Question

A convex mirror of radius of curvature $20cm$ forms an image which is half the size of the object. Determine the position of the object from the pole of the mirror.

Answer 1

Step 1: Given data:

$radiusofcurvature\left(R\right)=20cm$ and the size of the image is half the size of the object

$\Rightarrow h_i=\frac{1}{2}{h}_o$

Step 2: Formula used:

1. $Magnification,m=\frac{h_i}{h_o}=-\frac{v}{u},wherev=positionofimageandu=positionofobject$
2. $focallength,f=\frac{R}{2}$
3. $Mirrorformula,\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

Step 3: Finding the object distance:

$$\begin{gathered} Magnification,m=\frac{h_i}{h_o}=-\frac{v}{u} \\ As,h_i=\frac{1}{2}{h}_o \\ \Rightarrow \frac{h_i}{h_o}=\frac{1}{2} \\ \Rightarrow -\frac{v}{u}=\frac{1}{2} \\ So,v=-\frac{u}{2} \end{gathered}$$

Now, finding the focal length,

$f=\frac{R}{2}=\frac{20}{2}=10cm$

finding the position of object $\left(u\right)$

$$\begin{gathered} \frac{1}{f}=\frac{1}{v}+\frac{1}{u} \\ \Rightarrow \frac{1}{10}=-\frac{1}{{\displaystyle \frac{u}{2}}}+\frac{1}{u} \\ \Rightarrow \frac{1}{10}=\frac{-2+1}{u}=-\frac{1}{u} \\ \Rightarrow u=-10cm \end{gathered}$$

Hence, the position of the object is at $10cm$ from the pole of the mirror.