{"id":179447,"date":"2024-12-31T06:54:56","date_gmt":"2024-12-31T06:54:56","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=179447"},"modified":"2024-12-31T06:54:57","modified_gmt":"2024-12-31T06:54:57","slug":"a-series-lr-circuit-contains-an-emf-source-of-14-v-having-no-internal-resistance-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/12\/31\/a-series-lr-circuit-contains-an-emf-source-of-14-v-having-no-internal-resistance-2\/","title":{"rendered":"A series LR circuit contains an emf source of 14 V having no internal resistance"},"content":{"rendered":"\n<pre class=\"wp-block-code\"><code>     A series LR circuit contains an emf source of 14 V having no internal resistance, a resistor, a 34 H inductor having no appreciable resistance, and a switch. If the emf across the inductor is 80% of its maximum value 4.0 s after the switch is closed, what is the resistance of the resistor? a. 1.5 ? b. 1.9 ? c. 5.0 ? d. 14 ?\n\n<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark>\n<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>\n<\/strong><\/mark>To solve this problem, we use the equation for the voltage across the inductor in an LR circuit as a function of time:  \n\n\\&#91;\nV_L(t) = \\mathcal{E}(1 - e^{-t\/\\tau}),\n\\]\n\nwhere:  \n- \\(\\mathcal{E} = 14 \\, \\text{V}\\) (emf of the source),  \n- \\(\\tau = L\/R\\) is the time constant,  \n- \\(L = 34 \\, \\text{H}\\) is the inductance of the inductor,  \n- \\(R\\) is the resistance we need to find,  \n- \\(t = 4.0 \\, \\text{s}\\) is the given time.  \n\nThe emf across the inductor is 80% of its maximum value (\\(\\mathcal{E}\\)), so:  \n\\&#91;\nV_L(t) = 0.8\\mathcal{E} = 0.8 \\times 14 = 11.2 \\, \\text{V}.\n\\]\n\nSubstituting into the equation for \\(V_L(t)\\):  \n\\&#91;\n11.2 = 14(1 - e^{-t\/\\tau}).\n\\]\n\nDivide through by 14:  \n\\&#91;\n\\frac{11.2}{14} = 1 - e^{-t\/\\tau}.\n\\]\n\nSimplify:  \n\\&#91;\n0.8 = 1 - e^{-t\/\\tau}.\n\\]\n\nRearrange for \\(e^{-t\/\\tau}\\):  \n\\&#91;\ne^{-t\/\\tau} = 0.2.\n\\]\n\nTake the natural logarithm of both sides:  \n\\&#91;\n-\\frac{t}{\\tau} = \\ln(0.2).\n\\]\n\nSubstitute \\(t = 4.0 \\, \\text{s}\\):  \n\\&#91;\n-\\frac{4.0}{\\tau} = \\ln(0.2).\n\\]\n\nSolve for \\(\\tau\\):  \n\\&#91;\n\\tau = \\frac{4.0}{-\\ln(0.2)}.\n\\]\n\nNumerically:  \n\\&#91;\n\\ln(0.2) \\approx -1.609, \\quad \\tau = \\frac{4.0}{1.609} \\approx 2.49 \\, \\text{s}.\n\\]\n\nThe time constant \\(\\tau\\) is also defined as \\(\\tau = L\/R\\). Substituting \\(\\tau = 2.49 \\, \\text{s}\\) and \\(L = 34 \\, \\text{H}\\):  \n\\&#91;\nR = \\frac{L}{\\tau} = \\frac{34}{2.49} \\approx 13.65 \\, \\Omega.\n\\]\n\nThe closest option is **d. 14 \\(\\Omega\\)**.  \n\n### Explanation\nAn LR circuit's time constant (\\(\\tau\\)) determines the rate of change of current and voltage across components. The time constant is influenced by the inductance \\(L\\) and resistance \\(R\\). Here, we use the given percentage of maximum voltage (80%) to calculate \\(R\\), solving through exponential decay equations. This approach is foundational in electrical engineering to analyze transient responses in circuits.<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"","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-179447","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179447","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=179447"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179447\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=179447"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=179447"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=179447"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}