7.2 Mirrors

Again we use the small angle approximation. By inspection of the figure we see that $$2{\theta }_i=\alpha +\beta$$ and $${\theta }_i=\alpha +\gamma \text{.}$$ Now we multiply the second equation by two and subtract the first equation from itand we get: $$2\alpha +2\gamma -\alpha -\beta =0$$ or $$\alpha -\beta =-2\gamma \text{.}$$ Using the small angle approximation we see that this is $$\frac{h}{s_o}+\frac{h}{s_i}=\frac{-2h}{r}$$ where I have used the fact that $s_i$ is negative to the right of the mirror. So I can write the mirror equation as $$\frac{1}{s_o}+\frac{1}{s_i}=\frac{-2}{R}$$ or $$\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f}$$ where for a mirror $1/f=-2/R$