MHT CET · Physics · Ray Optics

An object is located on a wall, its image of equal size is to be obtained on a parallel wall with the help of a convex lens. The lens in placed at a distance ' \(\mathrm{d}\) ' in front of the second wall. The required focal length of the lens is

A less than \(\frac{\mathrm{d}}{4}\)
B more than \(\frac{\mathrm{d}}{4}\) but less than \(\frac{\mathrm{d}}{2}\)
C only \(\frac{\mathrm{d}}{4}\)
D only \(\frac{d}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(D) only \(\frac{d}{2}\)

Step-by-step Solution

Detailed explanation

Size of the image is equal to the size of the object. Hence image distance will be equal to object distance. Also if object distance \(u\) \(=2 \mathrm{f}\), Image distance will be \(2 \mathrm{f}\).
\(
\begin{aligned}
& \therefore \mathrm{u}=\mathrm{v}=\mathrm{d}=2 \mathrm{f} \\
& \therefore \mathrm{u}+\mathrm{v}=2 \mathrm{~d}=4 \mathrm{f} \\
& \therefore \mathrm{f}=\frac{\mathrm{d}}{2}
\end{aligned}
\)

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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