Relationship between the focal length and radius of curvature of a spherical mirror

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Fig. 1. ( a ) concave mirror ( b ) convex mirror.

Let C be the centre of curvature of the mirror.

Consider a ray parallel to the principal axis strinking the mirror at M, CM will be perpendicular to the mirror at M.

Let θ be the angle of incidence and MQ be the perpendicular from M on the principal axis, then

∠MCP = θ and ∠MFP =2θ


tanθ =MQ/CQ and tan2θ = MQ/FQ

for small θ, which is true for paraxial rays tanθ is aprox equal to θ and tan2θ is aprox equal to 2θ

Therefore from the above equation.


or FQ = CQ/2

Now, for small θ the point Q is very close to the point P therefore

FQ = f, CQ = R

Then from equation FQ = CQ/2

f = R/2

where f is focal length and R is radius of curvature.

Thus, In spherical mirror ( both concave and convex) focal length is equal to half of the radius of curvature.

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