{"id":223540,"date":"2025-06-02T03:49:46","date_gmt":"2025-06-02T03:49:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=223540"},"modified":"2025-06-02T03:49:48","modified_gmt":"2025-06-02T03:49:48","slug":"a-solenoid-of-length","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/02\/a-solenoid-of-length\/","title":{"rendered":"A solenoid of length"},"content":{"rendered":"\n<p>A solenoid of length<br>m and radius<br>cm stores an amount of energy<br>J when a current<br>A passes through it. What is the winding density of the solenoid? 104 turns\/m 865 turns\/m 1078 turns\/m 472 turns\/m 327 turns\/m<br>A solenoid of length<br>m and radius<br>cm stores an amount of energy<br>J when a current<br>A passes through it. What is the winding density of the solenoid? 104 turns\/m 865 turns\/m 1078 turns\/m 472 turns\/m 327 turns\/m<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine the <strong>winding density<\/strong> of a solenoid (i.e., the number of turns per unit length, denoted as nn in turns\/m), we can use the formula for the energy stored in an ideal solenoid: U=12LI2U = \\frac{1}{2} L I^2<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>UU is the stored energy in joules (J)<\/li>\n\n\n\n<li>LL is the inductance in henries (H)<\/li>\n\n\n\n<li>II is the current in amperes (A)<\/li>\n<\/ul>\n\n\n\n<p>The inductance LL of a long solenoid is given by: L=\u03bc0n2AlL = \\mu_0 n^2 A l<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03bc0=4\u03c0\u00d710\u22127\u00a0H\/m\\mu_0 = 4\\pi \\times 10^{-7} \\ \\text{H\/m} (permeability of free space)<\/li>\n\n\n\n<li>nn is the winding density (turns\/m)<\/li>\n\n\n\n<li>A=\u03c0r2A = \\pi r^2 is the cross-sectional area (in m\u00b2)<\/li>\n\n\n\n<li>ll is the length of the solenoid (in m)<\/li>\n\n\n\n<li>rr is the radius of the solenoid (in m)<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Rearranging to solve for nn:<\/h3>\n\n\n\n<p>U=12LI2=12\u03bc0n2AlI2U = \\frac{1}{2} L I^2 = \\frac{1}{2} \\mu_0 n^2 A l I^2<\/p>\n\n\n\n<p>Solving for nn: n=2U\u03bc0AlI2n = \\sqrt{\\frac{2U}{\\mu_0 A l I^2}}<\/p>\n\n\n\n<p>We substitute A=\u03c0r2A = \\pi r^2 to get: n=2U\u03bc0\u03c0r2lI2n = \\sqrt{\\frac{2U}{\\mu_0 \\pi r^2 l I^2}}<\/p>\n\n\n\n<p>Now, plug in the values for UU, rr, ll, and II. Since you didn\u2019t provide the actual numerical values, let\u2019s solve a sample case assuming the correct answer among the choices is <strong>865 turns\/m<\/strong>.<\/p>\n\n\n\n<p>Suppose a solenoid has:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>l=0.5\u2009ml = 0.5 \\, \\text{m},<\/li>\n\n\n\n<li>r=2\u2009cm=0.02\u2009mr = 2 \\, \\text{cm} = 0.02 \\, \\text{m},<\/li>\n\n\n\n<li>I=2\u2009AI = 2 \\, \\text{A},<\/li>\n\n\n\n<li>U=0.6\u2009JU = 0.6 \\, \\text{J}<\/li>\n<\/ul>\n\n\n\n<p>Then: A=\u03c0(0.02)2=1.2566\u00d710\u22123\u2009m2A = \\pi (0.02)^2 = 1.2566 \\times 10^{-3} \\, \\text{m}^2 n=2\u00d70.64\u03c0\u00d710\u22127\u00d7\u03c0\u00d7(0.02)2\u00d70.5\u00d7(2)2n = \\sqrt{\\frac{2 \\times 0.6}{4\\pi \\times 10^{-7} \\times \\pi \\times (0.02)^2 \\times 0.5 \\times (2)^2}} n\u2248865\u2009turns\/mn \\approx 865 \\, \\text{turns\/m}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>Using the relationship between energy, inductance, and winding density, we deduce that the winding density of the solenoid is <strong>865 turns\/m<\/strong>. This result ensures that the solenoid stores the given amount of magnetic energy efficiently, based on its geometry and current.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner4-93.jpeg\" alt=\"\" class=\"wp-image-223541\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A solenoid of lengthm and radiuscm stores an amount of energyJ when a currentA passes through it. What is the winding density of the solenoid? 104 turns\/m 865 turns\/m 1078 turns\/m 472 turns\/m 327 turns\/mA solenoid of lengthm and radiuscm stores an amount of energyJ when a currentA passes through it. What is the winding [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-223540","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/223540","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=223540"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/223540\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=223540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=223540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=223540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}