Question

Two similar thin equi-convex lenses, of focal length $f$ each, are kept coaxially in contact with each other such that the focal length of the combination is $F_1$ . when the space between the two lenses is filled with glycerine(which has the same refractive index($\mu =1.5$) as that of glass) then the equivalent focal length is $F_2$. Find ratio of $F_1$ and $F_2$.
A) $2:1$
B) $1:2$
C) $2:3$
D) $3:4$

Answer 1

Hint: when two or more lenses are joined together, the power of the lenses can be directly added. The space between two convex lenses is just like a concave lens so glycerine will take the form of a concave lens.

Complete Step by step answer:
Let the power of individual lens be $P$ and that of combination be $P_1$ and
$P_2$ for the first and second case respectively.
For the first case.
Power of equivalent lens can be written as
$P_1=P+P$
We know that,
$P={\displaystyle \frac{1}{f}}$
So the equation becomes,
${\displaystyle \frac{1}{F_1}}={\displaystyle \frac{1}{f}}+{\displaystyle \frac{1}{f}}$
$F_1={\displaystyle \frac{f}{2}}$
For second case,
It is given in the question that glycerine is added in between the two lenses. We know that the refractive index of glycerine is equal to the refractive index of glass, which means that glycerine also behaves just like a glass. When we add glycerine in between the two convex lenses, it takes the shape of a concave lens of the same focal length but negative in sign. Negative focal length means negative power.
$P_2=P+P+(-P)$
$$\begin{gathered} P_2=P \\ {\displaystyle \frac{1}{F_2}}={\displaystyle \frac{1}{f}} \\ {F}_2=f \end{gathered}$$
The ratio of $F_1$and $F_2$ is
${\displaystyle \frac{F_1}{F_2}}={\displaystyle \frac{{\displaystyle \frac{f}{2}}}{f}}$
${\displaystyle \frac{F_1}{F_2}}={\displaystyle \frac{1}{2}}$

Therefore, Option B) is the correct answer.

Additional information:
The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative focal length indicates that the system diverges light. A system with a shorter focal length bends the rays more sharply, bringing them to a focus in a shorter distance or diverging them more quickly.

Note:
When we added glycerine In between the two lenses, it takes the shape of a concave lens. Sometimes students ignore the negative sign in the focal length of the concave lens. The focal length is negative for a concave lens because it is measured in the opposite direction of the path of incident ray.