Where should an object be placed in front of a convex lens to obtain a real image of the same size as the object?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → At twice the focal length.
For a convex lens, the position and size of the image depend on where the object stands compared with the focus F and the point 2F (twice the focal length). When the object is at 2F, the image is formed at 2F on the other side, real, inverted and of the same size as the object.
Lens formula with \(u=-2f\): \(\displaystyle\frac{1}{v}=\frac{1}{f}+\frac{1}{u}=\frac{1}{f}-\frac{1}{2f}=\frac{1}{2f}\), so \(v=2f\)
Magnification \(\displaystyle m=\frac{v}{u}=\frac{2f}{-2f}=-1\): the image is inverted and as large as the object
Why the other options are incorrect
Option (1) At infinity — Incorrect. The rays arrive parallel and meet at F, giving a highly diminished, point-sized image.
Option (2) At the principal focus of the lens. — Incorrect. The refracted rays come out parallel, so the image forms at infinity and is highly enlarged.
Option (3) Between the optical center of the lens and its principal focal point — Incorrect. The image is virtual, erect and enlarged, as in a magnifying glass.