Question

You note that your prescription for new eyeglasses is $\mathbf{‐}\mathbf{4}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{D}$. What will their focal length be?

Short answer

The focal length of eyeglasses having power $\mathbf{‐}\mathbf{4}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{D}$ is $\mathit{f}\mathbf{=}\mathbf{‐}\mathbf{222}\mathbf{}\mathbf{mm}$ where the negative sign indicates that the lens is a concave lens.

Step-by-step solution

Concept Introduction

The focal length is the distance between a convex lens or a concave mirror and the focal point of a lens or mirror. It is the point at which two parallel light beams meet or converge. Depending on the nature of the lens and mirror (concave or convex), the focal length varies with the sign (positive or negative).

Information Provided

• Power of new eyeglasses:$\mathbf{‐}\mathbf{4}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{D}$

Focal length of glasses

The power in dioptres of lenses is inversely proportional to the focal length of the; where the power is given by,

$\mathit{P}\mathbf{=}\frac{\mathbf{1}}{\mathbf{f}}$

Hence, knowing the power of the reading glasses, then directly substitute in the above equation to find the focal length of the lens. Remember that since the power is given in the unit of dioptre, then substituting in the above equation, the obtained unit for the focal length will be in the unit of meter.

Hence, on substituting the value –

$$\begin{gathered} \mathit{f}\mathbf{=}\frac{\mathbf{1}}{\mathbf{P}} \\ \mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{D}} \\ \mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{222}\mathbf{}\mathbf{m} \\ \mathbf{}\mathbf{=}\mathbf{‐}\mathbf{0}\mathbf{.}\mathbf{222}\mathbf{\times }\mathbf{1000}\mathbf{}\mathbf{mm} \\ =‐222\mathrm{mm} \end{gathered}$$

Where the negative sign means that the lens is concave, a “diverging lens”.

Therefore, the value of focal length is $\mathit{f}\mathbf{=}\mathbf{‐}\mathbf{222}\mathbf{}\mathbf{mm}$