{"id":2055,"date":"2026-07-16T13:02:25","date_gmt":"2026-07-16T13:02:25","guid":{"rendered":"https:\/\/neetapp.in\/blog\/?p=2055"},"modified":"2026-07-16T13:02:26","modified_gmt":"2026-07-16T13:02:26","slug":"supplementary-units-nta-trap-ncert-concepts-mcqs","status":"publish","type":"post","link":"https:\/\/neetapp.in\/blog\/supplementary-units-nta-trap-ncert-concepts-mcqs\/","title":{"rendered":"Supplementary Units &#8211; NTA TRAP! \u2013 NCERT Concepts + MCQs"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Listen up, NEET aspirants<sup><\/sup>. Today, we are tackling a classic trap that the NTA loves to set in the Units and Measurements chapter<sup><\/sup>. Usually, if a physical quantity has a unit, it also has a dimensional formula<sup><\/sup>. But there is a massive exception to this standard rule: The Supplementary Physical Quantities<sup><\/sup>. Let us break them down.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The Dimensional Exception Principle<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In the SI system, there are two distinct supplementary units you must memorize<sup><\/sup>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Plane Angle ($d\\theta$)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>This is the 2D angle subtended at the center of a circle by an arc.<\/li>\n\n\n\n<li><strong>Formula:<\/strong> It is mathematically defined as the ratio of the arc length ($ds$) to the radius ($r$), giving us $d\\theta = \\frac{ds}{r}$.<\/li>\n\n\n\n<li><strong>Unit:<\/strong> Radian ($\\text{rad}$).<\/li>\n\n\n\n<li><strong>Pro-Tip:<\/strong> To convert degrees to radians, remember that $1^\\circ = \\frac{\\pi}{180} \\text{ radians}$. If you ever see a trig function without a degree symbol (e.g., $\\sin 15$), you must assume that the value is in radians.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Solid Angle ($d\\Omega$)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>This is the 3D equivalent, representing the angle subtended at the center of a sphere by a patch of surface area.<\/li>\n\n\n\n<li><strong>Formula:<\/strong> It is defined as the ratio of the intercepted area ($dA$) to the square of the radius ($r^2$), giving us $d\\Omega = \\frac{dA}{r^2}$.<\/li>\n\n\n\n<li><strong>Unit:<\/strong> Steradian ($\\text{sr}$).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The NEET High-Yield Secret:<\/strong> Look closely at those formulas<sup><\/sup>. Plane angle is length divided by length, and solid angle is area divided by area<sup><\/sup>. Because they are ratios of similar physical dimensions, the dimensions completely cancel out<sup><\/sup>! This makes their dimensional formula exactly $[M^0L^0T^0]$<sup><\/sup>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Memorize this fundamental rule right now: <strong>A physical quantity can have a unit without having a dimension, but a dimensionless quantity can only have a unit if it is a supplementary angle<\/strong><sup><\/sup>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The Golden Rule for Trigonometric Arguments<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Another high-yield mock paper concept tied to dimensions is the constraint of arguments<sup><\/sup>. The golden rule is this: <strong>the argument of any trigonometric function must be completely dimensionless<\/strong><sup><\/sup>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example, consider a wave equation represented by $y = A \\sin(aEt &#8211; bVx)$<sup><\/sup>. Because the argument of the sine function must be dimensionless, both the term $(aEt)$ and the term $(bVx)$ must have no dimensions (equaling $[M^0L^0T^0]$)<sup><\/sup>. If you know the dimensions of energy ($E$), time ($t$), velocity ($V$), and distance ($x$), you can easily calculate the unknown dimensions of $a$ and $b$ by equating them to one over the variables&#8217; dimensions<sup><\/sup>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Advanced Application: Solid Angle Flux in Electrostatics<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">This supplementary concept perfectly blends with Electrostatics\u2014a very common profile in top mock tests<sup><\/sup>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Imagine a point charge $q$ is placed on the axis of a disc, such that the disc subtends a cone of half-angle $\\theta = 45^\\circ$ at the position of the charge<sup><\/sup>. What is the electric flux ($\\phi$) passing through the disc<sup><\/sup>?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First, the solid angle ($\\Omega$) subtended by a cone of half-angle $\\theta$ is mathematically defined as:$$\\Omega = 2\\pi(1 &#8211; \\cos\\theta)\\text{ steradians}$$.<\/li>\n\n\n\n<li>Applying Gauss&#8217;s law, a charge $q$ radiates a total flux of $\\frac{q}{\\epsilon_0}$ across a full 3D sphere of $4\\pi$ steradians.<\/li>\n\n\n\n<li>To find the flux through just our disc, we multiply that total flux by the ratio of our solid angle to $4\\pi$.<\/li>\n\n\n\n<li>Therefore, $\\phi = \\frac{q}{\\epsilon_0}\\left(\\frac{\\Omega}{4\\pi}\\right)$.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Test Your Knowledge: NEET MCQs on Supplementary Units<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 1: The Dimensional Exception Principle (NEET 2022)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Plane angle and solid angle have:<sup><\/sup><\/p>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li>Both units and dimensions<\/li>\n\n\n\n<li>Units but no dimensions<\/li>\n\n\n\n<li>Dimensions but no units<\/li>\n\n\n\n<li>No units and no dimensions<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 2: Dimension Matching and Ratios (NEET 2024)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The quantities which have the same dimensions as those of solid angle are:<sup><\/sup><\/p>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li>Stress and angle<\/li>\n\n\n\n<li>Strain and arc<\/li>\n\n\n\n<li>Angular speed and stress<\/li>\n\n\n\n<li>Strain and angle<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 3: Solid Angle Flux Application (Aakash Test Series \/ NEET Mock Test)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A point charge $q$ is placed on the axis of a disc such that the disc subtends a cone of half-angle $\\theta = 45^\\circ$ at the position of the charge. The electric flux ($\\phi$) passing through the disc is:<sup><\/sup><\/p>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li>$\\frac{q}{2\\epsilon_0}$<\/li>\n\n\n\n<li>$\\frac{q}{\\epsilon_0}$<\/li>\n\n\n\n<li>$\\frac{\\sqrt{2}(\\sqrt{2}-1)q}{4\\epsilon_0}$<\/li>\n\n\n\n<li>$\\frac{q}{4\\epsilon_0}$<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 4: Argument Dimensionality Constraint (Mock Paper Concept)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In a wave equation represented by $y = A \\sin(aEt &#8211; bVx)$, where $y$ is displacement, $E$ is energy, $V$ is velocity, $t$ is time, and $x$ is distance, the dimensional formula of the ratio $\\frac{a}{b}$ is:<sup><\/sup><\/p>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li>$[M^1L^1T^1]$<\/li>\n\n\n\n<li>$[M^1L^2T^0]$<\/li>\n\n\n\n<li>$[M^{-1}L^0T^0]$<\/li>\n\n\n\n<li>$[MLT^{-2}]$<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 5: Classification in the SI System (NCERT Core Concept)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Which of the following is correct regarding the classification of Plane Angle and Solid Angle in the SI system?<sup><\/sup><\/p>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li>They are classified as Base Units because they are fundamental to geometry.<\/li>\n\n\n\n<li>They are classified as Derived Units because they are formed as ratios of other physical quantities.<\/li>\n\n\n\n<li>They are defined as Supplementary Units, which are dimensionless but possess unique SI units.<\/li>\n\n\n\n<li>They are classified as Improper Units because they lack dimensions.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Solutions<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution 1:<\/strong> Option (2) is correct. While physical quantities with units generally possess dimensions, supplementary quantities are the ultimate exception<sup><\/sup>. They possess units (radian and steradian) but are completely dimensionless ($[M^0L^0T^0]$)<sup><\/sup>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution 2:<\/strong> Option (4) is correct. Solid angle is dimensionless ($[M^0L^0T^0]$)<sup><\/sup>. Any other dimensionless quantities will share its dimensions<sup><\/sup>. Strain (change in dimension over original dimension) and plane angle (arc over radius) are both perfectly dimensionless ratios<sup><\/sup>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution 3:<\/strong> Option (3) is correct.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For $\\theta = 45^\\circ$, the solid angle is $\\Omega = 2\\pi(1 &#8211; \\cos 45^\\circ) = 2\\pi\\left(1 &#8211; \\frac{1}{\\sqrt{2}}\\right) = \\sqrt{2}\\pi(\\sqrt{2}-1)\\text{ sr}$.<\/li>\n\n\n\n<li>Substituting this into the flux ratio gives: $\\phi = \\frac{q}{\\epsilon_0}\\left(\\frac{\\Omega}{4\\pi}\\right) = \\frac{\\sqrt{2}(\\sqrt{2}-1)q}{4\\epsilon_0}$.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution 4:<\/strong> Option (3) is correct.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The argument of the sine function must be dimensionless.<\/li>\n\n\n\n<li>$[aEt] = [M^0L^0T^0] \\implies [a] = [E^{-1}t^{-1}] = [M^{-1}L^{-2}T^1]$.<\/li>\n\n\n\n<li>$[bVx] = [M^0L^0T^0] \\implies [b] = [V^{-1}x^{-1}] = [L^{-2}T^1]$.<\/li>\n\n\n\n<li>The ratio $\\left[\\frac{a}{b}\\right] = \\frac{[M^{-1}L^{-2}T^1]}{[L^{-2}T^1]} = [M^{-1}L^0T^0]$.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution 5:<\/strong> Option (3) is correct. NCERT explicitly notes that besides the seven base units, there are two supplementary units defined: plane angle (radian) and solid angle (steradian)<sup><\/sup>. They are dimensionless quantities that possess unique, official SI units<sup><\/sup>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Want a deeper dive into Units and Dimensions? Watch our full visual breakdown here!<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe title=\"Supplementary Quantities NTA TRAP! \u2013 Concepts + MCQs | Units &amp; Measurements NCERT Physics 11\" width=\"1170\" height=\"658\" src=\"https:\/\/www.youtube.com\/embed\/yg1XzM-Z2Ng?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Make sure to subscribe to the channel and visit NeetApp.in for more high-yield NEET Physics content!<\/strong><\/p>\n<style>\r\n.lwrp.link-whisper-related-posts{\r\n            \r\n            margin-top: 40px;\nmargin-bottom: 30px;\r\n        }\r\n        .lwrp .lwrp-title{\r\n            \r\n            \r\n        }.lwrp .lwrp-description{\r\n            \r\n            \r\n\r\n        }\r\n        .lwrp .lwrp-list-container{\r\n        }\r\n        .lwrp .lwrp-list-multi-container{\r\n            display: flex;\r\n        }\r\n        .lwrp .lwrp-list-double{\r\n            width: 48%;\r\n        }\r\n        .lwrp .lwrp-list-triple{\r\n            width: 32%;\r\n        }\r\n        .lwrp .lwrp-list-row-container{\r\n            display: flex;\r\n            justify-content: space-between;\r\n        }\r\n        .lwrp .lwrp-list-row-container .lwrp-list-item{\r\n            width: calc(25% - 20px);\r\n        }\r\n        .lwrp .lwrp-list-item:not(.lwrp-no-posts-message-item){\r\n            \r\n            max-width: 150px;\r\n        }\r\n        .lwrp .lwrp-list-item img{\r\n            max-width: 100%;\r\n            height: auto;\r\n            object-fit: cover;\r\n            aspect-ratio: 1 \/ 1;\r\n        }\r\n        .lwrp .lwrp-list-item.lwrp-empty-list-item{\r\n            background: initial !important;\r\n        }\r\n        .lwrp .lwrp-list-item .lwrp-list-link .lwrp-list-link-title-text,\r\n        .lwrp .lwrp-list-item .lwrp-list-no-posts-message{\r\n            \r\n            \r\n            \r\n            \r\n        }@media screen and (max-width: 480px) {\r\n            .lwrp.link-whisper-related-posts{\r\n                \r\n                \r\n            }\r\n            .lwrp .lwrp-title{\r\n                \r\n                \r\n            }.lwrp .lwrp-description{\r\n                \r\n                \r\n            }\r\n            .lwrp .lwrp-list-multi-container{\r\n                flex-direction: column;\r\n            }\r\n            .lwrp .lwrp-list-multi-container ul.lwrp-list{\r\n                margin-top: 0px;\r\n                margin-bottom: 0px;\r\n                padding-top: 0px;\r\n                padding-bottom: 0px;\r\n            }\r\n            .lwrp .lwrp-list-double,\r\n            .lwrp .lwrp-list-triple{\r\n                width: 100%;\r\n            }\r\n            .lwrp .lwrp-list-row-container{\r\n                justify-content: initial;\r\n                flex-direction: column;\r\n            }\r\n            .lwrp .lwrp-list-row-container .lwrp-list-item{\r\n                width: 100%;\r\n            }\r\n            .lwrp .lwrp-list-item:not(.lwrp-no-posts-message-item){\r\n                \r\n                max-width: initial;\r\n            }\r\n            .lwrp .lwrp-list-item .lwrp-list-link .lwrp-list-link-title-text,\r\n            .lwrp .lwrp-list-item .lwrp-list-no-posts-message{\r\n                \r\n                \r\n                \r\n                \r\n            };\r\n        }<\/style>\r\n<div id=\"link-whisper-related-posts-widget\" class=\"link-whisper-related-posts lwrp\">\r\n            <h3 class=\"lwrp-title\">Related Posts<\/h3>    \r\n        <div class=\"lwrp-list-container\">\r\n                                            <div class=\"lwrp-list-multi-container\">\r\n                    <ul class=\"lwrp-list lwrp-list-double lwrp-list-left\">\r\n                        <li class=\"lwrp-list-item\"><a href=\"https:\/\/neetapp.in\/blog\/re-neet-2026-june-21-paper-with-solutions-pdf-code-80\/\" class=\"lwrp-list-link\"><img width=\"480\" height=\"480\" src=\"https:\/\/neetapp.in\/blog\/wp-content\/uploads\/2026\/06\/ReNeet-2026-Solved-PDF-Thumb-1-585x585.jpg\" class=\"attachment-480x480 size-480x480 wp-post-image\" alt=\"\" srcset=\"https:\/\/neetapp.in\/blog\/wp-content\/uploads\/2026\/06\/ReNeet-2026-Solved-PDF-Thumb-1-585x585.jpg 585w, https:\/\/neetapp.in\/blog\/wp-content\/uploads\/2026\/06\/ReNeet-2026-Solved-PDF-Thumb-1-150x150.jpg 150w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><br><span class=\"lwrp-list-link-title-text\">Re-Neet 2026 June 21 Paper with Solutions PDF &#8211; 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Today, we are tackling a classic trap that the NTA loves to set in the Units and Measurements&hellip;<\/p>\n","protected":false},"author":1,"featured_media":2057,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[40,39],"tags":[89,85,90,92,86,94,87,83,59,84,88,91,82,93,74,81],"class_list":["post-2055","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-11-physics","category-physics","tag-aakash-mock-test-physics","tag-dimensionless-quantities","tag-electric-flux-solid-angle","tag-manim-physics-animation","tag-ncert-physics-class-11-chapter-2","tag-neet-2022-physics-solutions","tag-neet-2026-physics-solutions","tag-neet-2027-physics","tag-neet-physics","tag-neet-pyqs-solved","tag-nta-neet-syllabus","tag-physics-visual-learning","tag-plane-angle-and-solid-angle","tag-trignometric-argument-dimensions","tag-units-and-dimensions-neet","tag-units-and-measurements-class-11"],"_links":{"self":[{"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/posts\/2055","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/comments?post=2055"}],"version-history":[{"count":1,"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/posts\/2055\/revisions"}],"predecessor-version":[{"id":2058,"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/posts\/2055\/revisions\/2058"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/media\/2057"}],"wp:attachment":[{"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/media?parent=2055"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/categories?post=2055"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/neetapp.in\/blog\/wp-json\/wp\/v2\/tags?post=2055"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}