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F = R/2

  1. Apr 5, 2009 #1
    Prove how focus of a spherical mirror = Radius of curvature/2
     
  2. jcsd
  3. Apr 5, 2009 #2
    The quickest way (only approximate) is to use thin lens formulas:
    Use fo = object distance; fi = image distance; and ffl = focal length.
    So 1/fo + 1/fi = 1/ffl [basic thin lens formula]
    If both fo and fi = R (radius of curvature)
    then 1/R + 1/R = 1/ffl = 2/R
    So ffl = R/2
     
  4. Apr 5, 2009 #3
    but what if the object distance is not equal to image distance?
     
  5. Apr 6, 2009 #4

    jtbell

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    Staff: Mentor

    Does this help?

    Mirror Equation (in particular the bottom section)

    (I found this by a Google search for "focal length of a spherical mirror".)
     
  6. Apr 6, 2009 #5
    We know that if the object is at the center of radius of curvature R, then the image is also at the center of radius of curvature. So fo = fi = R

    Using the thin lens formula 1/fo + 1/fi = 1/ffl
    Substituting R we get
    1/R + 1/R = 1/ffl
    So 2/R = 1/ffl
    Now suppose the object is at infinity
    1/inf. + 1/fi = 1/ffl = 2/R
    then fi = R/2
     
    Last edited: Apr 6, 2009
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