{"id":155940,"date":"2024-10-16T17:57:43","date_gmt":"2024-10-16T17:57:43","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=155940"},"modified":"2024-10-16T17:57:46","modified_gmt":"2024-10-16T17:57:46","slug":"a-45-mh-ideal-inductor-is-connected-in-series-with-a-60-%cf%89-resistor-through-an-ideal-15-v-dc-power-supply-and-an-open-switch","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/16\/a-45-mh-ideal-inductor-is-connected-in-series-with-a-60-%cf%89-resistor-through-an-ideal-15-v-dc-power-supply-and-an-open-switch\/","title":{"rendered":"a 45-mh ideal inductor is connected in series with a 60-\u03c9 resistor through an ideal 15-v dc power supply and an open switch"},"content":{"rendered":"\n<p>a 45-mh ideal inductor is connected in series with a 60-\u03c9 resistor through an ideal 15-v dc power supply and an open switch. if the switch is closed at time t = 0 s, what is the current 7.0 ms later?<br>a.250 ma<\/p>\n\n\n\n<p>b.850 ma<\/p>\n\n\n\n<p>c.550 ma<\/p>\n\n\n\n<p>d.280 ma<\/p>\n\n\n\n<p>e.650 ma<\/p>\n\n\n\n<p><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><\/p>\n\n\n\n<p>The correct answer is: <strong>250 mA<\/strong><\/p>\n\n\n\n<p>To find the current through the circuit 7.0 ms after the switch is closed, we can use the formula for the current in an RL (resistor-inductor) circuit:<\/p>\n\n\n\n<p>[<br>I(t) = I_{\\text{max}} \\left(1 &#8211; e^{-\\frac{R}{L} t}\\right)<br>]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( I(t) ) is the current at time ( t ),<\/li>\n\n\n\n<li>( I_{\\text{max}} ) is the maximum steady-state current,<\/li>\n\n\n\n<li>( R ) is the resistance,<\/li>\n\n\n\n<li>( L ) is the inductance,<\/li>\n\n\n\n<li>( t ) is the time in seconds,<\/li>\n\n\n\n<li>( e ) is the base of the natural logarithm (approximately equal to 2.71828).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Calculate ( I_{\\text{max}} )<\/h3>\n\n\n\n<p>The maximum current ( I_{\\text{max}} ) can be calculated using Ohm&#8217;s law:<\/p>\n\n\n\n<p>[<br>I_{\\text{max}} = \\frac{V}{R}<br>]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( V = 15 \\, \\text{V} ) (the voltage of the power supply),<\/li>\n\n\n\n<li>( R = 60 \\, \\Omega ).<\/li>\n<\/ul>\n\n\n\n<p>Substituting the values:<\/p>\n\n\n\n<p>[<br>I_{\\text{max}} = \\frac{15 \\, \\text{V}}{60 \\, \\Omega} = 0.25 \\, \\text{A} \\text{ or } 250 \\, \\text{mA}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Find the time constant ( \\tau )<\/h3>\n\n\n\n<p>The time constant ( \\tau ) of the RL circuit is given by:<\/p>\n\n\n\n<p>[<br>\\tau = \\frac{L}{R}<br>]<\/p>\n\n\n\n<p>Substituting the given values:<\/p>\n\n\n\n<p>[<br>L = 45 \\, \\text{mH} = 0.045 \\, \\text{H}<br>]<br>[<br>R = 60 \\, \\Omega<br>]<\/p>\n\n\n\n<p>[<br>\\tau = \\frac{0.045}{60} = 0.00075 \\, \\text{s} = 0.75 \\, \\text{ms}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate the current at ( t = 7.0 \\, \\text{ms} )<\/h3>\n\n\n\n<p>Now, we can substitute ( t = 7.0 \\, \\text{ms} = 0.007 \\, \\text{s} ) into the current formula:<\/p>\n\n\n\n<p>[<br>I(0.007) = 0.25 \\left(1 &#8211; e^{-\\frac{60}{0.045} \\cdot 0.007}\\right)<br>]<\/p>\n\n\n\n<p>Calculating ( \\frac{R}{L} \\cdot t ):<\/p>\n\n\n\n<p>[<br>\\frac{60}{0.045} \\cdot 0.007 \\approx 9.333<br>]<\/p>\n\n\n\n<p>Now substituting back into the equation:<\/p>\n\n\n\n<p>[<br>I(0.007) = 0.25 \\left(1 &#8211; e^{-9.333}\\right) \\approx 0.25 \\left(1 &#8211; 0.000083\\right) \\approx 0.25 \\times 0.999917 \\approx 0.24998 \\, \\text{A}<br>]<\/p>\n\n\n\n<p>This means ( I(0.007) \\approx 0.25 \\, \\text{A} = 250 \\, \\text{mA} ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>Thus, the current 7.0 ms after closing the switch is approximately <strong>250 mA<\/strong>, which corresponds to option <strong>a<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>a 45-mh ideal inductor is connected in series with a 60-\u03c9 resistor through an ideal 15-v dc power supply and an open switch. if the switch is closed at time t = 0 s, what is the current 7.0 ms later?a.250 ma b.850 ma c.550 ma d.280 ma e.650 ma The Correct Answer and Explanation [&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 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