Question

A 4.5 cm needle is placed at 12 cm away from a convex mirror of a focal length of 15cm.

a) Give the location of the image and the magnification.

b) Explain what will happen if the needle is moved farther away from the mirror.

Answer 1

a)Step 1: Given data.

Height of object (h_o) = 4.5 cm.

Object distance (u) = -12 cm ( negative sign because of Cartesian sign convention)

Focal length of the mirror (f) = 15 cm.

Step 2: Finding the image distance.

1. According to mirror formulae

,$$\begin{gathered} \frac{1}{f}=\frac{1}{u}+\frac{1}{v} \\ v\to imagedis\tan ce \end{gathered}$$

. $$\begin{gathered} \frac{1}{v}=\frac{1}{f}-\frac{1}{u} \\ =\frac{1}{15}-\frac{1}{-12} \\ =\frac{1}{15}+\frac{1}{12} \\ =\frac{27}{180} \\ =\frac{3}{20} \\ v=\frac{20}{3} \\ =6.7cm \end{gathered}$$

Hence the location of the image is 20 cm in the back of the mirror.

Step 3: Calculating the magnification and size of image.

1. $$\begin{gathered} Magnification,M=\frac{h_i}{h_o}=-\frac{V}{U} \\ M=-\frac{6.7}{-12} \\ =0.56 \end{gathered}$$

2. $$\begin{gathered} M=\frac{h_i}{h_o}=0.56 \\ {h}_i=0.56\times h_o \\ =0.56\times 4.5 \\ =2.52cm \end{gathered}$$

Size of image = 2.52 cm

b) Explanation: The image distance also increases with the object distance, and size of the image will reduce with an increase in object distance.