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What are some ways of finding the center of curvature of concave and convex mirrors (spherical)?

For example, I can double the distance of focal length on the principle axis to find the center of curvature of concave mirror.

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Yes, the radius of curvature is twice the focal length from the pole (A in upper diagram, P in lower diagram) : $R=2f$.

  • You can obtain an approximate value of $f$ by finding the point at which a distant object is focussed, as suggested by the 2nd diagram. Or you can obtain $R$ as the distance at which an object is focussed at the same distance (inverted).
  • You can get a more accurate value of $f$ by plotting a graph of image vs object distances, in the form $1/v$ against $1/u$. The intercept is $1/f$.
  • You can use a special tool such as a spherometer to find radius of curvature directly.
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A concave mirror used for focusing light is parabolic, not spherical. There therefore isn't a "center of curvature".

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If you know the object distance and image distance, you can solve for the focal length f using the mirror equation. Then you will also know the center of curvature since it is 2 times f.

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