MCQ
Given a thin convex lens (refractive index $\mu_2$ ), kept in a liquid (refractive index $\mu_1, \mu_1<\mu_2$ ) having radii of curvature $\left|R_1\right|$ and $\left|R_2\right|$. Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?
  • A
    $\frac{\mu_1\left| R _1\right| \cdot\left| R _2\right|}{\mu_2\left(\left| R _1\right|+\left| R _2\right|\right)-\mu_1\left| R _1\right|}$
  • $\frac{\mu_1\left| R _1\right| \cdot\left| R _2\right|}{\mu_2\left(\left| R _1\right|+\left| R _2\right|\right)-\mu_1\left| R _2\right|}$
  • C
    $\frac{\mu_1\left| R _1\right| \cdot\left| R _2\right|}{\mu_2\left(2\left| R _1\right|+\left| R _2\right|\right)-\mu_1 \sqrt{\left| R _1\right| \cdot\left| R _2\right|}}$
  • D
    $\frac{\left(\mu_2+\mu_1\right)\left|R_1\right|}{\left(\mu_2-\mu_1\right)}$

Answer

Correct option: B.
$\frac{\mu_1\left| R _1\right| \cdot\left| R _2\right|}{\mu_2\left(\left| R _1\right|+\left| R _2\right|\right)-\mu_1\left| R _2\right|}$
(B)
Image
$\begin{array}{l}\frac{1}{ f _{ eq }}=\frac{2}{ f _{ L }}-\frac{1}{ f _{ m }} \\ f _{ m }=-\frac{\left| R _2\right|}{2} \\ \frac{1}{ f _{ L }}=\left(\frac{\mu_2}{\mu_1}-1\right)\left(\frac{1}{ R _1}+\frac{1}{ R _2}\right)\end{array}
$
$
\begin{array}{l}
\frac{1}{f_{eq}}=2\left(\frac{\mu_2-\mu_1}{\mu_1}\right)\left(\frac{R_1+R_2}{R_1 R_2}\right)+\frac{2}{R_2} \\
=\frac{2}{R_2}\left[\frac{\left(\mu_2-\mu_1\right)\left(R_1+R_2\right)+\mu_1 R_1}{\mu_1 R_1}\right] \\
=\frac{2}{R_2}\left[\frac{\mu_2 R_1+\mu_2 R_2-\mu_1 R_1-\mu_1 R_2+\mu_1 R_1}{\mu_1 R_1}\right] \\
\frac{1}{f_{eq}}=\frac{2\left[\mu_2 R_1+\mu_2 R_2-\mu_1 R_2\right]}{\mu_1 R_1 R_2}
\end{array}
$
For same size of image
$
\begin{array}{l}
u=2 f \\
u=\frac{\mu_1 R_1 R_2}{\mu_2 R_1+\mu_2 R_2-\mu_1 R_2}
\end{array}
$

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