Question 123: Ray Optics

Question

Given a thin convex lens (refractive index $\mu_2$ ), kept in a liquid (refractive index $\mu_1, \mu_1<\mu_2$ ) having radii of curvatures $\left|R_1\right|$ 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?

Accepted Answer

$$ \begin{aligned} & \frac{1}{\mathrm{f}_{\mathrm{eq}}}=\frac{2}{\mathrm{f}_{\mathrm{L}}}-\frac{1}{\mathrm{f}_{\mathrm{m}}} \\ & \mathrm{f}_{\mathrm{m}}=-\frac{\left|\mathrm{R}_2ight|}{2} \\ & \frac{1}{\mathrm{f}_{\mathrm{L}}}=\left(\frac{\mu_2}{\mu_1}-1ight)\left(\frac{1}{\mathrm{R}_1}+\frac{1}{\mathrm{R}_2}ight) \end{aligned} $$
$$
\begin{aligned}
&\begin{aligned}
& \frac{1}{\mathrm{f}_{\mathrm{eq}}}=2\left(\frac{\mu_2-\mu_1}{\mu_1}ight)\left(\frac{\mathrm{R}_1+\mathrm{R}_2}{\mathrm{R}_1 \mathrm{R}_2}ight)+\frac{2}{\mathrm{R}_2} \\
& =\frac{2}{\mathrm{R}_2}\left[\frac{\left(\mu_2-\mu_1ight)\left(\mathrm{R}_1+\mathrm{R}_2ight)+\mu_1 \mathrm{R}_1}{\mu_1 \mathrm{R}_1}ight] \\
& =\frac{2}{\mathrm{R}_2}\left[\frac{\mu_2 \mathrm{R}_1+\mu_2 \mathrm{R}_2-\mu_1 \mathrm{R}_1-\mu_1 \mathrm{R}_2+\mu_1 \mathrm{R}_1}{\mu_1 \mathrm{R}_1}ight] \\
& \frac{1}{\mathrm{f}_{\mathrm{eq}}}=\frac{2\left[\mu_2 \mathrm{R}_1+\mu_2 \mathrm{R}_2-\mu_1 \mathrm{R}_2ight]}{\mu_1 \mathrm{R}_1 \mathrm{R}_2}
\end{aligned}\\
&	ext {For same size of image }\\
&\mathrm{u}=2 \mathrm{f}\\
&\mathbf{u}=\frac{\mu_1 \mathbf{R}_1 \mathbf{R}_2}{\mu_2 \mathrm{R}_1+\mu_2 \mathrm{R}_2-\mu_1 \mathrm{R}_2}
\end{aligned}
$$

Answer

$\frac{\left(\mu_2+\mu_1\right)\left|R_1\right|}{\left(\mu_2-\mu_1\right)}$

Answer

$\frac{\mu_1\left|\mathrm{R}_1\right| \cdot\left|\mathrm{R}_2\right|}{\mu_2\left(2\left|\mathrm{R}_1\right|+\left|\mathrm{R}_2\right|\right)-\mu_1 \sqrt{\left|\mathrm{R}_1\right| \cdot\left|\mathrm{R}_2\right|}}$

Answer

$\frac{\mu_1\left|R_1\right| \cdot\left|R_2\right|}{\mu_2\left(\left|R_1\right|+\left|R_2\right|\right)-\mu_1\left|R_1\right|}$

Answer

$\frac{\mu_1\left|\mathrm{R}_1\right| \cdot\left|\mathrm{R}_2\right|}{\mu_2\left(\left|\mathrm{R}_1\right|+\left|\mathrm{R}_2\right|\right)-\mu_1\left|\mathrm{R}_2\right|}$

Ray Optics

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