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Question: What is the focal length of a convex lens of focal length \(30\,cm\) in contact with a concave lens ...

What is the focal length of a convex lens of focal length 30 cm30\,cm in contact with a concave lens of focal length 20 cm20\,cm? Is the system a converging or a diverging lens? Ignore the thickness of the lenses.

Explanation

Solution

To get the answer for this question, first understand what type of lens is called a converging lens and diverging lens, as depending on the type of lens it can be said which lens will have positive focal length or negative focal length. Once these details are known then we can say the final obtained focal length is for a converging or diverging lens.

Formula used:
1f=1f1+1f2\dfrac{1}{f}=\dfrac{1}{{{f}_{1}}}+\dfrac{1}{{{f}_{2}}}

Complete Step-by-Step solution:
We know that a convex lens is called a converging lens and the focal length of a converging lens is always positive and a concave lens is called a diverging lens and the focal length of a diverging lens is always negative.

Now,
Let the focal length of convex lens be denoted by f1{{f}_{1}},
the focal length of concave lens be denoted by f2{{f}_{2}},
and the focal length of the combined lens be denoted by ff.
So, the value of f1 and f2{{f}_{1}}\,\text{and}\,{{f}_{2}} from the question is:
f1=30 cm   and   f2=−20 cm{{f}_{1}}=30\,cm\,\,\,\text{and}\,\,\,{{f}_{2}}=-20\,cm

Hence, the focal length of combined lens is:
1f=1f1+1f2 ⇒1f=130+1(−20) ⇒1f=130−120 ⇒1f=20−3030×20 ⇒1f=−10600 ⇒1f=−160 ⇒f=−60 cm \begin{aligned} & \dfrac{1}{f}=\dfrac{1}{{{f}_{1}}}+\dfrac{1}{{{f}_{2}}} \\\ & \Rightarrow \dfrac{1}{f}=\dfrac{1}{30}+\dfrac{1}{(-20)} \\\ & \Rightarrow \dfrac{1}{f}=\dfrac{1}{30}-\dfrac{1}{20} \\\ & \Rightarrow \dfrac{1}{f}=\dfrac{20-30}{30\times 20} \\\ & \Rightarrow \dfrac{1}{f}=\dfrac{-10}{600} \\\ & \Rightarrow \dfrac{1}{f}=\dfrac{-1}{60} \\\ & \Rightarrow f=-60\,cm \\\ \end{aligned}

Since, the value of focal length is negative, the combination of lens is a diverging lens.
Therefore, the focal length of the combination of lens is 60 cm60\,cm and it is a diverging lens.

Note:
The combination of concave lens and convex lens is usually known as Telephoto Lens and the focal length of telephoto lens depends on the focal length of both the type of lenses. This type of question is normal in physics and it basically tests the understanding of the topic as the value of focal length plays an important role to identify the type of lens.