{"id":164608,"date":"2024-11-10T11:27:41","date_gmt":"2024-11-10T11:27:41","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=164608"},"modified":"2024-11-10T11:27:43","modified_gmt":"2024-11-10T11:27:43","slug":"you-charge-an-initially-uncharged-82-5mf-capacitor-through-a-42-3%cf%89-resistor-by-means-of-a-9-00-v-battery-having-negligible-internal-resistance","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/10\/you-charge-an-initially-uncharged-82-5mf-capacitor-through-a-42-3%cf%89-resistor-by-means-of-a-9-00-v-battery-having-negligible-internal-resistance\/","title":{"rendered":"You charge an initially uncharged 82.5mF capacitor through a 42.3\u03a9 resistor by means of a 9.00 V battery having negligible internal resistance."},"content":{"rendered":"\n<p>You charge an initially uncharged 82.5mF capacitor through a 42.3\u03a9 resistor by means of a 9.00 V battery having negligible internal resistance. Find the time constant \u03c4 of the circuit. What is the charge Q on the capacitor 1.79 time constants after the circuit is closed? What is the charge Q\u2080 after a long amount of time has passed?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Find the time constant, \u03c4<\/h3>\n\n\n\n<p>The time constant ( \\tau ) for an RC (Resistor-Capacitor) circuit is given by the formula:<\/p>\n\n\n\n<p>[<br>\\tau = R \\times C<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( R ) is the resistance in ohms (( \\Omega )),<\/li>\n\n\n\n<li>( C ) is the capacitance in farads (F).<\/li>\n<\/ul>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( R = 42.3 \\, \\Omega ),<\/li>\n\n\n\n<li>( C = 82.5 \\, \\text{mF} = 82.5 \\times 10^{-3} \\, \\text{F} ).<\/li>\n<\/ul>\n\n\n\n<p>Now, substitute these values into the formula:<\/p>\n\n\n\n<p>[<br>\\tau = (42.3 \\, \\Omega) \\times (82.5 \\times 10^{-3} \\, \\text{F}) = 3.49 \\, \\text{seconds}.<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate the charge on the capacitor after 1.79 time constants<\/h3>\n\n\n\n<p>The charge ( Q(t) ) on the capacitor at any time ( t ) during the charging process is given by:<\/p>\n\n\n\n<p>[<br>Q(t) = Q_{\\text{max}} \\left( 1 &#8211; e^{-t\/\\tau} \\right)<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( Q_{\\text{max}} ) is the maximum charge the capacitor can hold (after a long time),<\/li>\n\n\n\n<li>( t ) is the time in seconds,<\/li>\n\n\n\n<li>( \\tau ) is the time constant.<\/li>\n<\/ul>\n\n\n\n<p>We need to find the charge after 1.79 time constants, so ( t = 1.79 \\tau ).<\/p>\n\n\n\n<p>The maximum charge ( Q_{\\text{max}} ) on the capacitor is given by:<\/p>\n\n\n\n<p>[<br>Q_{\\text{max}} = C \\times V<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( C ) is the capacitance,<\/li>\n\n\n\n<li>( V ) is the voltage of the battery.<\/li>\n<\/ul>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( V = 9.00 \\, \\text{V} ),<\/li>\n\n\n\n<li>( C = 82.5 \\times 10^{-3} \\, \\text{F} ).<\/li>\n<\/ul>\n\n\n\n<p>The maximum charge ( Q_{\\text{max}} ) is:<\/p>\n\n\n\n<p>[<br>Q_{\\text{max}} = (82.5 \\times 10^{-3} \\, \\text{F}) \\times (9.00 \\, \\text{V}) = 0.7425 \\, \\text{C}.<br>]<\/p>\n\n\n\n<p>Now, substitute ( t = 1.79 \\tau = 1.79 \\times 3.49 \\, \\text{s} = 6.25 \\, \\text{s} ) into the equation for ( Q(t) ):<\/p>\n\n\n\n<p>[<br>Q(t) = 0.7425 \\, \\text{C} \\left( 1 &#8211; e^{-6.25\/3.49} \\right)<br>]<\/p>\n\n\n\n<p>First, calculate the exponent:<\/p>\n\n\n\n<p>[<br>\\frac{6.25}{3.49} \\approx 1.79<br>]<\/p>\n\n\n\n<p>Now, calculate the charge:<\/p>\n\n\n\n<p>[<br>Q(t) = 0.7425 \\, \\text{C} \\left( 1 &#8211; e^{-1.79} \\right) = 0.7425 \\, \\text{C} \\times (1 &#8211; 0.167) = 0.7425 \\, \\text{C} \\times 0.833 = 0.619 \\, \\text{C}.<br>]<\/p>\n\n\n\n<p>Thus, the charge on the capacitor after 1.79 time constants is approximately ( Q(t) = 0.619 \\, \\text{C} ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Find the charge after a long time (steady state)<\/h3>\n\n\n\n<p>After a long amount of time, the capacitor is fully charged, and the current stops. In this steady state, the charge on the capacitor is equal to ( Q_{\\text{max}} ).<\/p>\n\n\n\n<p>Thus, the charge on the capacitor after a long time is:<\/p>\n\n\n\n<p>[<br>Q_{\\infty} = Q_{\\text{max}} = 0.7425 \\, \\text{C}.<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of Answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The time constant ( \\tau = 3.49 \\, \\text{seconds} ),<\/li>\n\n\n\n<li>The charge on the capacitor after 1.79 time constants is ( Q(t) \\approx 0.619 \\, \\text{C} ),<\/li>\n\n\n\n<li>The charge on the capacitor after a long time is ( Q_{\\infty} = 0.7425 \\, \\text{C} ).<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>You charge an initially uncharged 82.5mF capacitor through a 42.3\u03a9 resistor by means of a 9.00 V battery having negligible internal resistance. Find the time constant \u03c4 of the circuit. What is the charge Q on the capacitor 1.79 time constants after the circuit is closed? What is the charge Q\u2080 after a long amount [&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-164608","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/164608","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=164608"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/164608\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=164608"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=164608"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=164608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}