Question

(a) Calculate the focal length of the mirror formed by the shiny back of a spoon that has a \(3\;cm\)radius of curvature.(b) What is its power in diopters?

Short answer

(a)Focal length of the mirror is\( - 1.50\;cm\).

(b)Power in diopter is \( - 66.7D\).

Step-by-step solution

Concept Introduction

The curvature's reciprocal,\(R\), is the radius of curvature in differential geometry. It is equal to the radius of the circular arc that, at that moment, most closely resembles the curve for a curve. The radius of curvature for surfaces is the diameter of a circle that, individually or in combination, best fits a normal section.

Information Provided

• The radius of curvature of spoon: \(3{\rm{ }}cm\)

Calculating focal length from radius

The focal length is related to the radius of curvature by the relation:

\(f = \dfrac{R}{2}\)

Where\(f\)represents focal length, and\(R\)represents radius of curvature.

The focal length has to have a minus sign since the image is formed by the "back" of the spoon.

\(\begin{aligned}f &= \dfrac{R}{2}\\ &= \dfrac{{ - 3.00\;cm}}{2}\\ &= - 1.50\;cm\end{aligned}\)

Hence, the focal length is \( - 1.50\;cm\).

Calculating the power of the mirror from the derived focal length

(b) The focal length is inversely proportional to the power.

That is,\(f = \dfrac{1}{P}\).

Where\(f\)represents focal length, and\(P\)represents power.

\(\begin{aligned}P &= \dfrac{1}{f}\\ &= \dfrac{1}{{ - 1.50 \times {{10}^{ - 2}}\;m}}\\ &= - 66.7D\end{aligned}\)

Therefore, power in dioptersis\( - 66.7D\).