Question

Two thin similar convex glass pieces are joined together front to front with its rear portion silvered such that a sharp image is formed 20 cm from the mirror. When the air between the glass pieces is replaced by water (${\mu }_w=4/3$), then the image formed from the mirror is at a distance of

• A. $8cm$
• B. $10cm$
• C. $6cm$
• D. $12cm$

Answer 1

The correct option is D $12cm$

$\begin{array}{c}\text{Initiall it is simply a concave}\\ \text{mirror with}f=-20cm\text{,}\\ R=-40cm\\ \text{Power of mirror}\left(P_m\right)=\frac{1}{20\times {10}^{-2}}\dots \\ P_m=5D\\ \text{Now, air is replaced with water}(\mu =4/3)\\ \text{inside the lens.}\\ \text{Focal length of lens}=\frac{R}{2(\mu -1)}\end{array}$

$\begin{array}{cc}f& =\frac{40}{2(\frac{4}{3}-1)}\\ & =\frac{7040\times 3}{2\times 1}\\ f& =60cm\\ \text{Power of lens}\left(P_L\right)& =\frac{1}{60\times {10}^{-2}}\\ & =\frac{10}{6}=\frac{5}{3}D\end{array}$

$\begin{array}{c}\text{Equivalent power}=2P_L+P_m\\ \{2P_L\text{is due to ray first strike lens and}\\ \text{also comes back after reflecting from mirron}\\ =2\times \frac{5}{3}+5\\ =>\frac{25}{3}D\end{array}$

$\begin{array}{cc}\text{focal length}& =\frac{1}{P\frac{}{}}\\ =\frac{1}{25/3}& =\frac{100}{25/3}=12cm\end{array}$

Hence (D) is correct option