How to Calculate Magnification

Note: A converging lens is wider in the middle than it is at the edges (like a magnifying glass.) A diverging lens is wider at the edges than it is in the middle (like a bowl). ^[1] Finding magnification is the same for both, with one important exception . Click here to go straight to the diverging lens exception.

1. 1
   Start with your equation and determine which variables you know. ^[2] Like with many other physics problems, a good way to approach magnification problems is to first write the equation you need to find your answer. From here, you can work backwards to find any pieces of the equation that you need. ^[3]

   • For example, let's say that a 6 centimeter tall action figure is placed half a meter away from a converging lens with a focal length of 20 centimeters. If we want to find the magnification , image size , and image distance , we can start by writing our equation like this:

     M = (h _i /h _o ) = -(d _i /d _o )

   • Right now, we know h _o (the height of the action figure) and d _o (the distance of the action figure from the lens.) We also know the focal length of the lens, which isn't in this equation. We need to find h _i , d _i , and M .
2. 2
   Use the lens equation to get d _i . If you know the distance of the object you're magnifying from the lens and the focal length of the lens, finding the distance of the image is easy with the lens equation. The lens equation is 1/f = 1/d _o + 1/d _i , where f = the focal length of the lens. ^[4]

   • In our example problem, we can use the lens equation to find d _i . Plug in your values for f and d _o and solve:

     1/f = 1/d _o + 1/d _i

     1/20 = 1/50 + 1/d _i

     5/100 - 2/100 = 1/d _i

     3/100 = 1/d _i

     100/3 = d _i = 33.3 centimeters

   • A lens's focal length is the distance from the center of the lens to the point where the rays of light converge in a focal point. If you've ever focused light through a magnifying glass to burn ants, you've seen this. In academic problems, this is often given to you. In real life, you can sometimes find this information labeled on the lens itself.
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3. 3
   Solve for h _i . Once you know d _o and d _i , you can find the height of the magnified image and the magnification of the lens. Notice the two equals signs in the magnification equation (M = (h _i /h _o ) = -(d _i /d _o )) — this means that all of the terms are equal to each other, so we can find M and h _i in whatever order we want. ^[5]

   • For our example problem, we can find h _i like this:

     (h _i /h _o ) = -(d _i /d _o )

     (h _i /6) = -(33.3/50)

     h _i = -(33.3/50) × 6

     h _i = -3.996 cm

   • Note that a negative height indicates that the image we see will be inverted (upside down).
4. 4
   Solve for M. You can solve for your final variable using either -(d _i /d _o ) or (h _i /h _o ). ^[6]

   • In our example, we would finally find M like this:

     M = (h _i /h _o )

     M = (-3.996/6) = -0.666

   • We also get the same answer if we use our d values:

     M = -(d _i /d _o )

     M = -(33.3/50) = -0.666

   • Note that magnification does not have a unit label.
5. 5
   Interpret your M value. Once you have a magnification value, you can predict several things about the image you would view through the lens. These are:

   • Its size. The bigger the absolute value of the M value, the bigger the object will seem under magnification. M values between 1 and 0 indicate that the object will look smaller.
   • Its orientation. Negative values indicate that the image of the object will be inverted.
   • In our example, our M value of -0.666 means that, under the conditions given, the image of the action figure will appear upside down and two-thirds its normal size.
6. 6
   For diverging lenses, use a negative focal length value. Even though diverging lenses look very different than converging lenses, you can find their magnification values using the same formulas as above. The one important exception here is that divergent lenses will have negative focal lengths. In a problem like the one above, this will affect the answer you get for d _i , so be sure to pay close attention. ^[7]

   • Let's re-do the example problem above, only this time, we'll say we're using a diverging lens with a focal length of -20 centimeters. All of the other starting values are the same.
   • First, we'll find d _i with the lens equation:

     1/f = 1/d _o + 1/d _i

     1/-20 = 1/50 + 1/d _i

     -5/100 - 2/100 = 1/d _i

     -7/100 = 1/d _i

     -100/7 = d _i = -14.29 centimeters

   • Now we'll find h _i and M with our new d _i value.

     (h _i /h _o ) = -(d _i /d _o )

     (h _i /6) = -(-14.29/50)

     h _i = -(-14.29/50) × 6

     h _i = 1.71 centimeters

     M = (h _i /h _o )

     M = (1.71/6) = 0.285
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