How Can R=2f Be Proven in Optics?

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SUMMARY

The relationship R=2f, where R represents the radius of curvature and f denotes the focal length, is established through the analysis of the parabola defined by the equation y=1/(4f)x². The focus of this parabola is located at (0, f), confirming that the focal length is indeed f. The curvature of the function is calculated using the formula |y"| / (1+(y')²)^(3/2), where y' and y" are derived as 1/(2f)x and 1/(2f), respectively. At x=0, the curvature simplifies to 1/(2f), leading to the conclusion that the radius of curvature is 2f.

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  • Understanding of parabolic equations and their properties
  • Familiarity with calculus, specifically derivatives and curvature
  • Knowledge of focal points in conic sections
  • Basic grasp of optical principles related to lenses and mirrors
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Hi... Do you know how to prove R=2f where R is the radius of curvature and f is the focal length?
 
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The parabola [itex]y= \frac{1}{4f}x^2[/itex] has focus at (0, f) so focal length f. The curvature of a function y= y(x) is given by
[tex]\frac {|y"|} {(1+(y')^2)^{ \frac {3}{2}}}[/tex]

In this case, y'= (1/(2f)x and y"= 1/(2f). At x= 0, the curvature is
1/(2f) and so the radius of curvature is 2f.
 
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Sorry, I can't see the image of the graph. Can you please send it again? thank you very much!
 

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