A dentist uses a mirror in front of a decayed tooth at a distance of 4cm from the tooth to get four times magnified image in the mirror. Use mirror formula to find the focal length and nature of the mirror used.
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Given; Magnification (m) = 4; Object distance (u) = -4cm; Image distance = v; Magnification (m) $=-\frac{\text{v}}{\text{u}}$$\Rightarrow4=-\frac{\text{v}}{-4}$
$\Rightarrow\text{v}=4\times4=16\text{cm}$
Image distance = 16cm. By mirror formula;$\frac{1}{\text{v}}+\frac{1}{\text{u}}=\frac{1}{\text{f}}$
$\Rightarrow\frac{1}{16}-\frac{1}{4}=\frac{1}{\text{f}}$
$\Rightarrow\frac{1-4}{16}=\frac{1}{\text{f}}$
$\Rightarrow-\frac{3}{16}=\frac{1}{\text{f}}$
$\Rightarrow\text{f}=-\frac{16}{3}=-5.33\text{cm}$
Hence the focal length is -5.33cm. Since the focal length is negative therefore the mirror is concave.
$\Rightarrow\text{v}=4\times4=16\text{cm}$
Image distance = 16cm. By mirror formula;$\frac{1}{\text{v}}+\frac{1}{\text{u}}=\frac{1}{\text{f}}$
$\Rightarrow\frac{1}{16}-\frac{1}{4}=\frac{1}{\text{f}}$
$\Rightarrow\frac{1-4}{16}=\frac{1}{\text{f}}$
$\Rightarrow-\frac{3}{16}=\frac{1}{\text{f}}$
$\Rightarrow\text{f}=-\frac{16}{3}=-5.33\text{cm}$
Hence the focal length is -5.33cm. Since the focal length is negative therefore the mirror is concave.
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