{"id":85826,"date":"2025-06-01T03:28:55","date_gmt":"2025-06-01T03:28:55","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=85826"},"modified":"2025-06-01T03:28:55","modified_gmt":"2025-06-01T03:28:55","slug":"when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/","title":{"rendered":"When a convex lens produces a real image of an object, the minimum dis"},"content":{"rendered":"

When a convex lens produces a real image of an object, the minimum distance between the object and image is equal to<\/p>\n

[amp_mcq option1=”the focal length of the convex lens” option2=”twice the focal length of the convex lens” option3=”four times the focal length of the convex lens” option4=”one half of the focal length of the convex lens” correct=”option3″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CDS-2 – 2018<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe minimum distance between an object and its real image formed by a convex lens is four times the focal length (4f).
\n<\/section>\n
\nFor a convex lens, a real image is formed when the object is placed outside the focal point (object distance |u| > f). The image formed is real and inverted. The lens formula is 1\/v – 1\/u = 1\/f. Using distances as positive values, 1\/v + 1\/|u| = 1\/f. Let D be the distance between the object and the image, D = |u| + v. To minimize D, we can express v in terms of |u| and f: 1\/v = 1\/f – 1\/|u| = (|u|-f)\/(f|u|), so v = f|u|\/(|u|-f). Thus, D = |u| + f|u|\/(|u|-f) = (|u|(|u|-f) + f|u|)\/(|u|-f) = (|u|\u00b2 – f|u| + f|u|)\/(|u|-f) = |u|\u00b2\/(|u|-f). Let x = |u|-f, so |u| = x+f. D = (x+f)\u00b2\/x = (x\u00b2 + 2xf + f\u00b2)\/x = x + 2f + f\u00b2\/x. For a real image, |u| must be greater than f, so x > 0. By AM-GM inequality, x + f\u00b2\/x \u2265 2\u221a(x * f\u00b2\/x) = 2f. The minimum value occurs when x = f\u00b2\/x, i.e., x\u00b2 = f\u00b2, so x = f (since x>0). This means |u|-f = f, so |u| = 2f. When |u|=2f, v = f(2f)\/(2f-f) = 2f\u00b2\/f = 2f. The minimum distance D = |u| + v = 2f + 2f = 4f. This happens when the object is placed at 2f, forming a real image at 2f.
\n<\/section>\n
\nIf the object is placed closer than the focal length (|u| < f), a virtual image is formed. As the object approaches the focal point from outside (from |u| > f), the real image moves further away from the lens towards infinity. As the object moves away from the lens, the real image moves towards the focal point. The minimum separation between object and a real image occurs specifically when the object is placed at 2f.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

When a convex lens produces a real image of an object, the minimum distance between the object and image is equal to [amp_mcq option1=”the focal length of the convex lens” option2=”twice the focal length of the convex lens” option3=”four times the focal length of the convex lens” option4=”one half of the focal length of the … <\/p>\n

Detailed SolutionWhen a convex lens produces a real image of an object, the minimum dis<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1088],"tags":[1114,1153,1128],"class_list":["post-85826","post","type-post","status-publish","format-standard","hentry","category-upsc-cds-2","tag-1114","tag-optics","tag-physics","no-featured-image-padding"],"yoast_head":"\nWhen a convex lens produces a real image of an object, the minimum dis<\/title>\n<meta name=\"description\" content=\"The minimum distance between an object and its real image formed by a convex lens is four times the focal length (4f). For a convex lens, a real image is formed when the object is placed outside the focal point (object distance |u| > f). The image formed is real and inverted. The lens formula is 1\/v - 1\/u = 1\/f. Using distances as positive values, 1\/v + 1\/|u| = 1\/f. Let D be the distance between the object and the image, D = |u| + v. To minimize D, we can express v in terms of |u| and f: 1\/v = 1\/f - 1\/|u| = (|u|-f)\/(f|u|), so v = f|u|\/(|u|-f). Thus, D = |u| + f|u|\/(|u|-f) = (|u|(|u|-f) + f|u|)\/(|u|-f) = (|u|\u00b2 - f|u| + f|u|)\/(|u|-f) = |u|\u00b2\/(|u|-f). Let x = |u|-f, so |u| = x+f. D = (x+f)\u00b2\/x = (x\u00b2 + 2xf + f\u00b2)\/x = x + 2f + f\u00b2\/x. For a real image, |u| must be greater than f, so x > 0. By AM-GM inequality, x + f\u00b2\/x \u2265 2\u221a(x * f\u00b2\/x) = 2f. The minimum value occurs when x = f\u00b2\/x, i.e., x\u00b2 = f\u00b2, so x = f (since x>0). This means |u|-f = f, so |u| = 2f. When |u|=2f, v = f(2f)\/(2f-f) = 2f\u00b2\/f = 2f. The minimum distance D = |u| + v = 2f + 2f = 4f. This happens when the object is placed at 2f, forming a real image at 2f.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"When a convex lens produces a real image of an object, the minimum dis\" \/>\n<meta property=\"og:description\" content=\"The minimum distance between an object and its real image formed by a convex lens is four times the focal length (4f). For a convex lens, a real image is formed when the object is placed outside the focal point (object distance |u| > f). The image formed is real and inverted. The lens formula is 1\/v - 1\/u = 1\/f. Using distances as positive values, 1\/v + 1\/|u| = 1\/f. Let D be the distance between the object and the image, D = |u| + v. To minimize D, we can express v in terms of |u| and f: 1\/v = 1\/f - 1\/|u| = (|u|-f)\/(f|u|), so v = f|u|\/(|u|-f). Thus, D = |u| + f|u|\/(|u|-f) = (|u|(|u|-f) + f|u|)\/(|u|-f) = (|u|\u00b2 - f|u| + f|u|)\/(|u|-f) = |u|\u00b2\/(|u|-f). Let x = |u|-f, so |u| = x+f. D = (x+f)\u00b2\/x = (x\u00b2 + 2xf + f\u00b2)\/x = x + 2f + f\u00b2\/x. For a real image, |u| must be greater than f, so x > 0. By AM-GM inequality, x + f\u00b2\/x \u2265 2\u221a(x * f\u00b2\/x) = 2f. The minimum value occurs when x = f\u00b2\/x, i.e., x\u00b2 = f\u00b2, so x = f (since x>0). This means |u|-f = f, so |u| = 2f. When |u|=2f, v = f(2f)\/(2f-f) = 2f\u00b2\/f = 2f. The minimum distance D = |u| + v = 2f + 2f = 4f. This happens when the object is placed at 2f, forming a real image at 2f.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T03:28:55+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"When a convex lens produces a real image of an object, the minimum dis","description":"The minimum distance between an object and its real image formed by a convex lens is four times the focal length (4f). For a convex lens, a real image is formed when the object is placed outside the focal point (object distance |u| > f). The image formed is real and inverted. The lens formula is 1\/v - 1\/u = 1\/f. Using distances as positive values, 1\/v + 1\/|u| = 1\/f. Let D be the distance between the object and the image, D = |u| + v. To minimize D, we can express v in terms of |u| and f: 1\/v = 1\/f - 1\/|u| = (|u|-f)\/(f|u|), so v = f|u|\/(|u|-f). Thus, D = |u| + f|u|\/(|u|-f) = (|u|(|u|-f) + f|u|)\/(|u|-f) = (|u|\u00b2 - f|u| + f|u|)\/(|u|-f) = |u|\u00b2\/(|u|-f). Let x = |u|-f, so |u| = x+f. D = (x+f)\u00b2\/x = (x\u00b2 + 2xf + f\u00b2)\/x = x + 2f + f\u00b2\/x. For a real image, |u| must be greater than f, so x > 0. By AM-GM inequality, x + f\u00b2\/x \u2265 2\u221a(x * f\u00b2\/x) = 2f. The minimum value occurs when x = f\u00b2\/x, i.e., x\u00b2 = f\u00b2, so x = f (since x>0). This means |u|-f = f, so |u| = 2f. When |u|=2f, v = f(2f)\/(2f-f) = 2f\u00b2\/f = 2f. The minimum distance D = |u| + v = 2f + 2f = 4f. This happens when the object is placed at 2f, forming a real image at 2f.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/","og_locale":"en_US","og_type":"article","og_title":"When a convex lens produces a real image of an object, the minimum dis","og_description":"The minimum distance between an object and its real image formed by a convex lens is four times the focal length (4f). For a convex lens, a real image is formed when the object is placed outside the focal point (object distance |u| > f). The image formed is real and inverted. The lens formula is 1\/v - 1\/u = 1\/f. Using distances as positive values, 1\/v + 1\/|u| = 1\/f. Let D be the distance between the object and the image, D = |u| + v. To minimize D, we can express v in terms of |u| and f: 1\/v = 1\/f - 1\/|u| = (|u|-f)\/(f|u|), so v = f|u|\/(|u|-f). Thus, D = |u| + f|u|\/(|u|-f) = (|u|(|u|-f) + f|u|)\/(|u|-f) = (|u|\u00b2 - f|u| + f|u|)\/(|u|-f) = |u|\u00b2\/(|u|-f). Let x = |u|-f, so |u| = x+f. D = (x+f)\u00b2\/x = (x\u00b2 + 2xf + f\u00b2)\/x = x + 2f + f\u00b2\/x. For a real image, |u| must be greater than f, so x > 0. By AM-GM inequality, x + f\u00b2\/x \u2265 2\u221a(x * f\u00b2\/x) = 2f. The minimum value occurs when x = f\u00b2\/x, i.e., x\u00b2 = f\u00b2, so x = f (since x>0). This means |u|-f = f, so |u| = 2f. When |u|=2f, v = f(2f)\/(2f-f) = 2f\u00b2\/f = 2f. The minimum distance D = |u| + v = 2f + 2f = 4f. This happens when the object is placed at 2f, forming a real image at 2f.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T03:28:55+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/","url":"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/","name":"When a convex lens produces a real image of an object, the minimum dis","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T03:28:55+00:00","dateModified":"2025-06-01T03:28:55+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The minimum distance between an object and its real image formed by a convex lens is four times the focal length (4f). For a convex lens, a real image is formed when the object is placed outside the focal point (object distance |u| > f). The image formed is real and inverted. The lens formula is 1\/v - 1\/u = 1\/f. Using distances as positive values, 1\/v + 1\/|u| = 1\/f. Let D be the distance between the object and the image, D = |u| + v. To minimize D, we can express v in terms of |u| and f: 1\/v = 1\/f - 1\/|u| = (|u|-f)\/(f|u|), so v = f|u|\/(|u|-f). Thus, D = |u| + f|u|\/(|u|-f) = (|u|(|u|-f) + f|u|)\/(|u|-f) = (|u|\u00b2 - f|u| + f|u|)\/(|u|-f) = |u|\u00b2\/(|u|-f). Let x = |u|-f, so |u| = x+f. D = (x+f)\u00b2\/x = (x\u00b2 + 2xf + f\u00b2)\/x = x + 2f + f\u00b2\/x. For a real image, |u| must be greater than f, so x > 0. By AM-GM inequality, x + f\u00b2\/x \u2265 2\u221a(x * f\u00b2\/x) = 2f. The minimum value occurs when x = f\u00b2\/x, i.e., x\u00b2 = f\u00b2, so x = f (since x>0). This means |u|-f = f, so |u| = 2f. When |u|=2f, v = f(2f)\/(2f-f) = 2f\u00b2\/f = 2f. The minimum distance D = |u| + v = 2f + 2f = 4f. This happens when the object is placed at 2f, forming a real image at 2f.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/when-a-convex-lens-produces-a-real-image-of-an-object-the-minimum-dis\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CDS-2","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-cds-2\/"},{"@type":"ListItem","position":3,"name":"When a convex lens produces a real image of an object, the minimum dis"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/85826","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=85826"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/85826\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=85826"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=85826"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=85826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}