{"id":223319,"date":"2025-06-01T12:57:42","date_gmt":"2025-06-01T12:57:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=223319"},"modified":"2025-06-01T12:57:44","modified_gmt":"2025-06-01T12:57:44","slug":"a-series-lr-circuit-contains-an-emf-source-of-14-v-with-no-internal-resistance-a-34-h-inductor-with-no-appreciable-resistance-and-a-switch","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/01\/a-series-lr-circuit-contains-an-emf-source-of-14-v-with-no-internal-resistance-a-34-h-inductor-with-no-appreciable-resistance-and-a-switch\/","title":{"rendered":"A series LR circuit contains an emf source of 14 V with no internal resistance, a 34 H inductor with no appreciable resistance, and a switch."},"content":{"rendered":"\n

A series LR circuit contains an emf source of 14 V with no internal resistance, a 34 H inductor with 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? 14 ohms 11 ohms 1.5 ohms 1.9 ohms 5.0 ohms
A series LR circuit contains an emf source of 14 V with no internal resistance, a 34 H inductor with 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? 14 ohms 11 ohms 1.5 ohms 1.9 ohms 5.0 ohms<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer: 1.5 ohms<\/strong><\/p>\n\n\n\n

\n\n\n\n

Explanation <\/strong><\/h3>\n\n\n\n

In a series LR (inductor-resistor) circuit connected to a DC voltage source, the inductor initially opposes the change in current due to its property of self-inductance. Over time, the current increases and eventually reaches a maximum steady value, at which point the inductor acts like a short circuit (offering no opposition).<\/p>\n\n\n\n

The key relationship for the emf across the inductor<\/strong> \u03b5L\\varepsilon_L at time tt after the switch is closed is given by: \u03b5L(t)=\u03b50\u22c5e\u2212Rt\/L\\varepsilon_L(t) = \\varepsilon_0 \\cdot e^{-Rt\/L}<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • \u03b5L(t)\\varepsilon_L(t) is the emf across the inductor at time tt,<\/li>\n\n\n\n
  • \u03b50=initial\u00a0emf=14\u2009V\\varepsilon_0 = \\text{initial emf} = 14 \\, \\text{V},<\/li>\n\n\n\n
  • RR is the resistance in ohms,<\/li>\n\n\n\n
  • LL is the inductance in henries,<\/li>\n\n\n\n
  • tt is the time in seconds.<\/li>\n<\/ul>\n\n\n\n

    We\u2019re told that 4.0 seconds after the switch is closed, the emf across the inductor is 80% of its maximum value<\/strong>, i.e., \u03b5L(4.0)=0.80\u22c514=11.2\u2009V\\varepsilon_L(4.0) = 0.80 \\cdot 14 = 11.2 \\, \\text{V}<\/p>\n\n\n\n

    Now substitute into the formula: 11.2=14\u22c5e\u2212R\u22c54\/3411.2 = 14 \\cdot e^{-R \\cdot 4 \/ 34}<\/p>\n\n\n\n

    Divide both sides by 14: 0.80=e\u22124R\/340.80 = e^{-4R\/34}<\/p>\n\n\n\n

    Take the natural logarithm of both sides: ln\u2061(0.80)=\u22124R34\\ln(0.80) = -\\frac{4R}{34} \u22120.2231=\u22124R34-0.2231 = -\\frac{4R}{34}<\/p>\n\n\n\n

    Multiply both sides by 34: \u22120.2231\u22c534=\u22124R-0.2231 \\cdot 34 = -4R \u22127.5854=\u22124R-7.5854 = -4R<\/p>\n\n\n\n

    Divide by -4: R=1.896\u22481.9\u2009\u03a9R = 1.896 \\approx \\boxed{1.9 \\, \\Omega}<\/p>\n\n\n\n

    However, note the error<\/strong>: We misinterpreted the decay formula. The voltage across the inductor<\/em> is actually proportional to the rate of change of current<\/strong>, which decreases over time. The correct exponential behavior for the inductor voltage<\/strong> after closing the switch is: \u03b5L=\u03b5\u22c5e\u2212Rt\/L\\varepsilon_L = \\varepsilon \\cdot e^{-Rt\/L}<\/p>\n\n\n\n

    So redoing the math properly: 0.80=e\u2212Rt\/L\u21d2ln\u2061(0.80)=\u2212Rt\/L\u21d2R=\u2212Ltln\u2061(0.80)=\u2212344ln\u2061(0.80)=8.5\u22c50.2231\u22481.90\u2009\u03a90.80 = e^{-Rt\/L} \\Rightarrow \\ln(0.80) = -Rt\/L \\Rightarrow R = -\\frac{L}{t} \\ln(0.80) = -\\frac{34}{4} \\ln(0.80) = 8.5 \\cdot 0.2231 \\approx 1.90 \\, \\Omega<\/p>\n\n\n\n

    So the correct answer is: 1.9 ohms<\/strong> \u2705<\/p>\n\n\n\n

    \n\n\n\n

    Final Answer:<\/strong> 1.9 ohms<\/strong><\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    A series LR circuit contains an emf source of 14 V with no internal resistance, a 34 H inductor with 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? 14 ohms 11 […]<\/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-223319","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/223319","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=223319"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/223319\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=223319"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=223319"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=223319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}