A thin rod of length $\frac{f}{3}$ is placed along the optic axis of a concave mirror of focal lengt

Question

A thin rod of length $\frac{f}{3}$ is placed along the optic axis of a concave mirror of focal length $f$ such that its image, which is real and elongated, just touches the rod. Twice the magnification is ____

Professor PC Thomas Classes · Answer

(1.5) : The question deals with lateral magnification when rod is placed along optic axis of concave mirror. Image of $A$ is formed at $A^{\prime}$. Image of $B$ is formed at $B$ itself. $B$ should therefore be at centre of curvature of mirror.\ $A B$ lies between $f$ and $2 f$ for real and elongated image.\ Object distance for $A=P A$.\ $	herefore \quad u=P B-A B=2 f-\frac{f}{3}=\frac{5 f}{3}$\ Formula for mirror : $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$\ $	herefore \quad \frac{1}{v}+\frac{1}{-5 f / 3}=\frac{1}{-f}$\ or $\quad v=-\frac{5}{2} f=-2.5 f \quad 	herefore P A^{\prime}=2.5 f$\ or $B A^{\prime}=P A^{\prime}-P B=2.5 f-2 f$\ or $B A^{\prime}=0.5 f$\ $	herefore \quad$ Magnification $=\frac{B A^{\prime}}{B A}$\ or $M=\frac{0.5 f}{f / 3}$ or $M=1.5$\