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Wize University Physics Textbook (Master) > Geometric Optics

Lensmaker's Equation

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Lensmaker's Equation


If each curved surface of the lens has a different radius of curvature, then the focal length can be found using the lensmaker's equation:

 1f=(n−1)(1R1−1R2) \boxed{ \ \dfrac{1}{f}=\big(n-1)\bigg(\dfrac{1}{R}_1-\dfrac{1}{R}_2\bigg) \ }
  • ff is the focal length of the lens
  • nn is the index of refraction of the material
  • R1R_1 is the radius of curvature of the side next to the object
  • R2R_2 is the radius of curvature of the side opposite to the object













→\to The focal length ff is positive for converging lenses and negative for diverging lenses.


→\to Each radius of curvature RR is positive if the surface is convex towards the object, and negative if the surface is concave towards the object.




Wize Concept
The radius of curvature is infinite for a plane mirror, which means that 1R=0\dfrac{1}{R}=0.

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Example: Exotic Material


A new exotic material was discovered to potentially replace glass. If this material is used instead of glass to make a lens with the same radii of curvature, how will the focal length of the lens change? (The index of refraction of glass is 1.51.5, and that of the new material is 1.21.2)


Let's write the lensmaker equation for each situation.

1f=(n−1)(1R1+1R2)\dfrac{1}{f} = (n-1)\Big(\dfrac{1}{R_1}+\dfrac{1}{R_2}\Big)


For glass we have:

1fglass=(1.5−1)(1R1+1R2)\dfrac{1}{f_{glass}} = (1.5-1)\Big(\dfrac{1}{R_1}+\dfrac{1}{R_2}\Big)


1fglass=0.5 (1R1+1R2)\dfrac{1}{f_{glass}} = 0.5\ \Big(\dfrac{1}{R_1}+\dfrac{1}{R_2}\Big)


For the new material we have:

1fnew=(1.2−1)(1R1+1R2)\dfrac{1}{f_{new}} = (1.2-1)\Big(\dfrac{1}{R_1}+\dfrac{1}{R_2}\Big)

1fnew=0.2 (1R1+1R2)\dfrac{1}{f_{new}} = 0.2\ \Big(\dfrac{1}{R_1}+\dfrac{1}{R_2}\Big)


Divide the two equations to get:

1fglass1fnew=0.50.2=2.5\dfrac{\dfrac{1}{f_{glass}}}{\dfrac{1}{f_{new}}} = \dfrac{0.5}{0.2}=2.5 (the R part cancels, since the R's are the same)


Rearrange:

fnewfglass=2.5   →   fnew=2.5 fglass{\dfrac{f_{new}}{f_{glass}}} = 2.5 \ \ \ \to \ \ \ f_{new}=2.5\ f_{glass}

Therefore the focal length with the new material will be 2.52.5 times larger than when glass is used.


Practice: One Side Flat


You want to make a divergent lens with a focal length of 11 m using glass (n=1.33n = 1.33). If one side of the lens is completely flat, what should the radius of curvature be for the other side?