{"id":919,"date":"2024-07-27T05:57:05","date_gmt":"2024-07-27T05:57:05","guid":{"rendered":"https:\/\/www.meniit.com\/study-material\/?p=919"},"modified":"2024-10-04T05:33:19","modified_gmt":"2024-10-04T05:33:19","slug":"gausss-law","status":"publish","type":"post","link":"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law","title":{"rendered":"Gauss&#8217;s Law"},"content":{"rendered":"<p style=\"text-align: justify;\">Gauss law relates the flux through a closed surface (a surface that encloses some volume) with charges present inside the surface.\u201c<br \/>\nConsider a point charge q surrounded by a spherical surface of radius <em>r<\/em> centered on the charge. The magnitude of the electric field everywhere<\/p>\n<p style=\"text-align: justify;\">on the surface on the sphere is E = k q\/r<sup>24<\/sup><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1076 aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaissian-surface.png\" alt=\"gaussian surface\" width=\"205\" height=\"140\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaissian-surface.png 205w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaissian-surface-95x65.png 95w\" sizes=\"auto, (max-width: 205px) 100vw, 205px\" \/><\/p>\n<p style=\"text-align: justify;\">The electric field is perpendicular to the spherical surface at all points on the surface. The electric flux through the surface is therefore EA, where A = 4\u03c0r\u00b2 is the surface area of the sphere:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-920 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-8.png\" alt=\"formula\" width=\"887\" height=\"411\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-8.png 887w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-8-300x139.png 300w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-8-768x356.png 768w\" sizes=\"auto, (max-width: 887px) 100vw, 887px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-921 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/note.png\" alt=\"note\" width=\"733\" height=\"139\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/note.png 733w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/note-300x57.png 300w\" sizes=\"auto, (max-width: 733px) 100vw, 733px\" \/><\/p>\n<p>&nbsp;<\/p>\n<div style=\"margin: 5px; padding: 10px; background-color: #fbdfed;\">\n<h4 id=\"example-solution\" style=\"color: #cc1d74;\"><strong>Example : <\/strong><\/h4>\n<p>Find out flux through the given Gaussian surface.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 \" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-9.png\" alt=\"formula\" width=\"214\" height=\"146\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/solution-7.png\" alt=\"solution\" width=\"923\" height=\"372\" \/><\/p>\n<\/div>\n<div style=\"margin: 5px; padding: 10px; background-color: #fbdfed;\">\n<h4 id=\"example-solution\" style=\"color: #cc1d74;\"><strong>Example : <\/strong><\/h4>\n<p>If a point charge q is placed at the centre of a cube, then find out flux through any one face of cube.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/solution-8.png\" alt=\"solution\" width=\"923\" height=\"372\" \/><\/p>\n<\/div>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_69_1 counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law\/#Applications-of-Gausss-Law\" title=\"Applications of Gauss&#8217;s Law\">Applications of Gauss&#8217;s Law<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law\/#Selection-of-Gaussian-Surface-to-calculate-Electric-Field\" title=\"Selection of Gaussian Surface to calculate Electric Field :\">Selection of Gaussian Surface to calculate Electric Field :<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law\/#Field-due-to-a-Point-Charge\" title=\"Field due to a Point Charge\">Field due to a Point Charge<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law\/#Field-due-to-Long-uniformly-charged-Hollow-Cylinder-of-Charge\" title=\"Field due to Long uniformly charged Hollow Cylinder of Charge\">Field due to Long uniformly charged Hollow Cylinder of Charge<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law\/#Field-due-to-Long-uniformly-charged-Solid-Cylinder-of-Charge\" title=\"Field due to Long uniformly charged Solid Cylinder of Charge\">Field due to Long uniformly charged Solid Cylinder of Charge<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law\/#Field-due-to-a-uniformly-charged-thin-spherical-shell\" title=\"Field due to a uniformly charged thin spherical shell\">Field due to a uniformly charged thin spherical shell<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law\/#Application\" title=\"Application :\">Application :<\/a><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Applications-of-Gausss-Law\"><\/span>Applications of Gauss&#8217;s Law<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Calculation of flux through surface ABCD by usage of symmetry :<\/strong><\/p>\n<p style=\"text-align: justify;\">(a) A charge q is placed at the corner of a cube<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-922 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram.png\" alt=\"diagram\" width=\"317\" height=\"247\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram.png 317w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-300x234.png 300w\" sizes=\"auto, (max-width: 317px) 100vw, 317px\" \/><\/p>\n<p style=\"text-align: justify;\">(b) By surrounding the charge with a series of cubes (total 8 cubes) such that the charge is at the centre of a larger cube, we have created an arrangement sufficiently symmetric to be able to solve for desired flux values.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-980 aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-10-300x66.png\" alt=\"formula \" width=\"300\" height=\"66\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-10-300x66.png 300w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-10.png 307w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-982 aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-11-300x80.png\" alt=\"formula\" width=\"300\" height=\"80\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-11-300x80.png 300w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-11.png 318w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-981 aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gauss-law.png\" alt=\"gauss law\" width=\"265\" height=\"205\" \/><\/p>\n<p>&nbsp;<\/p>\n<div style=\"margin: 5px; padding: 10px; background-color: #fbdfed;\">\n<h4 id=\"example-solution\" style=\"color: #cc1d74;\"><strong>Example : <\/strong><\/h4>\n<p>A point charge +q is placed at the centre of curvature of a hemisphere. Find flux through the hemispherical surface.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 \" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-1.png\" alt=\"diagram\" width=\"138\" height=\"106\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 \" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-5.png\" alt=\"diagram\" width=\"177\" height=\"174\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/solution-9.png\" alt=\"solution\" width=\"923\" height=\"372\" \/><\/p>\n<\/div>\n<div style=\"margin: 5px; padding: 10px; background-color: #fbdfed;\">\n<h4 id=\"example-solution\" style=\"color: #cc1d74;\"><strong>Example : <\/strong><\/h4>\n<p>A charge Q is placed at a distance a\/2 above the centre of a horizontal, square surface of edges as shown in figure. Find the flux of the electric field through the square surface<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-550 aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-2.png\" alt=\"diagram\" width=\"115\" height=\"91\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/solution-2.jpg\" alt=\"solution\" width=\"923\" height=\"372\" \/><\/p>\n<\/div>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Selection-of-Gaussian-Surface-to-calculate-Electric-Field\"><\/span>Selection of Gaussian Surface to calculate Electric Field :<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">With the help of extensive integration and considering a system which possess spherical, cylindrical<br \/>\nor planar symmetry we can calculate electric field by Gauss law.<\/p>\n<table>\n<tbody>\n<tr>\n<th class=\"thhead\">System<\/th>\n<th class=\"thhead\">Gaussian Surface<\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"345\">Point charge, Hollow Sphere, Solid Sphere<\/td>\n<td style=\"text-align: center;\" width=\"130\">Spherical<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"345\">Infinite Wire, Infinite Hollow Cylinder, Infinite Solid Cylinder<\/td>\n<td style=\"text-align: center;\" width=\"130\">Cylindrical<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"345\">Infinite Sheet<\/td>\n<td style=\"text-align: center;\" width=\"130\">Gaussian Pillbox<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\"><strong>Let us understand how to apply Gauss Law with different cases :<\/strong><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Field-due-to-a-Point-Charge\"><\/span>Field due to a Point Charge<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">First, we shall look for the symmetry of the field. Clearly the field is spherically symmetric. If we<br \/>\nenclose the charge in a sphere of radius R, the magnitude of electric field will be same at any point on the<br \/>\nsurface.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-997 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-surface.png\" alt=\"gaussian surface\" width=\"836\" height=\"332\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-surface.png 836w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-surface-300x119.png 300w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-surface-768x305.png 768w\" sizes=\"auto, (max-width: 836px) 100vw, 836px\" \/><\/p>\n<p><strong>Field due to an infinitely long straight uniformly charged wire<\/strong><\/p>\n<p style=\"text-align: justify;\">First, we shall look for the symmetry of electric field. Consider any point P. at a distance r from the line charge.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-999 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/charged-wired.png\" alt=\"charged wired\" width=\"698\" height=\"245\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/charged-wired.png 698w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/charged-wired-300x105.png 300w\" sizes=\"auto, (max-width: 698px) 100vw, 698px\" \/><\/p>\n<p style=\"text-align: justify;\">If a long linear charge distribution is kept along x-axis, at any point, field is directed radially away from x-axis. The field has a cylindrical symmetry.<br \/>\nTo find electric field, we enclose the distribution in a Gaussian cylinder of radius r and length l<\/p>\n<p>\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1001 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-3.png\" alt=\"\" width=\"674\" height=\"510\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-3.png 674w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-3-300x227.png 300w\" sizes=\"auto, (max-width: 674px) 100vw, 674px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Field-due-to-Long-uniformly-charged-Hollow-Cylinder-of-Charge\"><\/span>Field due to Long uniformly charged Hollow Cylinder of Charge<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Hollow cylinder with charge per unit length \u03bb<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1002 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/cylinder-of-charge.png\" alt=\"cylinder of charger\" width=\"403\" height=\"217\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/cylinder-of-charge.png 403w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/cylinder-of-charge-300x162.png 300w\" sizes=\"auto, (max-width: 403px) 100vw, 403px\" \/><\/p>\n<p>(a) Inside the cylinder (radial distance r &lt; R):<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1004 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-cylinder.png\" alt=\"gaussian cylinder\" width=\"423\" height=\"231\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-cylinder.png 423w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-cylinder-300x164.png 300w\" sizes=\"auto, (max-width: 423px) 100vw, 423px\" \/><\/p>\n<p style=\"text-align: justify;\">When we draw a Gaussian cylinder of radius r, we find that the charge enclosed by it is zero.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1006 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-12.png\" alt=\"formula\" width=\"317\" height=\"80\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-12.png 317w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-12-300x76.png 300w\" sizes=\"auto, (max-width: 317px) 100vw, 317px\" \/><\/p>\n<p style=\"text-align: justify;\">(b) Outside the cylinder (radial distance r &gt; R) :<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1007 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/outside-the-cylinder.png\" alt=\"outside the cylinder\" width=\"555\" height=\"378\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/outside-the-cylinder.png 555w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/outside-the-cylinder-300x204.png 300w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/outside-the-cylinder-95x65.png 95w\" sizes=\"auto, (max-width: 555px) 100vw, 555px\" \/><\/p>\n<p style=\"text-align: justify;\">Variation of E with r is also shown graphically here.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1008 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-6.png\" alt=\"diagram\" width=\"247\" height=\"178\" \/><\/p>\n<p>&nbsp;<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Field-due-to-Long-uniformly-charged-Solid-Cylinder-of-Charge\"><\/span>Field due to Long uniformly charged Solid Cylinder of Charge<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Let \u03c1 be the volumetric charge density and R be the radius of the cylinder. Here, the field is radially away from x-axis having a cylindrical symmetry.<\/p>\n<p style=\"text-align: justify;\"><strong>Case 1:<\/strong> r &gt; R. Consider a Gaussian cylinder of length \/ and radius r about x-axis. Electric field lines are directed radially away. Let E be the magnitude of electric field, then \u03c6 = E \u00d7 2\u03c0rl<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1009 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-7.png\" alt=\"diagram\" width=\"624\" height=\"434\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-7.png 624w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-7-300x209.png 300w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-7-95x65.png 95w\" sizes=\"auto, (max-width: 624px) 100vw, 624px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1011 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-8.png\" alt=\"diagram\" width=\"626\" height=\"388\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-8.png 626w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-8-300x186.png 300w\" sizes=\"auto, (max-width: 626px) 100vw, 626px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1013 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-9.png\" alt=\"diagram\" width=\"664\" height=\"181\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-9.png 664w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-9-300x82.png 300w\" sizes=\"auto, (max-width: 664px) 100vw, 664px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Field-due-to-a-uniformly-charged-thin-spherical-shell\"><\/span>Field due to a uniformly charged thin spherical shell<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Consider a shell having a charge Q uniformly distributed on its surface. The surface charge density is \u03c3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1015 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-13.png\" alt=\"formula\" width=\"423\" height=\"60\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-13.png 423w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-13-300x43.png 300w\" sizes=\"auto, (max-width: 423px) 100vw, 423px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1017 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-6-1.png\" alt=\"diagram\" width=\"194\" height=\"161\" \/><\/p>\n<h4>Field outside the shell:<\/h4>\n<p><strong>Case 1:<\/strong> Field outside the shell (radial distance r &gt; R) We enclose the shell in a Gaussian sphere.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1018 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-14.png\" alt=\"formula\" width=\"362\" height=\"230\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-14.png 362w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-14-300x191.png 300w\" sizes=\"auto, (max-width: 362px) 100vw, 362px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1019 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-sphere.png\" alt=\"gaussian sphere\" width=\"262\" height=\"208\" \/><\/p>\n<h4>Field inside the shell:<\/h4>\n<p><strong>Case 2:<\/strong> Field inside the shell (radial distance r &lt; R)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1021 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/field-inside-shell.png\" alt=\"field inside shell\" width=\"587\" height=\"256\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/field-inside-shell.png 587w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/field-inside-shell-300x131.png 300w\" sizes=\"auto, (max-width: 587px) 100vw, 587px\" \/><\/p>\n<h4>Field due to a uniformly charged Solid Sphere<\/h4>\n<p>Consider a uniform spherical charge distribution in which a charge Q is uniformly distributed over the volume of a sphere of radius R.<br \/>\nThe volumetric charge density is given by,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1025 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-15.png\" alt=\"formula\" width=\"446\" height=\"85\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-15.png 446w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-15-300x57.png 300w\" sizes=\"auto, (max-width: 446px) 100vw, 446px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1023 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/solid-sphere.png\" alt=\"solid sphere\" width=\"157\" height=\"128\" \/><\/p>\n<h4>Field outside the shell :<\/h4>\n<p><strong>Case 1:<\/strong> Field outside the sphere (radial distance r &gt; R). The field has spherical symmetry<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1026 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula.jpg\" alt=\"formula\" width=\"380\" height=\"215\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula.jpg 380w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-300x170.jpg 300w\" sizes=\"auto, (max-width: 380px) 100vw, 380px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1027 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-10.png\" alt=\"diagram\" width=\"268\" height=\"164\" \/><\/p>\n<h4>Field inside the shell :<\/h4>\n<p><strong>Case 2 :<\/strong> Field inside the sphere (radial distance r &lt; R).<br \/>\nConsider a Gaussian sphere inside the sphere of charge.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1028 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-16.png\" alt=\"formula\" width=\"401\" height=\"228\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-16.png 401w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-16-300x171.png 300w\" sizes=\"auto, (max-width: 401px) 100vw, 401px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1029 aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-11.png\" alt=\"diagram\" width=\"218\" height=\"259\" \/><\/p>\n<p><strong>Field due to a uniformly charged infinite plane sheet<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Non Conducting Sheet :<\/strong> Charge will be uniformly distributed on both the sides of the sheet. For Conducting sheets calculation will be different and we will learn that in Electrostatics of Conductors.<br \/>\n1. An infinitely large plane possesses a planar symmetry<br \/>\n2. Since the charge is uniformly distributed on the surface, the electric field must point perpendicular<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-1030 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-22-151141.png\" alt=\"\" width=\"630\" height=\"47\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-22-151141.png 630w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/Screenshot-2024-07-22-151141-300x22.png 300w\" sizes=\"auto, (max-width: 630px) 100vw, 630px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify;\">We choose our Gaussian surface to be a cylinder, which is often referred to as a &#8220;pillbox&#8221;. The pillbox all consists of three parts, two end-caps s<sub>1<\/sub> and s<sub>2<\/sub> and a curved side s<sub>3<\/sub>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1032 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-pillbox.png\" alt=\"gaussian pillbox \" width=\"591\" height=\"307\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-pillbox.png 591w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/gaussian-pillbox-300x156.png 300w\" sizes=\"auto, (max-width: 591px) 100vw, 591px\" \/><\/p>\n<p>3. Since the surface charge distribution is uniform the charge enclosed by the Gaussian &#8220;pillbox\u201d is q<sub>enc<\/sub> = \u03c3A where<br \/>\nA = A<sub>1<\/sub> = A<sub>2<\/sub> is the area of the end-caps.<\/p>\n<p>4. The total flux through the Gaussian pillbox is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1033 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-17.png\" alt=\"formula\" width=\"322\" height=\"79\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-17.png 322w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-17-300x74.png 300w\" sizes=\"auto, (max-width: 322px) 100vw, 322px\" \/><\/p>\n<p style=\"text-align: justify;\">Since the two ends are at the same distance from the plane, by symmetry, the magnitude of the electric field must be the same :<br \/>\nE<sub>1<\/sub> = E<sub>2<\/sub> = E. Hence, the total flux can be rewritten as \u03c6<sub>E<\/sub>= 2EA<\/p>\n<p style=\"text-align: justify;\">5. By applying Gauss&#8217;s law, we obtain<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1037 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-18.png\" alt=\"formula\" width=\"378\" height=\"231\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-18.png 378w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-18-300x183.png 300w\" sizes=\"auto, (max-width: 378px) 100vw, 378px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1038 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-19.png\" alt=\"formula\" width=\"568\" height=\"183\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-19.png 568w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/formula-19-300x97.png 300w\" sizes=\"auto, (max-width: 568px) 100vw, 568px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Application\"><\/span>Application :<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Two parallel sheets are given surface charge densities \u03c3<sub>1<\/sub> and \u03c3<sub>2<\/sub>. Electric fields in different regions are as shown<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1043 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-12.png\" alt=\"diagram\" width=\"320\" height=\"203\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-12.png 320w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-12-300x190.png 300w\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" \/><\/p>\n<p style=\"text-align: justify;\">Consider the following specific cases :<br \/>\n(i) When \u03c3<sub>1<\/sub> = \u03c3, \u03c3<sub>2<\/sub> = \u03c3, the situation will be like the one shown below<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1045 size-full aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-13.png\" alt=\"diagram \" width=\"196\" height=\"179\" \/><\/p>\n<p style=\"text-align: justify;\">(ii) When \u03c3<sub>1<\/sub> = \u03c3, \u03c3<sub>2<\/sub> = \u2013 \u03c3, the situation will be like the one shown below<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1049 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/diagram-14.png\" alt=\"diagram \" width=\"197\" height=\"168\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div style=\"margin: 5px; padding: 10px; background-color: #fbdfed;\">\n<h4 id=\"example-solution\" style=\"color: #cc1d74;\"><strong>Example : <\/strong><\/h4>\n<p>Two concentric shells of radii a and b carry charge q1 and q2 (uniformly distributed) respectively (a &lt; b). Calculate electric field at radial distance r from the centre for (i) r &lt; a (ii) b &gt; r &gt; a (iii) r &gt; b. Represent this field graphically.<\/p>\n<h4 style=\"color: #cc1d74;\"><strong>Solution : <\/strong><\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-550 alignnone\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/electric-field-1.png\" width=\"855\" height=\"732\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"margin: 5px; padding: 10px; background-color: #fbdfed;\">\n<h4 id=\"example-solution\" style=\"color: #cc1d74;\"><strong>Example : <\/strong><\/h4>\n<p>Figure shows a long thread along the axis of a long hollow cylinder. The charge per unit length of thread is \u03bb, while that of the cylinder is \u03bb, The radius of cylinder is R. What is the electric field at radial distance r from the axis for (i) r &lt; R (ii) r &gt; R?<\/p>\n<h4 style=\"color: #cc1d74;\"><strong>Solution : <\/strong><\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/electric-field-a.png\" width=\"923\" height=\"372\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"margin: 5px; padding: 10px; background-color: #fbdfed;\">\n<h4 id=\"example-solution\" style=\"color: #cc1d74;\"><strong>Example : <\/strong><\/h4>\n<p>A large plane conducting sheet is given a charge so that its surface charge density becomes \u03c3. Describe the electric field produced by it, inside the sheet and outside the sheet.<\/p>\n<h4 style=\"color: #cc1d74;\"><strong>Solution : <\/strong><\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-550 size-full\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/electric-field-b.png\" width=\"923\" height=\"372\" \/><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1060 aligncenter\" src=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/electric-field-c.png\" alt=\"electric-field-c\" width=\"895\" height=\"995\" srcset=\"https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/electric-field-c.png 895w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/electric-field-c-270x300.png 270w, https:\/\/www.meniit.com\/study-material\/wp-content\/uploads\/2024\/07\/electric-field-c-768x854.png 768w\" sizes=\"auto, (max-width: 895px) 100vw, 895px\" \/><\/p>\n<p>&nbsp;<\/p>\n<div class=\"newspaper-x-tags\"><strong>TAGS: <\/strong><span><a href=\"https:\/\/www.meniit.com\/study-material\/tag\/applications-of-gausss-law\" rel=\"tag\">Applications of Gauss&#8217;s Law<\/a><\/span><a href=\"https:\/\/www.meniit.com\/study-material\/tag\/field-due-to-a-point-charge\" rel=\"tag\">Field due to a Point Charge<\/a><\/span><a href=\"https:\/\/www.meniit.com\/study-material\/tag\/gauss-law\" rel=\"tag\">Gauss Law<\/a><\/span><a href=\"https:\/\/www.meniit.com\/study-material\/tag\/solid-cylinder-of-charge\" rel=\"tag\">Solid Cylinder of Charge<\/a><\/span><a href=\"https:\/\/www.meniit.com\/study-material\/tag\/spherical-shell\" rel=\"tag\">spherical shell<\/a> <\/div>\n","protected":false},"excerpt":{"rendered":"<p>Gauss law relates the flux through a closed surface (a surface that encloses some volume) with charges present inside the&nbsp;&nbsp;&#8230;.<a class=\"read_more\" href=\"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/gausss-law\" rel=\"nofollow\">Read More >><\/a><\/p>\n","protected":false},"author":5,"featured_media":1237,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"om_disable_all_campaigns":false,"rank_math_lock_modified_date":false,"footnotes":""},"categories":[255,256,253],"tags":[375,376,374,377,378],"class_list":["post-919","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11th-physics","category-class-xiith","category-jee","tag-applications-of-gausss-law","tag-field-due-to-a-point-charge","tag-gauss-law","tag-solid-cylinder-of-charge","tag-spherical-shell"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/posts\/919","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/comments?post=919"}],"version-history":[{"count":10,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/posts\/919\/revisions"}],"predecessor-version":[{"id":1604,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/posts\/919\/revisions\/1604"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/media\/1237"}],"wp:attachment":[{"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/media?parent=919"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/categories?post=919"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/tags?post=919"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}