{"id":222879,"date":"2025-06-01T04:13:40","date_gmt":"2025-06-01T04:13:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=222879"},"modified":"2025-06-01T04:13:42","modified_gmt":"2025-06-01T04:13:42","slug":"a-toroidal-solenoid-has-a-central-radius-of-0-50-m-and-a-cross-sectional-diameter-of-10-cm","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/01\/a-toroidal-solenoid-has-a-central-radius-of-0-50-m-and-a-cross-sectional-diameter-of-10-cm\/","title":{"rendered":"A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm. When a current passes through the coil of the solenoid, the magnetic field inside the solenoid at its CENTER has a magnitude of 2.0 \u00b5T. What is the largest value of the magnetic field inside the solenoid when this current is flowing? ( \\mu_0 = 4\\pi \\times 10^{-7} \\text{Tm\/A}) Note: Your answer is assumed to be reduced to the highest power possible. Your Answer: x10 Answer Hide hint for Question 2 The answer should be with two decimals and without units.<br>A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm. When a current passes through the coil of the solenoid, the magnetic field inside the solenoid at its CENTER has a magnitude of 2.0 \u00b5T. What is the largest value of the magnetic field inside the solenoid when this current is flowing? ( \\mu_0 = 4\\pi \\times 10^{-7} \\text{Tm\/A}) Note: Your answer is assumed to be reduced to the highest power possible. Your Answer: x10 Answer Hide hint for Question 2 The answer should be with two decimals and without units.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Correct Answer:<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2.22 \u00d7 10\u207b\u2076<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A <strong>toroidal solenoid<\/strong> is a solenoid shaped like a doughnut, where the magnetic field is confined within the core. The magnetic field at a point inside the toroid is given by: B=\u03bc0NI2\u03c0rB = \\frac{\\mu_0 N I}{2\\pi r}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>BB is the magnetic field at a radius rr from the center,<\/li>\n\n\n\n<li>\u03bc0=4\u03c0\u00d710\u22127\u00a0T\\cdotpm\/A\\mu_0 = 4\\pi \\times 10^{-7} \\ \\text{T\u00b7m\/A},<\/li>\n\n\n\n<li>NN is the total number of turns,<\/li>\n\n\n\n<li>II is the current,<\/li>\n\n\n\n<li>rr is the radial distance from the center of the toroid.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The magnetic field varies <strong>inversely with the radius<\/strong>, so the field is <strong>strongest at the inner edge<\/strong> and <strong>weakest at the outer edge<\/strong> of the toroid.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Given Data:<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Central radius of the toroid: rc=0.50\u00a0mr_c = 0.50 \\ \\text{m}<\/li>\n\n\n\n<li>Cross-sectional diameter: d=0.10\u00a0m\u21d2d = 0.10 \\ \\text{m} \\Rightarrow radius = 0.05\u00a0m0.05 \\ \\text{m}<\/li>\n\n\n\n<li>Magnetic field at center: Bcenter=2.0\u00a0\u03bcT=2.0\u00d710\u22126\u00a0TB_{\\text{center}} = 2.0 \\ \\mu\\text{T} = 2.0 \\times 10^{-6} \\ \\text{T}<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Find the Largest Magnetic Field:<\/strong><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>maximum magnetic field<\/strong> occurs at the <strong>smallest radius<\/strong> inside the toroid: rmin=rc\u2212d2=0.50\u22120.05=0.45&nbsp;mr_{\\text{min}} = r_c &#8211; \\frac{d}{2} = 0.50 &#8211; 0.05 = 0.45 \\ \\text{m}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The field at the center is at r=0.50&nbsp;mr = 0.50 \\ \\text{m}. Because B\u221d1rB \\propto \\frac{1}{r}, the field at r=0.45&nbsp;mr = 0.45 \\ \\text{m} is: Bmax=Bcenter\u22c5rcrmin=2.0\u00d710\u22126\u22c50.500.45B_{\\text{max}} = B_{\\text{center}} \\cdot \\frac{r_c}{r_{\\text{min}}} = 2.0 \\times 10^{-6} \\cdot \\frac{0.50}{0.45} Bmax=2.0\u00d710\u22126\u22c51.1\u203e=2.22\u00d710\u22126&nbsp;TB_{\\text{max}} = 2.0 \\times 10^{-6} \\cdot 1.\\overline{1} = 2.22 \\times 10^{-6} \\ \\text{T}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Final Answer:<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2.22 \u00d7 10\u207b\u2076<\/strong> (Expressed as <code>2.22<\/code> in the answer box with <code>x10\u207b\u2076<\/code> understood)<\/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-banner4-6.jpeg\" alt=\"\" class=\"wp-image-222880\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm. When a current passes through the coil of the solenoid, the magnetic field inside the solenoid at its CENTER has a magnitude of 2.0 \u00b5T. What is the largest value of the magnetic field inside the solenoid when [&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-222879","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222879","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=222879"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222879\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=222879"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=222879"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=222879"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}