{"id":1359,"date":"2017-10-27T16:32:00","date_gmt":"2017-10-27T16:32:00","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/rl-circuits\/"},"modified":"2017-11-08T03:26:58","modified_gmt":"2017-11-08T03:26:58","slug":"rl-circuits","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/rl-circuits\/","title":{"raw":"RL Circuits","rendered":"RL Circuits"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Calculate the current in an RL circuit after a specified number of characteristic time steps.<\/li>\n<li>Calculate the characteristic time of an RL circuit.<\/li>\n<li>Sketch the current in an RL circuit over time.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1169738093650\">We know that the current through an inductor [latex]L[\/latex] cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz\u2019s law). How long does the opposition last? Current <em data-effect=\"italics\"><em data-effect=\"italics\">will<\/em><\/em> flow and <em data-effect=\"italics\"><em data-effect=\"italics\">can<\/em><\/em> be turned off, but how long does it take? <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a> shows a switching circuit that can be used to examine current through an inductor as a function of time.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169737969272\">\n<div class=\"bc-figcaption figcaption\">(a) An <em data-effect=\"italics\">RL<\/em> circuit with a switch to turn current on and off. When in position 1, the battery, resistor, and inductor are in series and a current is established. In position 2, the battery is removed and the current eventually stops because of energy loss in the resistor. (b) A graph of current growth versus time when the switch is moved to position 1. (c) A graph of current decay when the switch is moved to position 2.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169737758284\" data-alt=\"Part a of the figure shows an inductor connected in series with a resistor. The arrangement is connected across a cell by an on and off switch with two positions. When in position one, the battery, resistor, and inductor are in series and a current is established. In position two, the battery is removed and the current stops eventually because of energy loss in the resistor. Part b of the diagram shows the graph when the switch is in position one. It shows a graph for current growth verses time. The current is along the Y axis and the time is along the X axis. The graph shows a smooth rise from origin to a maximum value I zero corresponding to Y axis and value four tau on X axis. Part c of the diagram shows the graph when the switch is in position two. It shows a graph for current decay verses time is shown. The current is along the Y axis and the time is along the X axis. The graph is decreasing curve from a value I zero on Y axis, touching the X axis at a point where value of time equals four tau.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_24_10_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows an inductor connected in series with a resistor. The arrangement is connected across a cell by an on and off switch with two positions. When in position one, the battery, resistor, and inductor are in series and a current is established. In position two, the battery is removed and the current stops eventually because of energy loss in the resistor. Part b of the diagram shows the graph when the switch is in position one. It shows a graph for current growth verses time. The current is along the Y axis and the time is along the X axis. The graph shows a smooth rise from origin to a maximum value I zero corresponding to Y axis and value four tau on X axis. Part c of the diagram shows the graph when the switch is in position two. It shows a graph for current decay verses time is shown. The current is along the Y axis and the time is along the X axis. The graph is decreasing curve from a value I zero on Y axis, touching the X axis at a point where value of time equals four tau.\" width=\"550\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1169737757267\">When the switch is first moved to position 1 (at <em data-effect=\"italics\">[latex]t=0[\/latex]<\/em>), the current is zero and it eventually rises to [latex]{I}_{0}=\\text{V\/R}[\/latex], where [latex]R[\/latex] is the total resistance of the circuit. The opposition of the inductor [latex]L[\/latex] is greatest at the beginning, because the amount of change is greatest. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. The opposing emf is proportional to the amount of change left. This is the hallmark of an exponential behavior, and it can be shown with calculus that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-314\">[latex]I={I}_{0}\\left(1-{e}^{-t\/\\tau }\\right)\\text{&nbsp;&nbsp;&nbsp;&nbsp;(turning on),}[\/latex]<\/div>\n<p id=\"import-auto-id1169737811419\">is the current in an <em data-effect=\"italics\">RL<\/em> circuit when switched on (Note the similarity to the exponential behavior of the voltage on a charging capacitor). The initial current is zero and approaches [latex]{I}_{0}=\\text{V\/R}[\/latex] with a <span data-type=\"term\" id=\"import-auto-id1169737897754\">characteristic time constant<\/span><br>\n[latex]\\tau [\/latex]<br>\nfor an <em data-effect=\"italics\">RL<\/em> circuit, given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\tau =\\frac{L}{R}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1169737803195\">where [latex]\\tau [\/latex] has units of seconds, since<br>\n[latex]\\text{1}\\phantom{\\rule{0.25em}{0ex}}\\text{H}\\text{=}\\text{1}\\phantom{\\rule{0.25em}{0ex}}\\text{\u03a9}\\text{\u00b7}\\text{s}[\/latex].<br>\n In the first period of time [latex]\\tau [\/latex], the current rises from zero to [latex]0\\text{.}\\text{632}{I}_{0}[\/latex], since [latex]I={I}_{0}\\left(1-{e}^{-1}\\right)={I}_{0}\\left(1-0\\text{.}\\text{368}\\right)=0\\text{.}\\text{632}{I}_{0}[\/latex]. The current will go 0.632 of the remainder in the next time [latex]\\tau [\/latex]. A well-known property of the exponential is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time [latex]\\tau [\/latex]. In just a few multiples of the time [latex]\\tau [\/latex], the final value is very nearly achieved, as the graph in <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a>(b) illustrates.<\/p>\n<p id=\"import-auto-id1169735754248\">The characteristic time [latex]\\tau [\/latex] depends on only two factors, the inductance [latex]L[\/latex] and the resistance [latex]R[\/latex]. The greater the inductance [latex]L[\/latex], the greater [latex]\\tau [\/latex] is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance [latex]R[\/latex], the greater [latex]\\tau [\/latex] is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases\u2014large [latex]L[\/latex] and small [latex]R[\/latex] \u2014more energy is stored in the inductor and more time is required to get it in and out.<\/p>\n<p id=\"import-auto-id1169737882691\">When the switch in <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a>(a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. But this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, [latex]\\left(\\text{1\/2}\\right){\\text{LI}}_{0}^{2}[\/latex], stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is [latex]{I}^{2}R[\/latex]. Once again the behavior is exponential, and<br>\n[latex]I[\/latex]<br>\n is found to be<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]I={I}_{0}{e}^{-t\/\\tau }\\text{&nbsp;&nbsp;&nbsp;&nbsp;(turning off).}[\/latex]<\/div>\n<p id=\"import-auto-id1169737711668\">(See <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a>(c).) In the first period of time [latex]\\tau =L\/R[\/latex] after the switch is closed, the current falls to 0.368 of its initial value, since [latex]I={I}_{0}{e}^{-1}=0\\text{.}\\text{368}{I}_{0}[\/latex]. In each successive time [latex]\\tau [\/latex], the current falls to 0.368 of the preceding value, and in a few multiples of [latex]\\tau [\/latex], the current becomes very close to zero, as seen in the graph in <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a>(c).<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1169738136126\">\n<div data-type=\"title\" class=\"title\">Calculating Characteristic Time and Current in an <em data-effect=\"italics\">RL<\/em> Circuit<\/div>\n<p id=\"import-auto-id1169737045564\">(a) What is the characteristic time constant for a 7.50 mH inductor in series with a [latex]\\text{3.00 \u03a9}[\/latex] resistor? (b) Find the current 5.00 ms after the switch is moved to position 2 to disconnect the battery, if it is initially 10.0 A.<\/p>\n<p id=\"import-auto-id1169737804462\"><strong>Strategy for (a)<\/strong><\/p>\n<p id=\"import-auto-id1169737787249\">The time constant for an <em data-effect=\"italics\">RL<\/em> circuit is defined by [latex]\\tau =L\/R[\/latex].<\/p>\n<p id=\"import-auto-id1169737713276\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1169738036454\">Entering known values into the expression for [latex]\\tau [\/latex] given in [latex]\\tau =L\/R[\/latex] yields<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\tau =\\frac{L}{R}=\\frac{7.50 mH}{3.00\\phantom{\\rule{0.25em}{0ex}}\\Omega }=2.50 ms.[\/latex]<\/div>\n<p id=\"import-auto-id1169738164481\"><strong>Discussion for (a)<\/strong><\/p>\n<p id=\"import-auto-id1169737723042\">This is a small but definitely finite time. The coil will be very close to its full current in about ten time constants, or about 25 ms.<\/p>\n<p id=\"import-auto-id1169736758585\"><strong>Strategy for (b)<\/strong><\/p>\n<p id=\"import-auto-id1169738010703\">We can find the current by using [latex]I={I}_{0}{e}^{-t\/\\tau }[\/latex], or by considering the decline in steps. Since the time is twice the characteristic time, we consider the process in steps.<\/p>\n<p id=\"import-auto-id1169736614228\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1169738189000\">In the first 2.50 ms, the current declines to 0.368 of its initial value, which is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}I&amp; =&amp; 0\\text{.}\\text{368}{I}_{0}=\\left(0\\text{.}\\text{368}\\right)\\left(\\text{10.0 A}\\right)\\\\ &amp; =&amp; 3\\text{.}\\text{68 A&nbsp;at&nbsp;}t=2\\text{.}\\text{50}\\text{&nbsp;ms.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1169738055826\">After another 2.50 ms, or a total of 5.00 ms, the current declines to 0.368 of the value just found. That is,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}I\\prime &amp; =&amp; 0\\text{.}\\text{368}I=\\left(0\\text{.}\\text{368}\\right)\\left(\\text{3.68 A}\\right)\\\\ &amp; =&amp; 1\\text{.}\\text{35}\\text{&nbsp;A&nbsp;at&nbsp;}t=5\\text{.}\\text{00}\\text{&nbsp;ms.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1169737972496\"><strong>Discussion for (b)<\/strong><\/p>\n<p id=\"import-auto-id1169737896000\">After another 5.00 ms has passed, the current will be 0.183 A (see <a href=\"#fs-id1169738227350\" class=\"autogenerated-content\">(Figure)<\/a>); so, although it does die out, the current certainly does not go to zero instantaneously.<\/p>\n<\/div>\n<p id=\"import-auto-id1169737717434\">In summary, when the voltage applied to an inductor is changed, the current also changes, <em data-effect=\"italics\"><em data-effect=\"italics\">but the change in current lags the change in voltage in an RL circuit<\/em><\/em>. In <a href=\"\/contents\/663b7dc1-df6f-4a85-bae8-10d20a358d01@4\">Reactance, Inductive and Capacitive<\/a>, we explore how an <em data-effect=\"italics\">RL<\/em> circuit behaves when a sinusoidal AC voltage is applied.<\/p>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1169737941056\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1169736611192\">\n<li id=\"import-auto-id1169737712209\">When a series connection of a resistor and an inductor\u2014an <em data-effect=\"italics\">RL<\/em> circuit\u2014is connected to a voltage source, the time variation of the current is\n<div data-type=\"equation\" class=\"equation\">[latex]I={I}_{0}\\left(1-{e}^{-t\/\\tau }\\right)\\text{&nbsp;&nbsp;&nbsp;&nbsp;(turning on).}[\/latex]<\/div>\n<p>        where [latex]{I}_{0}=V\/R[\/latex] is the final current.<\/p><\/li>\n<li id=\"import-auto-id1169737936637\">The characteristic time constant [latex]\\tau [\/latex] is [latex]\\tau =\\frac{L}{R}[\/latex] , where [latex]L[\/latex]  is the inductance and  [latex]R[\/latex] is the resistance.<\/li>\n<li id=\"import-auto-id1169738035430\">In the first time constant [latex]\\tau [\/latex], the current rises from zero to [latex]0\\text{.}\\text{632}{I}_{0}[\/latex], and 0.632 of the remainder in every subsequent time interval [latex]\\tau [\/latex].<\/li>\n<li id=\"import-auto-id1169738048163\">When the inductor is shorted through a resistor, current decreases as\n<div data-type=\"equation\" class=\"equation\">[latex]I={I}_{0}{e}^{-t\/\\tau }\\text{&nbsp;&nbsp;&nbsp;&nbsp;(turning off).}[\/latex]<\/div>\n<p>Here [latex]{I}_{0}[\/latex] is the initial current.<\/p><\/li>\n<li id=\"import-auto-id1169737060067\">Current falls to [latex]0\\text{.}\\text{368}{I}_{0}[\/latex] in the first time interval [latex]\\tau [\/latex], and 0.368 of the remainder toward zero in each subsequent time [latex]\\tau [\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1169738010703\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738227350\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738245588\">\n<p id=\"import-auto-id1169738210359\">If you want a characteristic <em data-effect=\"italics\">RL<\/em> time constant of 1.00 s, and you have a [latex]\\text{500 \u03a9}[\/latex] resistor, what value of self-inductance is needed?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737933054\">\n<p id=\"import-auto-id1169738164414\">500 H<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737723038\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736814918\">\n<p id=\"import-auto-id1169738117421\">Your <em data-effect=\"italics\">RL<\/em> circuit has a characteristic time constant of 20.0 ns, and a resistance of [latex]\\text{5.00 M\u03a9}[\/latex]. (a) What is the inductance of the circuit? (b) What resistance would give you a 1.00 ns time constant, perhaps needed for quick response in an oscilloscope?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737788008\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736620644\">\n<p>A large superconducting magnet, used for magnetic resonance imaging, has a 50.0 H inductance. If you want current through it to be adjustable with a 1.00 s characteristic time constant, what is the minimum resistance of system?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738094791\">\n<p id=\"import-auto-id1169736659279\">[latex]\\text{50.0 \u03a9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736581925\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736620836\">\n<p id=\"import-auto-id1169738110369\">Verify that after a time of 10.0 ms, the current for the situation considered in <a href=\"#fs-id1169738136126\" class=\"autogenerated-content\">(Figure)<\/a> will be 0.183 A as stated.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736596063\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736590714\">\n<p id=\"import-auto-id1169737770657\">Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H, and resistors ranging from<br>\n[latex]\\text{0.100 \u03a9}[\/latex] to<br>\n[latex]\\text{1.00 M\u03a9}[\/latex]. What is the range of characteristic <em data-effect=\"italics\">RL<\/em> time constants you can produce by connecting a single resistor to a single inductor?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738045475\">\n<p>[latex]1\\text{.}\\text{00}\u00d7{\\text{10}}^{\\text{\u201318}}\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex] to 0.100 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737812307\">\n<p id=\"import-auto-id1169738113687\">(a) What is the characteristic time constant of a 25.0 mH inductor that has a resistance of<br>\n[latex]\\text{4.00 \u03a9}[\/latex]? (b) If it is connected to a 12.0 V battery, what is the current after 12.5 ms?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737723042\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737785691\">\n<p id=\"import-auto-id1169738075685\">What percentage of the final current<br>\n[latex]{I}_{\\text{0}}[\/latex]<br>\n flows through an inductor [latex]L[\/latex] in series with a resistor [latex]R[\/latex], three time constants after the circuit is completed?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169736758585\">\n<p id=\"import-auto-id1169738251076\">95.0%<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738248980\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738223614\">\n<p id=\"import-auto-id1169737898346\">The 5.00 A current through a 1.50 H inductor is dissipated by a [latex]\\text{2.00 \u03a9}[\/latex] resistor in a circuit like that in <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a> with the switch in position 2. (a) What is the initial energy in the inductor? (b) How long will it take the current to decline to 5.00% of its initial value? (c) Calculate the average power dissipated, and compare it with the initial power dissipated by the resistor.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737787416\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736624109\">\n<p id=\"import-auto-id1169738250359\">(a) Use the exact exponential treatment to find how much time is required to bring the current through an 80.0 mH inductor in series with a<br>\n[latex]\\text{15.0 \u03a9}[\/latex]<br>\n resistor to 99.0% of its final value, starting from zero. (b) Compare your answer to the approximate treatment using integral numbers of [latex]\\tau [\/latex]. (c) Discuss how significant the difference is.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737768546\">\n<p id=\"import-auto-id1169738036121\">(a) 24.6 ms<\/p>\n<p id=\"import-auto-id1169737895428\">(b) 26.7 ms<\/p>\n<p id=\"import-auto-id1169738051862\">(c) 9% difference, which is greater than the inherent uncertainty in the given parameters.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737851564\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738011205\">\n<p id=\"import-auto-id1169737961873\">(a) Using the exact exponential treatment, find the time required for the current through a 2.00 H inductor in series with a<br>\n[latex]\\text{0.500 \u03a9}[\/latex]<br>\n resistor to be reduced to 0.100% of its original value. (b) Compare your answer to the approximate treatment using integral numbers of [latex]\\tau [\/latex]. (c) Discuss how significant the difference is.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1169738071203\">\n<dt>characteristic time constant<\/dt>\n<dd id=\"fs-id1169736581702\"> denoted by [latex]\\tau [\/latex], of a particular series <em data-effect=\"italics\">RL<\/em> circuit is calculated by [latex]\\tau =\\frac{L}{R}[\/latex], where [latex]L[\/latex]  is the inductance and  [latex]R[\/latex] is the resistance<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Calculate the current in an RL circuit after a specified number of characteristic time steps.<\/li>\n<li>Calculate the characteristic time of an RL circuit.<\/li>\n<li>Sketch the current in an RL circuit over time.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1169738093650\">We know that the current through an inductor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz\u2019s law). How long does the opposition last? Current <em data-effect=\"italics\"><em data-effect=\"italics\">will<\/em><\/em> flow and <em data-effect=\"italics\"><em data-effect=\"italics\">can<\/em><\/em> be turned off, but how long does it take? <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a> shows a switching circuit that can be used to examine current through an inductor as a function of time.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169737969272\">\n<div class=\"bc-figcaption figcaption\">(a) An <em data-effect=\"italics\">RL<\/em> circuit with a switch to turn current on and off. When in position 1, the battery, resistor, and inductor are in series and a current is established. In position 2, the battery is removed and the current eventually stops because of energy loss in the resistor. (b) A graph of current growth versus time when the switch is moved to position 1. (c) A graph of current decay when the switch is moved to position 2.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169737758284\" data-alt=\"Part a of the figure shows an inductor connected in series with a resistor. The arrangement is connected across a cell by an on and off switch with two positions. When in position one, the battery, resistor, and inductor are in series and a current is established. In position two, the battery is removed and the current stops eventually because of energy loss in the resistor. Part b of the diagram shows the graph when the switch is in position one. It shows a graph for current growth verses time. The current is along the Y axis and the time is along the X axis. The graph shows a smooth rise from origin to a maximum value I zero corresponding to Y axis and value four tau on X axis. Part c of the diagram shows the graph when the switch is in position two. It shows a graph for current decay verses time is shown. The current is along the Y axis and the time is along the X axis. The graph is decreasing curve from a value I zero on Y axis, touching the X axis at a point where value of time equals four tau.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_24_10_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows an inductor connected in series with a resistor. The arrangement is connected across a cell by an on and off switch with two positions. When in position one, the battery, resistor, and inductor are in series and a current is established. In position two, the battery is removed and the current stops eventually because of energy loss in the resistor. Part b of the diagram shows the graph when the switch is in position one. It shows a graph for current growth verses time. The current is along the Y axis and the time is along the X axis. The graph shows a smooth rise from origin to a maximum value I zero corresponding to Y axis and value four tau on X axis. Part c of the diagram shows the graph when the switch is in position two. It shows a graph for current decay verses time is shown. The current is along the Y axis and the time is along the X axis. The graph is decreasing curve from a value I zero on Y axis, touching the X axis at a point where value of time equals four tau.\" width=\"550\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1169737757267\">When the switch is first moved to position 1 (at <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7b41acc5cb99fb07aaa07b445eb2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/em>), the current is zero and it eventually rises to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1fdf56b8a10a3a74b01868a364e185ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#47;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the total resistance of the circuit. The opposition of the inductor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is greatest at the beginning, because the amount of change is greatest. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. The opposing emf is proportional to the amount of change left. This is the hallmark of an exponential behavior, and it can be shown with calculus that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-314\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3846bc49dae7ffc19b2cfbb6f624f058_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#61;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#123;&#101;&#125;&#94;&#123;&#45;&#116;&#47;&#92;&#116;&#97;&#117;&#32;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#32;&#32;&#32;&#40;&#116;&#117;&#114;&#110;&#105;&#110;&#103;&#32;&#111;&#110;&#41;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"246\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1169737811419\">is the current in an <em data-effect=\"italics\">RL<\/em> circuit when switched on (Note the similarity to the exponential behavior of the voltage on a charging capacitor). The initial current is zero and approaches <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1fdf56b8a10a3a74b01868a364e185ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#47;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> with a <span data-type=\"term\" id=\"import-auto-id1169737897754\">characteristic time constant<\/span><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/><br \/>\nfor an <em data-effect=\"italics\">RL<\/em> circuit, given by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac6b0bcfadbc0d29666e34bc0b5d13e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#82;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1169737803195\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> has units of seconds, since<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-81984090d6e01ce66ccb587a60effa85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&Omega;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&middot;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"60\" style=\"vertical-align: -1px;\" \/>.<br \/>\n In the first period of time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, the current rises from zero to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-38d150e3426666c7ca6f3b95955dadc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#50;&#125;&#123;&#73;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"55\" style=\"vertical-align: -3px;\" \/>, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0438ff820a705e6251fe6613c59df8b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#61;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#123;&#101;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#50;&#125;&#123;&#73;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"330\" style=\"vertical-align: -7px;\" \/>. The current will go 0.632 of the remainder in the next time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>. A well-known property of the exponential is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>. In just a few multiples of the time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, the final value is very nearly achieved, as the graph in <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a>(b) illustrates.<\/p>\n<p id=\"import-auto-id1169735754248\">The characteristic time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> depends on only two factors, the inductance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> and the resistance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. The greater the inductance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, the greater <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, the greater <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases\u2014large <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> and small <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> \u2014more energy is stored in the inductor and more time is required to get it in and out.<\/p>\n<p id=\"import-auto-id1169737882691\">When the switch in <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a>(a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. But this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ec3888472254de786ff82edf4134c36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#47;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#73;&#125;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"67\" style=\"vertical-align: -4px;\" \/>, stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bf66bd6354707b6d049820e77725a717_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#73;&#125;&#94;&#123;&#50;&#125;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: 0px;\" \/>. Once again the behavior is exponential, and<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><br \/>\n is found to be<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0fa92e01ed518827a2047f3b92695cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#61;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#123;&#101;&#125;&#94;&#123;&#45;&#116;&#47;&#92;&#116;&#97;&#117;&#32;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#32;&#32;&#32;&#40;&#116;&#117;&#114;&#110;&#105;&#110;&#103;&#32;&#111;&#102;&#102;&#41;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"195\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1169737711668\">(See <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a>(c).) In the first period of time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5781d2bfee04d7be6020a63b01513a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;&#61;&#76;&#47;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -5px;\" \/> after the switch is closed, the current falls to 0.368 of its initial value, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e6a86eb7e7c26aad37091d34bfe0fb86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#61;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#123;&#101;&#125;&#94;&#123;&#45;&#49;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#56;&#125;&#123;&#73;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -3px;\" \/>. In each successive time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, the current falls to 0.368 of the preceding value, and in a few multiples of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, the current becomes very close to zero, as seen in the graph in <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a>(c).<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1169738136126\">\n<div data-type=\"title\" class=\"title\">Calculating Characteristic Time and Current in an <em data-effect=\"italics\">RL<\/em> Circuit<\/div>\n<p id=\"import-auto-id1169737045564\">(a) What is the characteristic time constant for a 7.50 mH inductor in series with a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-35f3115f697c79ab99b6e4c954f2018d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#46;&#48;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> resistor? (b) Find the current 5.00 ms after the switch is moved to position 2 to disconnect the battery, if it is initially 10.0 A.<\/p>\n<p id=\"import-auto-id1169737804462\"><strong>Strategy for (a)<\/strong><\/p>\n<p id=\"import-auto-id1169737787249\">The time constant for an <em data-effect=\"italics\">RL<\/em> circuit is defined by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5781d2bfee04d7be6020a63b01513a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;&#61;&#76;&#47;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<p id=\"import-auto-id1169737713276\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1169738036454\">Entering known values into the expression for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> given in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5781d2bfee04d7be6020a63b01513a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;&#61;&#76;&#47;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -5px;\" \/> yields<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b9c93c3b1b2676acd66dcfbfcaa61419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#82;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#46;&#53;&#48;&#32;&#109;&#72;&#125;&#123;&#51;&#46;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;&#125;&#61;&#50;&#46;&#53;&#48;&#32;&#109;&#115;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"208\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1169738164481\"><strong>Discussion for (a)<\/strong><\/p>\n<p id=\"import-auto-id1169737723042\">This is a small but definitely finite time. The coil will be very close to its full current in about ten time constants, or about 25 ms.<\/p>\n<p id=\"import-auto-id1169736758585\"><strong>Strategy for (b)<\/strong><\/p>\n<p id=\"import-auto-id1169738010703\">We can find the current by using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b8eb032261443d6a0a437955ed372c81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#61;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#123;&#101;&#125;&#94;&#123;&#45;&#116;&#47;&#92;&#116;&#97;&#117;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"87\" style=\"vertical-align: -3px;\" \/>, or by considering the decline in steps. Since the time is twice the characteristic time, we consider the process in steps.<\/p>\n<p id=\"import-auto-id1169736614228\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1169738189000\">In the first 2.50 ms, the current declines to 0.368 of its initial value, which is<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e806b2307fced54738cedeb24ca6d51e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#73;&#38;&#32;&#61;&#38;&#32;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#56;&#125;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#46;&#48;&#32;&#65;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#32;&#65;&#32;&#97;&#116;&#32;&#125;&#116;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#109;&#115;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"256\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1169738055826\">After another 2.50 ms, or a total of 5.00 ms, the current declines to 0.368 of the value just found. That is,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fd2ae31d7e989b3456976b9377797bcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#73;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#38;&#32;&#61;&#38;&#32;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#56;&#125;&#73;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#46;&#54;&#56;&#32;&#65;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#65;&#32;&#97;&#116;&#32;&#125;&#116;&#61;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#109;&#115;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"254\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1169737972496\"><strong>Discussion for (b)<\/strong><\/p>\n<p id=\"import-auto-id1169737896000\">After another 5.00 ms has passed, the current will be 0.183 A (see <a href=\"#fs-id1169738227350\" class=\"autogenerated-content\">(Figure)<\/a>); so, although it does die out, the current certainly does not go to zero instantaneously.<\/p>\n<\/div>\n<p id=\"import-auto-id1169737717434\">In summary, when the voltage applied to an inductor is changed, the current also changes, <em data-effect=\"italics\"><em data-effect=\"italics\">but the change in current lags the change in voltage in an RL circuit<\/em><\/em>. In <a href=\"\/contents\/663b7dc1-df6f-4a85-bae8-10d20a358d01@4\">Reactance, Inductive and Capacitive<\/a>, we explore how an <em data-effect=\"italics\">RL<\/em> circuit behaves when a sinusoidal AC voltage is applied.<\/p>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1169737941056\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1169736611192\">\n<li id=\"import-auto-id1169737712209\">When a series connection of a resistor and an inductor\u2014an <em data-effect=\"italics\">RL<\/em> circuit\u2014is connected to a voltage source, the time variation of the current is\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0397fac88fd7224885ddb4ea1343c4fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#61;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#123;&#101;&#125;&#94;&#123;&#45;&#116;&#47;&#92;&#116;&#97;&#117;&#32;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#32;&#32;&#32;&#40;&#116;&#117;&#114;&#110;&#105;&#110;&#103;&#32;&#111;&#110;&#41;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"246\" style=\"vertical-align: -7px;\" \/><\/div>\n<p>        where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-231686fc00a3a1e56641a9954895174a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#61;&#86;&#47;&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -5px;\" \/> is the final current.<\/p>\n<\/li>\n<li id=\"import-auto-id1169737936637\">The characteristic time constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-06444a4679c4069aeb6c15fe07381b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/> , where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>  is the inductance and  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the resistance.<\/li>\n<li id=\"import-auto-id1169738035430\">In the first time constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, the current rises from zero to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-38d150e3426666c7ca6f3b95955dadc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#50;&#125;&#123;&#73;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"55\" style=\"vertical-align: -3px;\" \/>, and 0.632 of the remainder in every subsequent time interval <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>.<\/li>\n<li id=\"import-auto-id1169738048163\">When the inductor is shorted through a resistor, current decreases as\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0fa92e01ed518827a2047f3b92695cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#61;&#123;&#73;&#125;&#95;&#123;&#48;&#125;&#123;&#101;&#125;&#94;&#123;&#45;&#116;&#47;&#92;&#116;&#97;&#117;&#32;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#32;&#32;&#32;&#40;&#116;&#117;&#114;&#110;&#105;&#110;&#103;&#32;&#111;&#102;&#102;&#41;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"195\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4093fe819f69ef3bfc49cef36758c7a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#73;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> is the initial current.<\/p>\n<\/li>\n<li id=\"import-auto-id1169737060067\">Current falls to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdb81dae22739b49abb7ca85eaaa0256_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#56;&#125;&#123;&#73;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"55\" style=\"vertical-align: -3px;\" \/> in the first time interval <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, and 0.368 of the remainder toward zero in each subsequent time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>.<\/li>\n<\/ul>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1169738010703\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738227350\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738245588\">\n<p id=\"import-auto-id1169738210359\">If you want a characteristic <em data-effect=\"italics\">RL<\/em> time constant of 1.00 s, and you have a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4adb59d4b364f93efd3c63b30dc4a008_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: 0px;\" \/> resistor, what value of self-inductance is needed?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737933054\">\n<p id=\"import-auto-id1169738164414\">500 H<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737723038\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736814918\">\n<p id=\"import-auto-id1169738117421\">Your <em data-effect=\"italics\">RL<\/em> circuit has a characteristic time constant of 20.0 ns, and a resistance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9a70bf68bb98b5ddb048a5d40226e79b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&#48;&#32;&#77;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"53\" style=\"vertical-align: -1px;\" \/>. (a) What is the inductance of the circuit? (b) What resistance would give you a 1.00 ns time constant, perhaps needed for quick response in an oscilloscope?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737788008\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736620644\">\n<p>A large superconducting magnet, used for magnetic resonance imaging, has a 50.0 H inductance. If you want current through it to be adjustable with a 1.00 s characteristic time constant, what is the minimum resistance of system?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738094791\">\n<p id=\"import-auto-id1169736659279\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1dda6b705847660e691936b00150b633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#46;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736581925\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736620836\">\n<p id=\"import-auto-id1169738110369\">Verify that after a time of 10.0 ms, the current for the situation considered in <a href=\"#fs-id1169738136126\" class=\"autogenerated-content\">(Figure)<\/a> will be 0.183 A as stated.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736596063\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736590714\">\n<p id=\"import-auto-id1169737770657\">Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H, and resistors ranging from<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-02630e5443d2170dabb4989af79e6765_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#49;&#48;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/> to<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b0d720864002e084e5497406912c7756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#48;&#48;&#32;&#77;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/>. What is the range of characteristic <em data-effect=\"italics\">RL<\/em> time constants you can produce by connecting a single resistor to a single inductor?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738045475\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-69d3d4b8b1781e80ce4467b3684de2db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#56;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"79\" style=\"vertical-align: -1px;\" \/> to 0.100 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737812307\">\n<p id=\"import-auto-id1169738113687\">(a) What is the characteristic time constant of a 25.0 mH inductor that has a resistance of<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-956af898020ec2d9bf3caf4effe5ec52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#46;&#48;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/>? (b) If it is connected to a 12.0 V battery, what is the current after 12.5 ms?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737723042\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737785691\">\n<p id=\"import-auto-id1169738075685\">What percentage of the final current<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7bfb9cc31fbdc5acae7770646a2b918b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#73;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/><br \/>\n flows through an inductor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> in series with a resistor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, three time constants after the circuit is completed?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169736758585\">\n<p id=\"import-auto-id1169738251076\">95.0%<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738248980\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738223614\">\n<p id=\"import-auto-id1169737898346\">The 5.00 A current through a 1.50 H inductor is dissipated by a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d570de15027e8206b7b384fab1df2354_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#48;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> resistor in a circuit like that in <a href=\"#import-auto-id1169737969272\" class=\"autogenerated-content\">(Figure)<\/a> with the switch in position 2. (a) What is the initial energy in the inductor? (b) How long will it take the current to decline to 5.00% of its initial value? (c) Calculate the average power dissipated, and compare it with the initial power dissipated by the resistor.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737787416\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736624109\">\n<p id=\"import-auto-id1169738250359\">(a) Use the exact exponential treatment to find how much time is required to bring the current through an 80.0 mH inductor in series with a<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31c4bfc04558b8426d92dad6213cb0ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#46;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><br \/>\n resistor to 99.0% of its final value, starting from zero. (b) Compare your answer to the approximate treatment using integral numbers of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>. (c) Discuss how significant the difference is.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737768546\">\n<p id=\"import-auto-id1169738036121\">(a) 24.6 ms<\/p>\n<p id=\"import-auto-id1169737895428\">(b) 26.7 ms<\/p>\n<p id=\"import-auto-id1169738051862\">(c) 9% difference, which is greater than the inherent uncertainty in the given parameters.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737851564\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738011205\">\n<p id=\"import-auto-id1169737961873\">(a) Using the exact exponential treatment, find the time required for the current through a 2.00 H inductor in series with a<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42374360f5bee543976e6d7e1ebf6cc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#53;&#48;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><br \/>\n resistor to be reduced to 0.100% of its original value. (b) Compare your answer to the approximate treatment using integral numbers of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>. (c) Discuss how significant the difference is.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1169738071203\">\n<dt>characteristic time constant<\/dt>\n<dd id=\"fs-id1169736581702\"> denoted by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac03dc06d394ae61cf07e44856527651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, of a particular series <em data-effect=\"italics\">RL<\/em> circuit is calculated by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-06444a4679c4069aeb6c15fe07381b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>  is the inductance and  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the resistance<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1359","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1290,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1359","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1359\/revisions"}],"predecessor-version":[{"id":1360,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1359\/revisions\/1360"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1290"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1359\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1359"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1359"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1359"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1359"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}