KCET · Physics · Ray Optics

Two thin biconvex lenses have focal lengths \(f_{1}\) and \(f_{2}\). A third thin biconcave lens has focal length of \(f_{3}\). If the two biconvex lenses are in contact, then the total power of the lenses is \(P_{1}\). If the first convex lens is in contact with the third lens, then the total power is \(P_{2}\). If the second lens is in contact with the third lens, the total power is \(P_{3}\), then

A \(P_{1}=\frac{f_{1} f_{2}}{f_{1}-f_{2}}, P_{2}=\frac{f_{1} f_{3}}{f_{3}-f_{1}}\) and \(P_{3}=\frac{f_{2} f_{3}}{f_{3}-f_{2}}\)
B \(P_{1}=\frac{f_{1}-f_{2}}{f_{1} f_{2}}, P_{2}=\frac{f_{3}-f_{1}}{f_{3}+f_{1}}\) and \(P_{3}=\frac{f_{3}-f_{2}}{f_{2} f_{3}}\)
C \(P_{1}=\frac{f_{1}-f_{2}}{f_{1} f_{2}}, P_{2}=\frac{f_{3}-f_{1}}{f_{1} f_{3}}\) and \(P_{3}=\frac{f_{3}-f_{2}}{f_{2} f_{3}}\)
D \(P_{1}=\frac{f_{1}+f_{2}}{f_{1} f_{2}}, P_{2}=\frac{f_{3}-f_{1}}{f_{1} f_{3}}\) and \(P_{3}=\frac{f_{3}-f_{2}}{f_{2} f_{3}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(P_{1}=\frac{f_{1}+f_{2}}{f_{1} f_{2}}, P_{2}=\frac{f_{3}-f_{1}}{f_{1} f_{3}}\) and \(P_{3}=\frac{f_{3}-f_{2}}{f_{2} f_{3}}\)

Step-by-step Solution

Detailed explanation

According to cartesian sign conversion,
Focal length of first biconvex lens \(=f_{1}\)
Focal length of second biconvex lens \(=f_{2}\)
Focal length of third biconcave lens \(=-f_{3}\)
\(\therefore\) Focal length of combination of first and second lenses is
\(\frac{1}{f}=\frac{1}{f_{1}}+\frac{1}{f_{2}}=\frac{f_{1}+f_{2}}{f_{1} f_{2}}\)
As, power of lens, \(P_{1}=\frac{1}{f}=\frac{f_{1}+f_{2}}{f_{1} f_{2}}\)
Similarly, power of combination of first and third lenses is
\(P_{2}=\frac{1}{f}=\frac{1}{f_{1}}-\frac{1}{f_{3}}=\frac{f_{3}-f_{1}}{f_{1} f_{3}}\)
and for combination of second and third lenses is
\(P_{3}=\frac{1}{f}=\frac{1}{f_{2}}-\frac{1}{f_{3}}=\frac{f_{3}-f_{2}}{f_{2} f_{3}}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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