Two concave refracting surfaces of equal radii of curvature and...

$Two concave refracting surfaces of equal radii of curvature and refractive index 1.5 face each other in air as shown in$
Two concave refracting surfaces of equal radii of curvature and refractive index 1.5 face each other in air as shown in figure. A point object O is placed midway, between P and B . The separation between the images of

O

, formed by each refracting surface is :

Options

A0.214 R
B0.411 R
C0.124 R
D0.114 R

Correct answer

D. 0.114 R

Step-by-step solution

For B

\begin{aligned} & \frac{\mu_2}{\mathrm{~V}}-\frac{\mu_1}{\mathrm{u}}=\frac{\mu_2-\mu_1}{\mathrm{R}} \\ & \frac{1.5}{\mathrm{~V}}+\frac{1}{\frac{R}{2}}=\frac{0.5}{-\mathrm{R}} \\ & \frac{1.5}{\mathrm{~V}}=-\frac{1}{2 \mathrm{R}}-\frac{2}{\mathrm{R}} \\ & \frac{1.5}{\mathrm{~V}}=\frac{-5}{2 \mathrm{R}} \Rightarrow V_B=-0.6 \mathrm{R} \end{aligned}

For A

\begin{aligned} & \frac{1.5}{V}+\frac{2}{3 R}=\frac{0.5}{-R} \\ & \frac{1.5}{V}=-\frac{1}{2 R}-\frac{2}{3 R} \\ & \frac{1.5}{V}=-\frac{7}{6 R} \\ & V_A=-\frac{9}{7} R \end{aligned}

Distance between images

=2 \mathrm{R}-\left(0.6 \mathrm{R}+\frac{9}{7} \mathrm{R}\right)=0.114 \mathrm{R}

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