Question

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

Answer 1

Answer: 6.67

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,
$u_1=-20cm$
$\therefore \frac{1}{v_1}=\frac{1}{f}-\frac{1}{u_1}$
$\therefore \frac{1}{v_1}=\frac{1}{-10}-\frac{1}{-20}$
$\therefore v_1=-20cm$
Negative sign indicates image is on the same side as that of object.
Alternate method
As one end of object lies at $2F$, its image from concave mirror will also get formed at $2F$ on same side, i.e., $|u|=|v|$.
Similarly for the other end of the object,
$u_2=-(20+20)=-40cm$
$\therefore \frac{1}{v_2}=\frac{1}{f}-\frac{1}{u_2}$
$\therefore \frac{1}{v_2}=\frac{1}{-10}-\frac{1}{-40}$
$\therefore v_2=\frac{-40}{3}cm$
$\therefore$ Length of the image $=v_2-v_1$
$=\frac{-40}{3}-(-20)$
$=6.67cm$