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Convex Mirrors


basic diagram

ray diagrams

proof r = 2f




Basic Ray Diagram



basic ray diagram for a convex mirror



The basic ray diagram for a convex mirror introduces a number of important terms:



aperture - the diameter of the circular mirror


pole - where the principal axis meets the mirror surface


centre of curvature - the centre of the sphere that the mirror forms part


radius of curvature (r) - radius of the sphere


principal axis - the line through the centre of curvature and the pole of the mirror


focal length (f) - equal to half the radius of curvature f = r/2




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Ray Diagrams


Ray diagrams are constructed by taking the path of three distinct rays from a point on the object:



X) a ray parallel to the principal axis reflected through F (the principal focus)


Y) a ray passing through C which is then reflected back along its original path


Z) a ray passing through F, which is then reflected parallel to the principal axis



note - the convex mirror is considered to be so thin as to be represented by a vertical line



construction rays for a convex mirror




For all the object positions listed below,



object between f and the mirror

object at f

object between f and 2f

object at 2f

object at infinity



. . . the ray diagrams are virtually the same as in the diagram above. Hence the result is the same. The image produced is virtual, upright and diminished.



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Proof of r = 2f



proof of r equals  2f for convex mirror



The angles in the triangle(CFX) are vertically opposite and corresponding angles from the angles of reflection and incidence.


From the diagram it is apparent that XF does not equal PF.


However, when point X is closer to the principal axis, XF approaches the size of PF.


So for rays close to the principal axis, we can say that XF = PF.


In other words,


XC = PF + FC


r = f + f


r = 2f





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