Question

A thin symmetrical double convex lens of refractive index \[{\mu _2} = \]1.5 is placed between a medium of refractive index ${\mu _1} = $1.4 to the left and another medium of refractive index ${\mu _3} = $1.6 to the right. Then, the system behaves as.
A. A convex lens
B. A concave lens
C. A glass plate
D. A convo concave lens

Answer 1

Hint: We can consider the refraction through the lens as a refraction through a spherical surface twice i.e. once from the front side of the lens and then from the second circular surface of the lens. Refraction through a circular surface is given as$ - \dfrac{{{\mu _1}}}{u} + \dfrac{{{\mu _1}}}{v} = \dfrac{{{\mu _2} - {\mu _1}}}{R}$.
If the rays coming from infinity meet at infinity the lens acts as a glass plate.

Complete step by step answer:
Let us consider two cases for refraction through the two surfaces of the lens. We know that the equation for refraction through a spherical surface is given by as-$ - \dfrac{{{\mu _1}}}{u} + \dfrac{{{\mu _1}}}{v} = \dfrac{{{\mu _2} - {\mu _1}}}{R}$ -----------(1)
Case 1: When the rays travel from first medium$\left( {{\mu _1} = 1.4} \right)$ to the first surface of the lens$\left( {{\mu _2} = 1.5} \right)$.
Substituting the values in equation 1,
$
   - \dfrac{{1.4}}{{ - \infty }} + \dfrac{{1.5}}{v} = \dfrac{{1.5 - 1.4}}{R} \\
 \Rightarrow 0 + \dfrac{{1.5}}{v} = \dfrac{{0.1}}{R} \\
\Rightarrow v = 15R \\
 $
Case 2: When the rays travel from the second surface of the lens$\left( {{\mu _1} = 1.5} \right)$ to the second medium$\left( {{\mu _2} = 1.6} \right)$. The image distance found in case 1 is used as object distance for this case, u$ = 15R$.
Substituting the values in equation 1.
$
   - \dfrac{{1.5}}{{15R}} + \dfrac{{1.6}}{V} = \dfrac{{1.6 - 1.5}}{{ - R}} \\
\Rightarrow - \dfrac{{0.1}}{R} + \dfrac{{1.6}}{V} = - \dfrac{{0.1}}{R} \\
\Rightarrow \dfrac{{1.6}}{V} = 0 \\
\Rightarrow \dfrac{1}{V} = 0 \Rightarrow V = \infty \\
 $
Therefore the final image distance is $V = \infty $.

Hence the correct option is C the system acts as a glass plate.

Note: Refraction through a lens with two different mediums on its either side can be considered as refraction from a spherical surface twice. The sign of object distance, image distance and focal length should be taken carefully according to the convention to avoid error in answer.