Question

A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror.

Answer 1

Hint:We are given the convex mirror- the convex mirror is also known as the diverging mirror as it diverges a parallel beam of light. We are provided with the height of the object, object distance, focal length of the convex mirror. We can use the mirror formula to arrive at our answer.

Complete step by step answer:
Height of the object, \[{{h}_{0}}=4.5cm\]
Object distance, considering the sign conventions, \[u=-12cm\]
The focal length of a convex mirror, \[f=15cm\]
Using the mirror formula,
\[\begin{align}
& \dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u} \\
&\Rightarrow \dfrac{1}{15}=\dfrac{1}{v}-\dfrac{1}{12} \\
&\Rightarrow \dfrac{1}{v}=\dfrac{1}{15}+\dfrac{1}{12} \\
&\Rightarrow \dfrac{1}{v}=\dfrac{4+5}{60} \\
&\therefore v=-6.7cm \\
\end{align}\]
Hence, the image of the needle is 6.7 cm away from the mirror. Also, it is on the other side of the mirror.
Now to find about the nature and size of the image we have to use the concept of magnification,
\[m=\dfrac{-v}{u}=\dfrac{6.7}{12}=0.56\]
The magnification comes out to be positive and is smaller than one. Thus, the image formed is diminished and is erect and the nature of the image is virtual. If the needle is moved farther from the mirror, the size of the image will reduce gradually.

Note: Always remember that the focal length of concave mirror is positive and the focal length of convex mirror is negative. The magnification if positive means the image is inverted and if the magnification is negative means the image is erect.