The location of the image can also be calculated using an equation called the lens maker's equation. First we need to define the distance to the object as d_o, the distance to the image as d_i, and the focal length of the lens as f. The following image shows these distances labeled. Also, notice that in the image I have put labels for the height of the object h_o and the height of the image h_i. We will use these to define magnification.

The Lens Maker's Equation:

1 over f equals 1 over d sub i plus 1 over d sub o 1 f = 1 d_i + 1 d_o

Magnification Equation:

M equals h i sub over h sub o equals negative d sub i over d sub o M = h_i h_o = – d_i d_o

Example:
I put a 3 cm tall object 15 cm from a lens with focal length of 10 cm, find the location and height of the image.

Givens:

Check Your Answer

h_o = 3 cm
d_o = 15 cm
f = 10 cm
d_i = ?
h_i = ?

Equation:

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1 f = 1 d_i + 1 d_o
1 10 = 1 d_i + 1 15
3 30 = 1 d_i + 2 30 ← the least common denominator here would be 30
1 30 = 1 d_i
d_i = 30 cm

Answer:

Check Your Answer

The image is 30 cm from the lens.

Now for the height of the image:

Check Your Answer

M = - di do	M = hi ho
M = - 30 15	-2 = hi 3
M = -2	hi = -6 cm

Answer:

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The image is 6 cm tall and inverted.