Question

A short linear object of length $20\, cm$ lies along the axis of a concave mirror of focal length $10 \,cm$ at a distance $20 \,cm$ from the pole of the mirror. The length of the image is equal to _______$cm$.

Answer 1

From mirror formula,
$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$
As, the object lies along the axis, for the end nearer to the mirror,
$u _{1} =-20\, cm $
$\therefore \frac{1}{ v _{1}} =\frac{1}{ f }-\frac{1}{ u _{1}} $
$\therefore \frac{1}{ v _{1}} =\frac{1}{-10}-\frac{1}{-20}$
$\therefore v _{1} =-20\, cm$
Negative sign indicates image is on the same side as that of object.
Alternate method
As one end of object lies at $2 F$, its image from concave mirror will also get formed at $2 F$ on same side, i.e., $| u |=| v |$.
Similarly for the other end of the object,
$u _{2} =-(20+20)=-40\, cm$
$\therefore \frac{1}{ v _{2}} =\frac{1}{ f }-\frac{1}{ u _{2}} $
$\therefore \frac{1}{ v _{2}} =\frac{1}{-10}-\frac{1}{-40} $
$\therefore v _{2} =\frac{-40}{3} cm$
$ \therefore $ Length of the image $=v_{2}-v_{1} $
$=\frac{-40}{3}-(-20) $
$=6.67\, cm $