**Lens Math**

Learn how to use equations with lens math to determine information about the image like height and distance from the lens.

**Lens Math Variables**

Name |
Variable |
Unit |
Unit Abbreviation |

focal length |
f |
centimeters or meters |
cm or m |

distance to image |
d_{i} |
centimeters or meters |
cm or m |

distance to object |
d_{o} |
centimeters or meters |
cm or m |

magnification |
M |
times |
x |

height of image |
h_{i} |
centimeters or meters |
cm or m |

height of object |
h_{o} |
centimeters or meters |
cm or m |

**Virtual and Real Sides of a Lens**

Light travels through a lens unlike a mirror which reflects. Light travels to the real side given a positive (+) which is on the opposite side of the object. If the image appears on the real side it will be given a positive value. If the image appears on the virtual side, side of the object, it is virtual and given a negative symbol.

**Lens Math Facts**

**For all lenses:****d**no matter what, an object is always real_{o}is always positive +

**For a concave lens:****(only produce, virtual, upright, reduced images)****f is always negative (-)**- concave lens focal distances will be negative during math since the curve of the front side is toward the virtual side.

**d**_{i}is always negative (-)- because concave lenses only produce virtual upright images

**For a convex lens:****f is always positive (+)**- convex lens focal distances will be positive since the curve of the front side is toward the real side.
**d**. You will determine this by the solution for_{i}can be positive or negative**d**._{i}- If
**d**is a positive in your solution, the image is real and inverted._{i } - If
**d**is a negative in your solution, the image is virtual and upright._{i }

- If

**Lens Equation** **Tips**

**d _{o}**

- First of all the object is always real and
**d**will always be positive, never make this negative and if a solution is negative, you did something wrong._{o }

**d _{i}**

- If a problem says you have a virtual image, make the
**d**negative. If a problem says that the image appears on the same side as the object, make the_{i}**d**negative._{i} - A real image is always inverted, a virtual image is always upright.

**f**

- If you have a concave lens the f will always be negative
- If you have a convex lens the f will always be positive

This equation can be used three ways

**(h**_{i /}**h**_{o }**) =****(d**_{i /}**-d**_{o }**)****M=****h**_{i /}**h**_{o}**M= -d**_{i /}**d**_{o}

**M**

- you can solve for
**M**with**h**and_{i }**h**_{o}**M=****h**_{i /}**h**_{o}

- you can solve for
**M**with**d**and_{i }**do****M= -d**_{i /}**d**_{o}

- If
**M is 1**the image is not magnified - If
**M is less than 1**the image is reduced - If
**M is greater than 1**the image is enlarged - If
**M is negative**the image is real and inverted - If
**M is positive**the image is virtual and upright

**Example Problems**

Use the video provided for extra support through these questions.

**1. An object is 5.00 meters away from a convex lens, which produces a real image 1.00 meters away. What is the focal length of the lens?
**

**2. An object is 5.00 meters away from a convex lens, which produces a real image 1.00 meters away. What is the magnification of the image?
**

**3. An object is 6.0 centimeters away from a convex lens, which produces a virtual image 7.0 centimeters away. What is the focal length of the lens?
**

**4. A 2 cm tall object is 6.0 centimeters away from a convex lens, which produces a virtual image 7.0 centimeters away. What is the height of the image?
**

**5. An 8 cm tall object is 12.0 centimeters away from a concave lens, which produces a virtual image 7.0 centimeters away. What is the focal length of the lens?
**

**6. An 8 cm tall object is 12.0 centimeters away from a concave lens, which produces a virtual image 7.0 centimeters away. What is the height of the image?
**

**7. Find the image distance for a convex lens with an object distance of 15 cm and a focal length of 30 cm.
**

**8. Given d _{i }= -18 cm, d_{o}= 6 cm, and M=3, describe the image.**

**9. Find the focal length for a convex lens with an object distance of 60 cm and an real image distance of 15 cm.
**

**10. Find the distance an object is away from a convex lens with an real image formed 60 cm away and a focal length of 20 cm.
**

**PhET Lens Lab**

Return to the previous section by clicking here to complete a PhET lens lab that includes lens math.

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