A convex lens has a focal length f. An object is placed between infinity and 2f from the lens on its axis. The image formed is located

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BlackTom AI · Accepted Answer

Using the thin-lens equation 1/f = 1/do + 1/di is the key concept; it relates focal length f, object distance do, and image distance di for a convex lens. 1. Relevant knowledge points: - Thin-lens formula: 1/f = 1/do + 1/di. - For a real object (do positive) and a real image (di positive), a convex lens forms real inverted images when do > f. - Special cases: object at infinity → image at f; object at 2f → image at 2f; object between f and 2f → image beyond 2f; object beyond 2f → image between f and 2f. 2. Step-by-step reasoning: Given the object lies between infinity and 2f, its distance do satisfies do > 2f (finite but greater than 2f). Use the lens formula and solve for di: 1/f = 1/do + 1/di => 1/di = 1/f - 1/do = (do - f)/(f do) => di = f do/(do - f). 3. Intermediate algebra and inequalities: Show di > f: di = f do/(do - f) > f ⇔ do/(do - f) > 1, which is true because do > do - f (f>0). Show di < 2f: di < 2f ⇔ f do/(do - f) < 2f ⇔ do/(do - f) < 2 ⇔ do < 2(do - f) ⇔ do < 2do - 2f ⇔ 0 < do - 2f ⇔ do > 2f. This last inequality holds by the problem statement. Thus f < di < 2f. 4. Why the correct answer is correct: Algebraically and conceptually, when the object is beyond 2f (i.e., between 2f and infinity), the image distance falls between f and 2f on the opposite side of the lens. The image is real, inverted, and reduced in size. 5. Why other choices are incorrect: - At 2f: occurs only if object is exactly at 2f, not for all positions between 2f and infinity. - At f: occurs only in the limit as object → infinity. - Between the lens and f: that region corresponds to virtual, upright, magnified images when the object is inside f, not the present case. 6. Conclusion: Therefore the image is located between f and 2f, confirming the provided correct answer.

A convex lens has a focal length f. An object is placed between infinity and 2f from the lens on its axis. The image formed is locatedSingle choice

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