## 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 di centimeters or meters cm or m distance to object do centimeters or meters cm or m magnification M times x height of image hi centimeters or meters cm or m height of object ho 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:
• do is always positive + no matter what, an object is always real
• 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.
• di 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.
• di can be positive or negative.  You will determine this by the solution for di.
• If di is a positive in your solution, the image is real and inverted.
• If di is a negative in your solution, the image is virtual and upright.

### Lens EquationTips

do

• First of all the object is always real and do will always be positive, never make this negative and if a solution is negative, you did something wrong.

di

• If a problem says you have a virtual image, make the di negative.  If a problem says that the image appears on the same side as the object, make the di negative.
• 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

• (hi / h) = (di / -d)
• M= hi / ho
• M= -di / do

M

• you can solve for M with hi and ho
• M= hi / ho
• you can solve for M with di and do
• M= -di / do
• 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 di = -18 cm, do= 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.