Question

Some telephoto cameras use a mirror rather than a lens. What radius of curvature mirror is needed to replace a$\mathbf{800}\mathbf{}\mathbf{mm}$focal length telephoto lens?

Short answer

The radius of curvature of the mirror is$\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{m}$

Step-by-step solution

Definition of the radius of curvature

The radius of the curvature lens is double the focal length of the lens.

Calculating radius of curvature

Given values:

$\mathit{f}\mathbf{=}\mathbf{800}\mathbf{}\mathbf{mm}$

First, we have to convert the unit into an SI unit:

$$\begin{gathered} \mathit{f}\mathbf{=}\mathbf{800}\mathbf{.}\left(1\mathrm{mm}\times \frac{1\times {10}^{‐3}m}{1\mathrm{mm}}\right) \\ \mathbf{=}\mathbf{800}\mathbf{\times }{\mathbf{10}}^{\mathbf{‐}\mathbf{3}}\mathbf{}\mathbf{m} \end{gathered}$$

The radius of curvature of the mirror can be written as:

$\mathit{R}\mathbf{=}\mathbf{2}\mathit{f}$

Substituting the value of focal length in the above equation, we get,

role="math" localid="1654076860828" $$\begin{gathered} \mathit{R}\mathbf{=}\mathbf{2}\mathbf{\times }\mathbf{800}\mathbf{\times }{\mathbf{10}}^{\mathbf{‐}\mathbf{3}}\mathbf{}\mathbf{m} \\ \mathbf{=}\mathbf{1600}\mathbf{\times }{\mathbf{10}}^{\mathbf{‐}\mathbf{3}}\mathbf{}\mathbf{m} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{m} \end{gathered}$$

Hence, the radius of curvature of the mirror is 1.6m