing through converging and diverging lenses. Students will also learn how to do ray tracing diagrams and calculate image distances and magnification using the lens' maker's equation.

### Key Equations

$\frac{1}{f} = \frac{1}{d_o} +\frac{1}{d_i} \;$ ; The len's maker's equation

Where f is the focal length of the lens, $d_o$ is the distance of the object from the lens and $d_i$ is the distance the image is formed from the lens.

$M = \frac{-d_i}{d_0}$

The size of an object’s image is larger (or smaller) than the object itself by its magnification, $M$ . The level of magnification is proportional to the ratio of $d_i$ and $d_o$ . An image that is double the size of the object would have magnification $M = 2$ .

### Guidance

• For lenses, the distance from the center of the lens to the focus is $f$ . Focal lengths for foci behind the lens are positive in sign. The distance from the center of the lens to the object in question is $d_0$ , where distances to the left of the lens are positive in sign. The distance from the center of the lens to the image is $d_i$ . This number is positive for real images (formed to the right of the lens), and negative for virtual images (formed to the left of the lens).
• Lenses , made from curved pieces of glass, focus or de-focus light as it passes through. Lenses that focus light are called converging lenses, and these are the ones used to make telescopes and cameras. Lenses that de-focus light are called diverging lenses.
• Lenses can be used to make visual representations, called images .
• The focal length , $f$ , of a lens or mirror is the distance from the surface of the lens to the place where the light is focused. This is called the focal point or focus . For diverging lenses, the focal length is negative.
• For converging lens, one can find the focal point by simply holding a piece of paper near the lens until a distant image is formed. The distance from the paper to the lens is the focal point.

• When light rays converge behind a lens, a real image is formed. Real images are useful in that you can place photographic film at the physical location of the real image, expose the film to the light, and make a two-dimensional representation of the world, a photograph.

• When light rays diverge behind a lens, a virtual image is formed. A virtual image is a manifestation of your brain (it traces the diverging rays backwards and forms an image), like the person you see “behind” a mirror’s surface when you brush your teeth (there's obviously no real light focused behind a mirror!). Since virtual images aren’t actually “anywhere,” you can’t place photographic film anywhere to capture them.
• Real images are upside-down, or inverted . You can make a real image of an object by putting it farther from a mirror or lens than the focal length. Virtual images are typically right-side-up. You can make virtual images by moving the lens closer to the object than the focal length.

In ray tracing problems, you will do a careful ray tracing with a ruler (including the extrapolation of rays for virtual images). It is best if you can use different colors for the three different ray tracings. When sketching diverging rays, you should use dotted lines for the extrapolated lines in front of a lens in order to produce the virtual image. When comparing measured distances and heights to calculated distances and heights, values within $10$ % are considered “good.” Use the Table ( below ) as your guide.

Mirror type Ray tracings

Converging lenses

( convex )

Ray #1: Leaves tip, travels parallel to optic axis, refracts and travels through to the focus.

Ray #2: Leaves tip, travels through focus on same side, travels through lens, and exits lens parallel to optic axis on opposite side.

Ray #3: Leaves tip, passes straight through center of lens and exits without bending.

Diverging lenses

( concave )

Ray #1: Leaves tip, travels parallel to optic axis, refracts OUTWARD by lining up with focus on the SAME side as the candle.

Ray #2: Leaves tip, heads toward the focus on the OPPOSITE side, and emerges parallel from the lens.

Ray #3: Leaves tip, passes straight through the center of lens and exits without bending.

#### Example 1

You have a converging lens of focal length 2 units. If you place an object 5 units away from the lens, (a) draw a ray diagram of the situation to estimate where the image will be and (b) list the charactatistics of the image. Finally (c) calculate the position of the image. A diagram of the situation is shown below.

##### Solution

(a): To draw the ray diagram, we'll follow the steps laid out above for converging lenses.

First we draw the a ray that travels parallel to the principle axis and refracts through the focus on the other side (the red ray). Next we draw a ray through the focus on the same side that refracts out parallel (the green ray). Finally we draw the ray that travels straight through the center of the lens without refracting (the blue ray). The result is shown below.

(b): Based on the ray diagram and the initial position of the object, we know that the image is a real, inverted, and smaller than the original object.

(c): To calculate the exact position of the object, we can use the lens maker's equation.

$\frac{1}{f}&=\frac{1}{d_o} + \frac{1}{d_i}\\\frac{1}{d_i}&=\frac{1}{f} - \frac{1}{d_o}\\\frac{1}{d_i}&=\frac{1}{2\;\text{units}} - \frac{1}{5\;\text{units}}\\\frac{1}{d_i}&=\frac{3}{10\;\text{units}}\\d_i&=3.33\;\text{units}\\$

### Simulation

Note: this simulation only shows the effects of a convex lens

### Time for Practice

1. Consider a converging lens with a focal length equal to three units and an object placed outside the focal point.
1. Carefully trace three rays coming off the top of the object and form the image.
2. Measure $d_o$ and $d_i$ .
3. Use the mirror/lens equation to calculate $d_i$ .
4. Find the percent difference between your measured $d_i$ and your calculated $d_i$ .
5. Measure the magnification $M$ and compare it to the calculated magnification.
2. Consider a converging lens with a focal length equal to three units, but this time with the object placed inside the focal point.
1. Carefully trace three rays coming off the top of the object and form the image.
2. Measure $d_o$ and $d_i$ .
3. Use the mirror/lens equation to calculate $d_i$ .
4. Find the percent difference between your measured $d_i$ and your calculated $d_i$ .
5. Measure the magnification $M$ and compare it to the calculated magnification.
3. Consider a diverging lens with a focal length equal to four units.
1. Carefully trace three rays coming off the top of the object and show where they converge to form the image.
2. Measure $d_o$ and $d_i$ .
3. Use the mirror/lens equation to calculate $d_i$ .
4. Find the percent difference between your measured $d_i$ and your calculated $d_i$ .
5. Measure the magnification $M$ and compare it to the calculated magnification.
4. A piece of transparent goo falls on your paper. You notice that the letters on your page appear smaller than they really are. Is the goo acting as a converging lens or a diverging lens? Explain. Is the image you see real or virtual? Explain.
5. An object is placed $30 \;\mathrm{mm}$ in front of a lens. An image of the object is located $90 \;\mathrm{mm}$ behind the lens.
1. Is the lens converging or diverging? Explain your reasoning.
2. What is the focal length of the lens?
6. Little Red Riding Hood ( aka $R-$ Hood) gets to her grandmother’s house only to find the Big Bad Wolf ( aka BBW) in her place. $R-$ Hood notices that BBW is wearing her grandmother’s glasses and it makes the wolf’s eyes look magnified (bigger).
1. Are these glasses for near-sighted or far-sighted people? For full credit, explain your answer thoroughly. You may need to consult some resources online.
2. Create a diagram of how these glasses correct a person’s vision.

1. c. $3$ units e. $- 2/3$
2. c. -6 (so 6 units on left side) e. 3 times bigger
3. c. $-2.54$ units (2.54 units on the left) e. $M=.36$
4. .
5. b. $22.5 \;\mathrm{mm}$
6. .