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Derive the following.
Derive R = 2f for a spherical mirror, where the symbols have their usual meaning.
Derive $R = 2f$ for a spherical mirror, where the symbols have their usual meaning.
To show: $R = 2f$
Answer
Answer
R = 2f for a concave mirror (small aperture).
- Let a ray MN parallel to the principal axis strike a concave mirror of pole P and centre of curvature C at M; after reflection it passes through the focus F. CM is the normal, so the angle of incidence is $\angle CMN = i$.
- Since MN is parallel to the axis, $\angle MCP = i$ (alternate angles) and by the law of reflection $\angle CMF = i$.
- In triangle CMF, $\angle FCM = \angle FMC = i$, so $FC = FM$.
- For a small aperture, M is close to P, so $FM \approx FP = f$.
- Then $CP = CF + FP = f + f = 2f$.
- As $CP = R$, $R = 2f$.