{"id":1767,"date":"2020-12-03T12:16:34","date_gmt":"2020-12-03T06:46:34","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=1767"},"modified":"2020-12-03T15:30:13","modified_gmt":"2020-12-03T10:00:13","slug":"focal-length-related-radius-curvature","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/focal-length-related-radius-curvature\/","title":{"rendered":"How is Focal Length related to Radius of Curvature?"},"content":{"rendered":"<h2><span style=\"color: #00ccff;\"><strong>How is Focal Length related to Radius of Curvature?<\/strong><\/span><\/h2>\n<p><span style=\"color: #993366;\"><strong>Mirror formula<\/strong><\/span><br \/>\n<span style=\"color: #993366;\"><strong>Definition :<\/strong> <\/span>The equation relating the object distance (u) the image distance (v) and the mirror focal length (f) is called the mirror formula.<\/p>\n\\(\\frac { 1 }{ v } +\\frac { 1 }{ u } =\\frac { 1 }{ f } \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\\)\n<p><span style=\"color: #993366;\"><strong>Assumptions made :<\/strong><\/span><\/p>\n<ol>\n<li>The mirror has a small aperture.<\/li>\n<li>The object lies close to principal axis of the mirror.<\/li>\n<li>The incident rays make small angles with the mirror surface or the principal axis.<\/li>\n<\/ol>\n<p><span style=\"color: #993366;\"><strong>Linear magnification For spherical mirrors<\/strong><\/span><\/p>\n<p><span style=\"color: #993366;\"><strong>Definition :<\/strong><\/span> The ratio of the size of the image, as formed by reflection from the mirror to the size of the object, is called linear magnification produced by the mirror. It is represented by the symbol m.<\/p>\n\\(m=\\frac { -v }{ u } =\\frac { Hieght\\quad of\\quad image }{ Hieght\\quad of\\quad object } \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \\)\n<p><span style=\"color: #993366;\"><strong>Power of mirror<\/strong><\/span><br \/>\nPower of a mirror [in Diopters] <strong>= <\/strong><strong>\u00a0<\/strong>\\(\\frac { 1 }{ f(in\\quad metre) } \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\\)<\/p>\n<p><strong>People also ask<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/cbselibrary.com\/what-is-reflection-of-light\/\">What is Reflection of Light?<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/law-of-reflection-of-light\/\">What is the Law of Reflection of Light?<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/total-internal-reflection\/\">What do you mean by Total Internal Reflection?<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/applications-total-internal-reflection\/\">Applications of Total Internal Reflection<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/image-reflection-by-plane-mirror\/\">Image Reflection by a Plane Mirror<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/type-image-formed-by-plane-mirror\/\">Which Type of Image is Formed by a Plane Mirror?<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/image-formed-by-reflection-real-or-virtual\/\">Is an Image formed by Reflection Real or Virtual<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/reflection-light-spherical-mirror\/\">Reflection of Light from Spherical Mirror<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/concave-and-convex-mirrors\/\">What are Concave and Convex Mirrors?<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/image-formed-spherical-mirror\/\">How is the Image Formed by a Spherical Mirror?<\/a><\/li>\n<\/ul>\n<h2><span style=\"color: #00ccff;\"><strong>Focal Length related to Radius of Curvature Example Problems with Solutions<\/strong><\/span><\/h2>\n<p><strong><span style=\"color: #993366;\">Example 1:<\/span> \u00a0 \u00a0<\/strong>An object is placed in front of a plane mirror. If the mirror is moved away from the object through a distance x, by how much distance will the image move?<br \/>\n<strong><span style=\"color: #008000;\">Solution:<\/span> \u00a0 \u00a0<\/strong>Suppose the object O was initially at a distance d from the plane mirror M as shown in fig. The image formed at O&#8217; is at a distance d behind the mirror. Now, the mirror is shifted by a distance x to M&#8217; such that the distance of the object from M&#8217; becomes d + x. The image now formed at O&#8221;\u00a0which is also at a distance d + x from M&#8217;.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-95842 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-1.png\" alt=\"How is Focal Length related to Radius of Curvature 1\" width=\"271\" height=\"154\" \/>So, OM = MO&#8217; = d<br \/>\nOM&#8217; = M&#8217;O&#8221; = d + x<br \/>\nThus, \u00a0\u00a0 OO&#8221; = OM&#8217; + M&#8217;O&#8221; = 2(d + x) \u00a0 &#8230;(1)<br \/>\nwhen\u00a0\u00a0\u00a0\u00a0 OO&#8217; = OM +\u00a0 MO&#8217; = 2d \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &#8230;(2)<br \/>\n\u2234 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 O&#8217;O&#8221; = OO&#8221; \u2013 OO&#8217;<br \/>\n= 2(d + x) \u2013 2d<br \/>\n= 2x<br \/>\nThus, the image is shifted from O&#8217; to O&#8221; by a distance 2x.<\/p>\n<p><strong><span style=\"color: #993366;\">Example 2:<\/span> \u00a0 \u00a0<\/strong>An insect is at a distance of 1.5m from a plane mirror. Calculate the following?<br \/>\n(i) Distance at which the image of the insect is formed.<br \/>\n(ii) distance between the insect and its image.<br \/>\n<strong><span style=\"color: #008000;\">Solution:<\/span> \u00a0 \u00a0<\/strong><br \/>\n(i) The distance of insect from the mirror<br \/>\n= 1.5 m<br \/>\n\u2234 The distance of insect from the mirror is also equal to 1.5 m. The image is formed at 1.5 m behind the mirror.<br \/>\n(ii) The distance between the insect and\u00a0\u00a0 image<br \/>\n= 1.5 + 1.5 = 3m<\/p>\n<p><strong><span style=\"color: #993366;\">Example 3: \u00a0 \u00a0<\/span><\/strong>A concave mirror is made up by cutting a portion of a hollow glass sphere of radius 30 cm. Calculate the focal length of the mirror.<br \/>\n<strong><span style=\"color: #008000;\">Solution: \u00a0 \u00a0<\/span><\/strong>The radius of curvature of the mirror = 30 cm<br \/>\nThus, the focal length of the mirror<br \/>\n=\\(\\frac { 30 cm }{ 2\u00a0} \u00a0\\) = 15 cm<\/p>\n<p><strong><span style=\"color: #993366;\">Example 4:<\/span> \u00a0 \u00a0<\/strong>An object is placed at a distance of 15 cm from a concave mirror of focal length 10 cm. Find the position of the image.<br \/>\n<strong><span style=\"color: #008000;\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-95844 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-2.png\" alt=\"How is Focal Length related to Radius of Curvature 2\" width=\"305\" height=\"218\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-2.png 305w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-2-300x214.png 300w\" sizes=\"(max-width: 305px) 100vw, 305px\" \/>Solution: \u00a0 \u00a0<\/span><\/strong><br \/>\nWe have u = \u201315 cm and <em>f<\/em> = \u201310 cm<br \/>\nUsing the relation,\u00a0\\(\\frac { 1 }{ v } +\\frac { 1 }{ u } =\\frac { 1 }{ f }\\) \u00a0we get<br \/>\n\\(\\frac { 1 }{ v } +\\frac { 1 }{ -15\u00a0} =\\frac { 1 }{ -10\u00a0} \\)<br \/>\nor \u00a0\\(\\frac { 1 }{ v } =\\frac { 1 }{ 15 } -\\frac { 1 }{ 10 } =-\\frac { 1 }{ 30 } \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\\)<br \/>\nor \u00a0v = \u201330 cm<br \/>\nSo the image will be formed 30 cm from the mirror. Since n has a negative sign, the image is formed to the left of the mirror, i.e. in front of the mirror as shown in fig.<\/p>\n<p><strong><span style=\"color: #993366;\">Example 5:<\/span> \u00a0 \u00a0<\/strong>A 3 cm long object is placed perpendicular to the principal axis of a concave mirror. The distance of the object from the mirror is 15 cm, and its image is formed 30 cm from the mirror on the same side of the mirror as the object . Calculate the height of the image formed.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-95846 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-3.png\" alt=\"How is Focal Length related to Radius of Curvature 3\" width=\"305\" height=\"217\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-3.png 305w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-3-300x213.png 300w\" sizes=\"(max-width: 305px) 100vw, 305px\" \/><br \/>\n<strong><span style=\"color: #008000;\">Solution: \u00a0 \u00a0<\/span><\/strong>Here u = \u201315 cm and n = \u201330 cm<br \/>\nSize of the object, h = 2 cm<br \/>\nMagnification, \u00a0 \\(m=\\frac { h&#8217; }{ h } =-\\frac { v }{ u } \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \\)<br \/>\nor \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\\(\\frac { h&#8217; }{ h } =-\\frac { (-30) }{ (-15) } =2 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \\)<br \/>\nor \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0h&#8217; = \u2013 2 \u00d7 h = \u2013 2 \u00d7 3<br \/>\n= \u2013 6 cm<br \/>\nSo the height of the image is 6 cm. The minus sign shows that it is on the lower side of the principal axis, i.e. the image is inverted.<\/p>\n<p><strong><span style=\"color: #993366;\">Example 6:<\/span> \u00a0 \u00a0<\/strong>A 1.4 cm long object is placed perpendicular to the principal axis of a convex mirror of focal length 15 cm at a distance of 10 cm from it. Calculate the following :<br \/>\n(i) location of the image<br \/>\n(ii) height of the image<br \/>\n(iii) nature of the image<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-95848 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-4.png\" alt=\"How is Focal Length related to Radius of Curvature 4\" width=\"299\" height=\"194\" \/><br \/>\n<strong><span style=\"color: #008000;\">Solution:<\/span> \u00a0 \u00a0<\/strong><br \/>\n(i) For a convex mirror, focal length is positive.<br \/>\nTherefore, f = +15 cm and u = \u201310 cm<br \/>\nUsing the relation, \\(\\frac { 1 }{ v } +\\frac { 1 }{ u } =\\frac { 1 }{ f } \\), we get<br \/>\n\\(\\frac { 1 }{ v } +\\frac { 1 }{ -10 } =\\frac { 1 }{ 15 } \\)<br \/>\nor \\(\\frac { 1 }{ v } =\\frac { 1 }{ 15 } +\\frac { 1 }{ 10 } =\\frac { 5 }{ 30 } =\\frac { 1 }{ 6 } \\)<br \/>\nor v = 6 cm<br \/>\nSince v is positive, the image is formed to the right of the mirror at a distance 6 cm from it.<br \/>\n(ii) Magnification,<br \/>\nor \\(m=\\frac { h&#8217; }{ h } =-\\frac { v }{ u } \\)<br \/>\n\\(\\frac { h&#8217; }{ h } =\\frac { (-6) }{ (-10) } =+0.6 \\)<br \/>\nor h&#8217; = + 0.6 \u00d7 h<sub>0<\/sub> = 0.6 \u00d7 1.4<br \/>\n= 0.84 cm.<br \/>\nThus, the height of the image is 0.84 cm.<br \/>\n(iii) Since h&#8217; is positive, the image will be on the same side of the principal axis as the object.<br \/>\nHence, the image is virtual, erect and diminished.<\/p>\n<p><strong><span style=\"color: #993366;\">Example 7: \u00a0\u00a0<\/span><\/strong> An object is placed at a distance of 40 cm from a convex mirror of focal length 30 cm. Find the position of image and its nature.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-95852 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/How-is-Focal-Length-related-to-Radius-of-Curvature-5.png\" alt=\"How is Focal Length related to Radius of Curvature 5\" width=\"299\" height=\"193\" \/><br \/>\n<strong><span style=\"color: #008000;\">Solution:<\/span><\/strong> Here, object distance, u = \u201340 cm<br \/>\nFocal length of convex mirror, f = +30 cm<br \/>\nNow, using mirror formula, \\(\\frac { 1 }{ v } +\\frac { 1 }{ u } =\\frac { 1 }{ f } \\) we get<br \/>\n\\(\\frac { 1 }{ v } +\\frac { 1 }{ -40 } =\\frac { 1 }{ 30 } \\)<br \/>\nor \\(\\frac { 1 }{ v } =\\frac { 1 }{ 40 } +\\frac { 1 }{ 30 } =\\frac { 7 }{ 120 } \\)<br \/>\nor v\u00a0= \\(\\frac { 120 }{ 7 } \\)<br \/>\nThe positive sign shows that the image is formed on the right, i.e. behind the mirror.<br \/>\nNow, magnification,<br \/>\n\u21d2 \\(m=-\\frac { v }{ u } =-\\frac { 120 }{ 7\\times (-40) } =+\\frac { 3 }{ 7 } \\)<br \/>\nSince, the magnification is positive, the image is erect. Thus, the image is formed 17.1 cm behind the mirror. The image is virtual, erect and diminished.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>How is Focal Length related to Radius of Curvature? Mirror formula Definition : The equation relating the object distance (u) the image distance (v) and the mirror focal length (f) is called the mirror formula. Assumptions made : The mirror has a small aperture. The object lies close to principal axis of the mirror. 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Mirror formula Definition : The equation relating the object distance (u) the image distance (v) and the mirror focal length (f) is called the mirror formula. Assumptions made : The mirror has a small aperture. The object lies close to principal axis of the mirror. The ... 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