Question

When a thin convex lens is put in contact with a thin concave lens of the same focal length, the resultant combination has focal length equal to,
$\begin{align}
  & (A)\dfrac{f}{2} \\
 & (B)2f \\
 & (C)zero \\
 & (D)\infty \\
\end{align}$

Answer 1

Hint: We know that power of combination of two lenses is the sum of individual power of two lenses. Since power is the reciprocal of the focal length, substitute the value of focal lengths of two lenses in the equation of power of combination. Thus we get a new equation. Then by rearranging them we will get the resultant focal length of combination of two lenses.
Formula used:
${{P}_{combination}}={{P}_{1}}+{{P}_{2}}$
where, ${{P}_{1}}$ and ${{P}_{2}}$ are the powers of individual lenses.
$P=\dfrac{1}{f}$
where, $f$ is the focal length.
Then,
$\dfrac{1}{{{F}_{combination}}}=\dfrac{1}{{{f}_{1}}}+\dfrac{1}{{{f}_{2}}}$
where, ${{f}_{1}}$ and ${{f}_{2}}$ are the focal lengths of convex and concave lenses respectively.

Complete answer:
We know that,
$\dfrac{1}{{{F}_{combination}}}=\dfrac{1}{{{f}_{1}}}+\dfrac{1}{{{f}_{2}}}$
where, ${{f}_{1}}$ and ${{f}_{2}}$ are the focal lengths of convex and concave lenses respectively.
By using sign convention in case of a convex lens focal length is positive and for a concave lens focal length is negative.
Then the equation becomes,
$\dfrac{1}{{{F}_{combination}}}=\dfrac{1}{{{f}_{1}}}+\dfrac{1}{-{{f}_{2}}}$
Given that the focal length of convex and concave lenses are the same. So let it be $f$,
$\Rightarrow \dfrac{1}{{{F}_{combination}}}=\dfrac{1}{f}-\dfrac{1}{f}$
$\Rightarrow \dfrac{1}{{{F}_{combination}}}=0$
$\Rightarrow {{F}_{combination}}=\dfrac{1}{0}$
$\Rightarrow {{F}_{combination}}=\infty $

So, the correct answer is “Option D”.

Additional Information:
If the parallel paraxial beam of light is incident on a convex lens then, making some angles with the principal axis, the reflected rays would converge from a point in a plane through F normal to the principal axis. This is called the focal plane of the mirror. The distance between the focus and the pole P is called the focal length of the mirror. But in case of a concave lens the incident beam diverges from the surface of the concave lens.

Note:
The convex lens always converges from a point from the plane through F normal to the principal axis. This is known as the focal plane of the mirror. Hence focal length of a convex lens is always positive. But in case of a concave lens the incident beam diverges from the surface of the concave lens. Hence, the focal length of a concave mirror is always negative.