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An object 4 cm in height, is placed at 15 cm in front of a concave mirror of focal length 10 cm. At what distance from the mirror should a screen be placed to obtain a sharp image of the object. Calculate the height of the image.

Answer:

Given:   \[u=-15\text{ }cm,\text{ }f=-10\text{ }cm,\,\,{{h}_{i}}=4\,cm\]

Using the mirror formula

\[\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\]

\[\frac{1}{v}-\frac{1}{15}=\frac{-1}{10}\]

\[\frac{1}{v}=\frac{-1}{10}+\frac{1}{15}\]

\[\frac{1}{v}=\frac{-15+10}{150}=\frac{-5}{150}\]

\[v=-30\,cm\]

Thus, to obtain a sharp image of the object, the screen should be placed in front of the mirror at a distance of 30 cm.

Now,            \[m=\frac{-v}{u}=\frac{\text{height of image}}{\text{height of object}}\]

\[m=-\left( \frac{-30}{-15} \right)\]\[\Rightarrow m=-2\]

\[-2=\frac{{{h}_{1}}}{4}\] \[\Rightarrow {{h}_{1}}=-8\,cm\]

‹         Height of the image is 8 cm.