Reflection of Light
4.0 Spherical mirror formulae
4.0 Spherical mirror formulae
For concave mirror
$CB$ is normal to the mirror at $B$.
From the laws of reflection i.e. angle of incidence is equal to the angle of reflection. So, $$\angle OBC = \angle IBC = \theta $$
In $\Delta OBC$, $$\alpha + \theta = \beta \quad ...(i)$$
In $\Delta IBC$, $$\beta + \theta = \gamma \quad ...(ii)$$
From equation $(i)$ and $(ii)$ we get, $$\alpha + \gamma = 2\beta \quad ...(iii)$$
Due to paraxial approximation,
\[\left. \begin{gathered} \alpha \approx \tan \alpha \hspace{1em} \\ \beta \approx \tan \beta \hspace{1em} \\ \gamma \approx \tan \gamma \hspace{1em} \\ P' \approx P \hspace{1em} \\ \end{gathered} \right\}\quad ...(iv)\]
From equation $(iii)$ and $(iv)$ we get, $$\tan \alpha + \tan \gamma = 2\tan \beta $$ or $$\frac{{BP'}}{{OP'}} + \frac{{BP'}}{{IP'}} = 2\left( {\frac{{BP'}}{{CP'}}} \right)\quad ...(v)$$
Applying sign convention,
$$\begin{equation} \begin{aligned} OP' = - u \\ IP' = - v \\ CP' = - R \\\end{aligned} \end{equation} $$
So, $$\begin{equation} \begin{aligned} \frac{{BP'}}{{ - u}} + \frac{{BP'}}{{ - v}} = 2\left( {\frac{{BP'}}{{ - R}}} \right) \\ \frac{1}{v} + \frac{1}{u} = \frac{2}{R} \\\end{aligned} \end{equation} $$ As $f = \frac{R}{2}$. So, $$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$
For convex mirror
In $\Delta CMO$, $$\alpha + \beta = \theta \quad ...(i)$$
In $\Delta IMO$, $$\alpha + \gamma = 2\theta \quad ...(ii)$$
From equation $(i)$ and $(ii)$ we get, $$\alpha + 2\beta = \gamma $$ or $$\gamma - \alpha = 2\beta \quad ...(iii)$$
Due to paraxial approximation,
\[\left. \begin{gathered} \alpha \approx \tan \alpha \hspace{1em} \\ \beta \approx \tan \beta \hspace{1em} \\ \gamma \approx \tan \gamma \hspace{1em} \\ \end{gathered} \right\}\quad ...(iv)\]
From equation $(iii)$ and $(iv)$ we get, $$\tan \gamma - \tan \alpha = 2\tan \beta $$ or $$\frac{{MP}}{{IP}} - \frac{{MP}}{{OP}} = 2\left( {\frac{{MP}}{{CP}}} \right)\quad ...(v)$$
Applying sign convention,
$$\begin{equation} \begin{aligned} PI = + v \\ OP = - u \\ PC = + R \\\end{aligned} \end{equation} $$
So, $$\begin{equation} \begin{aligned} \frac{{MP}}{v} - \frac{{MP}}{{ - u}} = 2\left( {\frac{{MP}}{R}} \right) \\ \frac{1}{v} + \frac{1}{u} = \frac{2}{R} \\\end{aligned} \end{equation} $$ As $f = \frac{R}{2}$. So, $$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$
Note:
- The focal length of a spherical mirror of radius $R$ is negative for concave mirror or converging mirror.
- The focal length of a spherical mirror of radius $R$ is positive for convex mirror or diverging mirror.
