{"id":4816,"date":"2014-12-11T02:29:23","date_gmt":"2014-12-11T02:29:23","guid":{"rendered":"https:\/\/courses.candelalearning.com\/colphysics\/?post_type=chapter&#038;p=4816"},"modified":"2016-11-03T18:42:05","modified_gmt":"2016-11-03T18:42:05","slug":"22-9-magnetic-fields-produced-by-currents-amperes-law","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-physics\/chapter\/22-9-magnetic-fields-produced-by-currents-amperes-law\/","title":{"raw":"Magnetic Fields Produced by Currents: Ampere\u2019s Law","rendered":"Magnetic Fields Produced by Currents: Ampere\u2019s Law"},"content":{"raw":"<div>\r\n<div>\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<div>\r\n<ul>\r\n\t<li>Calculate current that produces a magnetic field.<\/li>\r\n\t<li>Use the right hand rule 2 to determine the direction of current or the direction of magnetic field loops.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nHow much current is needed to produce a significant magnetic field, perhaps as strong as the Earth\u2019s field? Surveyors will tell you that overhead electric power lines create magnetic fields that interfere with their compass readings. Indeed, when Oersted discovered in 1820 that a current in a wire affected a compass needle, he was not dealing with extremely large currents. How does the shape of wires carrying current affect the shape of the magnetic field created? We noted earlier that a current loop created a magnetic field similar to that of a bar magnet, but what about a straight wire or a toroid (doughnut)? How is the direction of a current-created field related to the direction of the current? Answers to these questions are explored in this section, together with a brief discussion of the law governing the fields created by currents.\r\n<div title=\"Magnetic Field Created by a Long Straight Current-Carrying Wire: Right Hand Rule 2\">\r\n<div>\r\n<div>\r\n<div>\r\n<h2>Magnetic Field Created by a Long Straight Current-Carrying Wire: Right Hand Rule 2<\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nMagnetic fields have both direction and magnitude. As noted before, one way to explore the direction of a magnetic field is with compasses, as shown for a long straight current-carrying wire in Figure 1. Hall probes can determine the magnitude of the field. The field around a long straight wire is found to be in circular loops. The <em> right hand rule 2<\/em> (RHR-2) emerges from this exploration and is valid for any current segment\u2014<em>point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops<\/em> created by it.\r\n<div title=\"Figure 22.38.\">\r\n<div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"250\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20110611\/Figure_23_09_01a.jpg\" alt=\"Figure a shows a vertically oriented wire with current I running from bottom to top. Magnetic field lines circle the wire counter-clockwise as view from the top. Figure b illustrates the right hand rule 2. The thumb points up with current I. The fingers curl around counterclockwise as viewed from the top.\" width=\"250\" height=\"839\" \/> Figure 1. (a) Compasses placed near a long straight current-carrying wire indicate that field lines form circular loops centered on the wire. (b) Right hand rule 2 states that, if the right hand thumb points in the direction of the current, the fingers curl in the direction of the field. This rule is consistent with the field mapped for the long straight wire and is valid for any current segment.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe <em> magnetic field strength (magnitude) produced by a long straight current-carrying wire<\/em> is found by experiment to be\r\n<p style=\"text-align: center;\">[latex]B=\\frac{{\\mu}_{0}I}{2\\pi r}\\left(\\text{long straight wire}\\right)\\\\[\/latex],<\/p>\r\nwhere <em>I<\/em> is the current, <em>r<\/em> is the shortest distance to the wire, and the constant\u00a0[latex]{\\mu }_{0}=4\\pi \\times 10^{-7}\\text{T}\\cdot\\text{ m\/A}\\\\[\/latex]\u00a0is the <em> permeability of free space<\/em>. (<em>\u03bc<\/em><sub>0<\/sub> is one of the basic constants in nature. We will see later that <em>\u03bc<\/em><sub>0<\/sub> is related to the speed of light.) Since the wire is very long, the magnitude of the field depends only on distance from the wire <em>r<\/em>, not on position along the wire.\r\n<div title=\"Example 22.6. Calculating Current that Produces a Magnetic Field\">\r\n<div title=\"Example 22.6. Calculating Current that Produces a Magnetic Field\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 1. Calculating Current that Produces a Magnetic Field<\/h3>\r\n<div>\r\n\r\nFind the current in a long straight wire that would produce a magnetic field twice the strength of the Earth\u2019s at a distance of 5.0 cm from the wire.\r\n<h4><strong>Strategy<\/strong><\/h4>\r\nThe Earth\u2019s field is about\u00a05.0 \u00d7 10<sup>\u22125\u00a0<\/sup>T, and so here <em>B<\/em> due to the wire is taken to be\u00a0<span id=\"MathJax-Span-91782\" class=\"mrow\"><span id=\"MathJax-Span-91783\" class=\"semantics\"><span id=\"MathJax-Span-91784\" class=\"mrow\"><span id=\"MathJax-Span-91785\" class=\"mrow\"><span id=\"MathJax-Span-91786\" class=\"mrow\"><span id=\"MathJax-Span-91787\" class=\"mrow\"><span id=\"MathJax-Span-91788\" class=\"mn\">1<\/span><span id=\"MathJax-Span-91789\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-91790\" class=\"mrow\"><span id=\"MathJax-Span-91791\" class=\"mn\">0\u00a0<\/span><span id=\"MathJax-Span-91792\" class=\"mo\">\u00d7\u00a0<\/span><span id=\"MathJax-Span-91793\" class=\"msup\"><span id=\"MathJax-Span-91794\" class=\"mtext\">10<\/span><sup><span id=\"MathJax-Span-91795\" class=\"mrow\"><span id=\"MathJax-Span-91796\" class=\"mrow\"><span id=\"MathJax-Span-91797\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-91798\" class=\"mn\">4\u00a0<\/span><\/span><\/span><\/sup><\/span><\/span><span id=\"MathJax-Span-91799\" class=\"mspace\"><\/span><span id=\"MathJax-Span-91800\" class=\"mn\">T<\/span><\/span><\/span><span id=\"MathJax-Span-91801\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span>. The equation [latex]B=\\frac{\\mu_{0}I}{2\\pi r}\\\\[\/latex]\u00a0can be used to find <em>I<\/em>, since all other quantities are known.\r\n<h4><strong>Solution<\/strong><\/h4>\r\nSolving for <em>I<\/em> and entering known values gives\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}I&amp; =&amp; \\frac{2\\pi rB}{\\mu _{0}}=\\frac{2\\pi\\left(5.0\\times 10^{-2}\\text{ m}\\right)\\left(1.0\\times 10^{-4}\\text{ T}\\right)}{4\\pi \\times 10^{-7}\\text{ T}\\cdot\\text{m\/A}}\\\\ &amp; =&amp; 25\\text{ A}\\end{array}\\\\[\/latex]<\/p>\r\n\r\n<h4><strong>Discussion<\/strong><\/h4>\r\nSo a moderately large current produces a significant magnetic field at a distance of 5.0 cm from a long straight wire. Note that the answer is stated to only two digits, since the Earth\u2019s field is specified to only two digits in this example.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Ampere\u2019s Law and Others\">\r\n<div>\r\n<div>\r\n<div>\r\n<h2>Ampere\u2019s Law and Others<\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe magnetic field of a long straight wire has more implications than you might at first suspect. <em>Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment.<\/em> The formal statement of the direction and magnitude of the field due to each segment is called the <em> Biot-Savart law<\/em>. Integral calculus is needed to sum the field for an arbitrary shape current. This results in a more complete law, called <em> Ampere\u2019s law<\/em>, which relates magnetic field and current in a general way. Ampere\u2019s law in turn is a part of <em> Maxwell\u2019s equations<\/em>, which give a complete theory of all electromagnetic phenomena. Considerations of how Maxwell\u2019s equations appear to different observers led to the modern theory of relativity, and the realization that electric and magnetic fields are different manifestations of the same thing. Most of this is beyond the scope of this text in both mathematical level, requiring calculus, and in the amount of space that can be devoted to it. But for the interested student, and particularly for those who continue in physics, engineering, or similar pursuits, delving into these matters further will reveal descriptions of nature that are elegant as well as profound. In this text, we shall keep the general features in mind, such as RHR-2 and the rules for magnetic field lines listed in <a title=\"22.3. Magnetic Fields and Magnetic Field Lines\" href=\".\/chapter\/22-3-magnetic-fields-and-magnetic-field-lines\/\" target=\"_blank\">Magnetic Fields and Magnetic Field Lines<\/a>, while concentrating on the fields created in certain important situations.\r\n<div>\r\n<div class=\"textbox learning-objectives\">\r\n<h3><strong>Making Connections: Relativity<\/strong><\/h3>\r\n<div>Hearing all we do about Einstein, we sometimes get the impression that he invented relativity out of nothing. On the contrary, one of Einstein\u2019s motivations was to solve difficulties in knowing how different observers see magnetic and electric fields.<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Magnetic Field Produced by a Current-Carrying Circular Loop\">\r\n<div>\r\n<div>\r\n<div>\r\n<h2>Magnetic Field Produced by a Current-Carrying Circular Loop<\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe magnetic field near a current-carrying loop of wire is shown in Figure 2. Both the direction and the magnitude of the magnetic field produced by a current-carrying loop are complex. RHR-2 can be used to give the direction of the field near the loop, but mapping with compasses and the rules about field lines given in <a title=\"22.3. Magnetic Fields and Magnetic Field Lines\" href=\".\/chapter\/22-3-magnetic-fields-and-magnetic-field-lines\/\" target=\"_blank\">Magnetic Fields and Magnetic Field Lines<\/a> are needed for more detail. There is a simple formula for the <em> magnetic field strength at the center of a circular loop<\/em>. It is\r\n<p style=\"text-align: center;\">[latex]B=\\frac{\\mu_{0}I}{2R}\\left(\\text{at center of loop}\\right)\\\\[\/latex],<\/p>\r\nwhere <em>R<\/em> is the radius of the loop. This equation is very similar to that for a straight wire, but it is valid <em>only<\/em> at the center of a circular loop of wire. The similarity of the equations does indicate that similar field strength can be obtained at the center of a loop. One way to get a larger field is to have <em>N<\/em> loops; then, the field is <em>B\u00a0<\/em>=\u00a0<em>N\u03bc<\/em><sub>0<\/sub><em>I<\/em>\/(2<em>R<\/em>). Note that the larger the loop, the smaller the field at its center, because the current is farther away.\r\n<div title=\"Figure 22.39.\">\r\n<div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"450\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20110618\/Figure_23_09_02a.jpg\" alt=\"Figure a illustrates use of the right hand rule 2 to determine the direction of the magnetic field around a current-carrying loop. The right hand thumb points in the direction of I while the fingers curl around in the direction of B. Figure b shows the magnetic field lines circling the wire, as viewed from the side.\" width=\"450\" height=\"505\" \/> Figure 2. (a) RHR-2 gives the direction of the magnetic field inside and outside a current-carrying loop. (b) More detailed mapping with compasses or with a Hall probe completes the picture. The field is similar to that of a bar magnet.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Magnetic Field Produced by a Current-Carrying Solenoid\">\r\n<div>\r\n<div>\r\n<div>\r\n<h2>Magnetic Field Produced by a Current-Carrying Solenoid<\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nA <em> solenoid<\/em> is a long coil of wire (with many turns or loops, as opposed to a flat loop). Because of its shape, the field inside a solenoid can be very uniform, and also very strong. The field just outside the coils is nearly zero. Figure 3\u00a0shows how the field looks and how its direction is given by RHR-2.\r\n<div title=\"Figure 22.40.\">\r\n<div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"550\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20110619\/Figure_23_09_03a.jpg\" alt=\"A diagram of a solenoid. The current runs up from the battery on the left side and spirals around with the solenoid wire such that the current runs upward in the front sections of the solenoid and then down the back. An illustration of the right hand rule 2 shows the thumb pointing up in the direction of the current and the fingers curling around in the direction of the magnetic field. A length wise cutaway of the solenoid shows magnetic field lines densely packed and running from the south pole to the north pole, through the solenoid. Lines outside the solenoid are spaced much farther apart and run from the north pole out around the solenoid to the south pole.\" width=\"550\" height=\"479\" \/> Figure 3. (a) Because of its shape, the field inside a solenoid of length l is remarkably uniform in magnitude and direction, as indicated by the straight and uniformly spaced field lines. The field outside the coils is nearly zero. (b) This cutaway shows the magnetic field generated by the current in the solenoid.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the <em> magnetic field strength inside a solenoid<\/em> is simply\r\n<p style=\"text-align: center;\">[latex]B={\\mu }_{0}nI\\left(\\text{inside a solenoid}\\right)\\\\[\/latex],<\/p>\r\nwhere <em>n<\/em> is the number of loops per unit length of the solenoid (<em>n\u00a0<\/em>=\u00a0<em>N<\/em>\/<em>l<\/em>, with <em>N<\/em> being the number of loops and <em>l<\/em> the length). Note that <em>B<\/em> is the field strength anywhere in the uniform region of the interior and not just at the center. Large uniform fields spread over a large volume are possible with solenoids, as Example 2\u00a0implies.\r\n<div title=\"Example 22.7. Calculating Field Strength inside a Solenoid\">\r\n<div title=\"Example 22.7. Calculating Field Strength inside a Solenoid\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 2. Calculating Field Strength inside a Solenoid<\/h3>\r\n<div>\r\n\r\nWhat is the field inside a 2.00-m-long solenoid that has 2000 loops and carries a 1600-A current?\r\n<h4><strong>Strategy<\/strong><\/h4>\r\nTo find the field strength inside a solenoid, we use [latex]B={\\mu }_{0}nI\\\\[\/latex]. First, we note the number of loops per unit length is\r\n<p style=\"text-align: center;\">[latex]n=\\frac{N}{l}=\\frac{2000}{2.00\\text{ m}}=1000\\text{ m}^{-1}=10{\\text{ cm}}^{-1}\\\\[\/latex].<\/p>\r\n<strong>Solution<\/strong> Substituting known values gives\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}B &amp; =&amp; {\\mu}_{0}nI=\\left(4\\pi \\times 10^{-7}\\text{ T}\\cdot\\text{m\/A}\\right)\\left(1000\\text{ m}^{-1}\\right)\\left(1600\\text{ A}\\right)\\\\ &amp; =&amp; 2.01\\text{ T}\\end{array}\\\\[\/latex]<\/p>\r\n\r\n<h4><strong>Discussion<\/strong><\/h4>\r\nThis is a large field strength that could be established over a large-diameter solenoid, such as in medical uses of magnetic resonance imaging (MRI). The very large current is an indication that the fields of this strength are not easily achieved, however. Such a large current through 1000 loops squeezed into a meter\u2019s length would produce significant heating. Higher currents can be achieved by using superconducting wires, although this is expensive. There is an upper limit to the current, since the superconducting state is disrupted by very large magnetic fields.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThere are interesting variations of the flat coil and solenoid. For example, the toroidal coil used to confine the reactive particles in tokamaks is much like a solenoid bent into a circle. The field inside a toroid is very strong but circular. Charged particles travel in circles, following the field lines, and collide with one another, perhaps inducing fusion. But the charged particles do not cross field lines and escape the toroid. A whole range of coil shapes are used to produce all sorts of magnetic field shapes. Adding ferromagnetic materials produces greater field strengths and can have a significant effect on the shape of the field. Ferromagnetic materials tend to trap magnetic fields (the field lines bend into the ferromagnetic material, leaving weaker fields outside it) and are used as shields for devices that are adversely affected by magnetic fields, including the Earth\u2019s magnetic field.\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h2><strong>PhET Explorations: Generator<\/strong><\/h2>\r\n<div>Generate electricity with a bar magnet! Discover the physics behind the phenomena by exploring magnets and how you can use them to make a bulb light.<\/div>\r\n<\/div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<a href=\"http:\/\/phet.colorado.edu\/sims\/faraday\/generator_en.jnlp\"><img src=\"http:\/\/phet.colorado.edu\/sims\/faraday\/generator-600.png\" alt=\"Generator screenshot\" width=\"300\" height=\"197\" \/><\/a> Click to download the simulation. Run using Java.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Section Summary<\/h2>\r\n<ul class=\"itemizedlist\">\r\n\t<li class=\"listitem\">The strength of the magnetic field created by current in a long straight wire is given by\r\n<div id=\"m42382-eip-807\" class=\"equation\" title=\"Equation 22.36.\">\r\n<div class=\"mediaobject\" style=\"text-align: center;\">[latex]B=\\frac{{\\mu}_{0}I}{2\\pi r}\\left(\\text{long straight wire}\\right)\\\\[\/latex]<\/div>\r\n<\/div>\r\nwhere <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>I<\/em><\/span><\/span> is the current, <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>r<\/em><\/span><\/span> is the shortest distance to the wire, and the constant[latex]{\\mu}_{0}=4\\pi \\times 10^{-7}\\text{ T}\\cdot\\text{ m\/A}\\\\[\/latex]\u00a0is the permeability of free space.<\/li>\r\n\t<li class=\"listitem\">The direction of the magnetic field created by a long straight wire is given by right hand rule 2 (RHR-2): <span class=\"emphasis\"><em>Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops<\/em><\/span> created by it.<\/li>\r\n\t<li class=\"listitem\">The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampere\u2019s law.<\/li>\r\n\t<li class=\"listitem\">The magnetic field strength at the center of a circular loop is given by\r\n<div id=\"m42382-eip-430\" class=\"equation\" title=\"Equation 22.37.\">\r\n<div class=\"mediaobject\" style=\"text-align: center;\">[latex]B=\\frac{\\mu_{0}I}{2R}\\left(\\text{at center of loop}\\right)\\\\[\/latex]<\/div>\r\n<\/div>\r\nwhere <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>R<\/em><\/span><\/span> is the radius of the loop. This equation becomes\u00a0<span id=\"MathJax-Span-94193\" class=\"mrow\"><span id=\"MathJax-Span-94194\" class=\"mi\">B\u00a0<\/span><span id=\"MathJax-Span-94195\" class=\"mo\">=<em>\u00a0<\/em><\/span><span id=\"MathJax-Span-94196\" class=\"msub\"><em><span id=\"MathJax-Span-94197\" class=\"mi\">\u03bc<\/span><\/em><sub><span id=\"MathJax-Span-94198\" class=\"mrow\"><span id=\"MathJax-Span-94199\" class=\"mn\">0<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-94200\" class=\"mrow\"><em><span id=\"MathJax-Span-94201\" class=\"mstyle\"><span id=\"MathJax-Span-94202\" class=\"mrow\"><span id=\"MathJax-Span-94203\" class=\"mrow\"><span id=\"MathJax-Span-94204\" class=\"mtext\">nI<\/span><\/span><\/span><\/span><\/em><span id=\"MathJax-Span-94205\" class=\"mo\">\/<\/span><span id=\"MathJax-Span-94206\" class=\"mo\">(<\/span><\/span><span id=\"MathJax-Span-94207\" class=\"mn\">2<\/span><em><span id=\"MathJax-Span-94208\" class=\"mi\">R<\/span><\/em><span id=\"MathJax-Span-94209\" class=\"mo\">)<\/span>\u00a0for a flat coil of <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>N<\/em><\/span><\/span> loops. RHR-2 gives the direction of the field about the loop. A long coil is called a solenoid.<\/li>\r\n\t<li class=\"listitem\">The magnetic field strength inside a solenoid is\r\n<div id=\"m42382-eip-942\" class=\"equation\" title=\"Equation 22.38.\">\r\n<div class=\"mediaobject\" style=\"text-align: center;\">[latex]B={\\mu }_{0}\\text{nI}\\left(\\text{inside a solenoid}\\right)\\\\[\/latex]<\/div>\r\n<\/div>\r\nwhere <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>n<\/em><\/span><\/span> is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<section>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Conceptual Questions<\/h3>\r\n<div>\r\n<div>\r\n\r\n1. Make a drawing and use RHR-2 to find the direction of the magnetic field of a current loop in a motor (such as in Figure 1 from <a href=\".\/chapter\/22-8-torque-on-a-current-loop-motors-and-meters\/\" target=\"_blank\">Torque on a Current Loop<\/a>). Then show that the direction of the torque on the loop is the same as produced by like poles repelling and unlike poles attracting.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl><dt>right hand rule 2 (RHR-2):<\/dt><dd>a rule to determine the direction of the magnetic field induced by a current-carrying wire: Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops<\/dd><\/dl><dl><dt>magnetic field strength (magnitude) produced by a long straight current-carrying wire:<\/dt><dd>defined as [latex]B=\\frac{\\mu_{0}I}{2\\pi r}\\\\[\/latex], where <em>I\u00a0<\/em>is the current,<em> r<\/em>\u00a0is the shortest distance to the wire, and\u00a0<em><span id=\"MathJax-Span-94293\" class=\"mi\">\u03bc<\/span><\/em><sub><span id=\"MathJax-Span-94294\" class=\"mrow\"><span id=\"MathJax-Span-94295\" class=\"mn\">0<\/span><\/span><\/sub> is the permeability of free space<\/dd><\/dl><dl><dt>permeability of free space:<\/dt><dd>the measure of the ability of a material, in this case free space, to support a magnetic field; the constant [latex]\\mu_{0}=4\\pi \\times 10^{-7}T\\cdot \\text{m\/A}\\\\[\/latex]<\/dd><\/dl><dl><dt>magnetic field strength at the center of a circular loop:<\/dt><dd>defined as [latex]B=\\frac{{\\mu }_{0}I}{2R}\\\\[\/latex] where <em>R<\/em>\u00a0is the radius of the loop<\/dd><\/dl><dl><dt>solenoid:<\/dt><dd>a thin wire wound into a coil that produces a magnetic field when an electric current is passed through it<\/dd><\/dl><dl><dt>magnetic field strength inside a solenoid:<\/dt><dd>defined as [latex]B={\\mu }_{0}\\text{nI}\\\\[\/latex] where <em>n<\/em>\u00a0is the number of loops per unit length of the solenoid\u00a0<em><span id=\"MathJax-Element-5847-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-94377\" class=\"math\"><span id=\"MathJax-Span-94378\" class=\"mrow\"><span id=\"MathJax-Span-94379\" class=\"semantics\"><span id=\"MathJax-Span-94380\" class=\"mrow\"><span id=\"MathJax-Span-94381\" class=\"mrow\"><span id=\"MathJax-Span-94382\" class=\"mrow\"><span id=\"MathJax-Span-94383\" class=\"mrow\"><span id=\"MathJax-Span-94385\" class=\"mrow\"><span id=\"MathJax-Span-94386\" class=\"mi\">n\u00a0<\/span><span id=\"MathJax-Span-94387\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-94388\" class=\"mrow\"><span id=\"MathJax-Span-94389\" class=\"mi\">N<\/span><span id=\"MathJax-Span-94390\" class=\"mo\">\/<\/span><span id=\"MathJax-Span-94391\" class=\"mi\">l<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/em>, with <em><span id=\"MathJax-Element-5848-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-94392\" class=\"math\"><span id=\"MathJax-Span-94393\" class=\"mrow\"><span id=\"MathJax-Span-94394\" class=\"semantics\"><span id=\"MathJax-Span-94395\" class=\"mrow\"><span id=\"MathJax-Span-94396\" class=\"mrow\"><span id=\"MathJax-Span-94397\" class=\"mrow\"><span id=\"MathJax-Span-94398\" class=\"mi\">N<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/em> being the number of loops and<em> l\u00a0<\/em>the length)<\/dd><\/dl><dl><dt>Biot-Savart law:<\/dt><dd>a physical law that describes the magnetic field generated by an electric current in terms of a specific equation<\/dd><\/dl><dl><dt>Ampere\u2019s law:<\/dt><dd>the physical law that states that the magnetic field around an electric current is proportional to the current; each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment<\/dd><\/dl><dl><dt>Maxwell\u2019s equations:<\/dt><dd>a set of four equations that describe electromagnetic phenomena<\/dd><\/dl><\/div>","rendered":"<div>\n<div>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<div>\n<ul>\n<li>Calculate current that produces a magnetic field.<\/li>\n<li>Use the right hand rule 2 to determine the direction of current or the direction of magnetic field loops.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>How much current is needed to produce a significant magnetic field, perhaps as strong as the Earth\u2019s field? Surveyors will tell you that overhead electric power lines create magnetic fields that interfere with their compass readings. Indeed, when Oersted discovered in 1820 that a current in a wire affected a compass needle, he was not dealing with extremely large currents. How does the shape of wires carrying current affect the shape of the magnetic field created? We noted earlier that a current loop created a magnetic field similar to that of a bar magnet, but what about a straight wire or a toroid (doughnut)? How is the direction of a current-created field related to the direction of the current? Answers to these questions are explored in this section, together with a brief discussion of the law governing the fields created by currents.<\/p>\n<div title=\"Magnetic Field Created by a Long Straight Current-Carrying Wire: Right Hand Rule 2\">\n<div>\n<div>\n<div>\n<h2>Magnetic Field Created by a Long Straight Current-Carrying Wire: Right Hand Rule 2<\/h2>\n<\/div>\n<\/div>\n<\/div>\n<p>Magnetic fields have both direction and magnitude. As noted before, one way to explore the direction of a magnetic field is with compasses, as shown for a long straight current-carrying wire in Figure 1. Hall probes can determine the magnitude of the field. The field around a long straight wire is found to be in circular loops. The <em> right hand rule 2<\/em> (RHR-2) emerges from this exploration and is valid for any current segment\u2014<em>point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops<\/em> created by it.<\/p>\n<div title=\"Figure 22.38.\">\n<div>\n<div>\n<div style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20110611\/Figure_23_09_01a.jpg\" alt=\"Figure a shows a vertically oriented wire with current I running from bottom to top. Magnetic field lines circle the wire counter-clockwise as view from the top. Figure b illustrates the right hand rule 2. The thumb points up with current I. The fingers curl around counterclockwise as viewed from the top.\" width=\"250\" height=\"839\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. (a) Compasses placed near a long straight current-carrying wire indicate that field lines form circular loops centered on the wire. (b) Right hand rule 2 states that, if the right hand thumb points in the direction of the current, the fingers curl in the direction of the field. This rule is consistent with the field mapped for the long straight wire and is valid for any current segment.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The <em> magnetic field strength (magnitude) produced by a long straight current-carrying wire<\/em> is found by experiment to be<\/p>\n<p style=\"text-align: center;\">[latex]B=\\frac{{\\mu}_{0}I}{2\\pi r}\\left(\\text{long straight wire}\\right)\\\\[\/latex],<\/p>\n<p>where <em>I<\/em> is the current, <em>r<\/em> is the shortest distance to the wire, and the constant\u00a0[latex]{\\mu }_{0}=4\\pi \\times 10^{-7}\\text{T}\\cdot\\text{ m\/A}\\\\[\/latex]\u00a0is the <em> permeability of free space<\/em>. (<em>\u03bc<\/em><sub>0<\/sub> is one of the basic constants in nature. We will see later that <em>\u03bc<\/em><sub>0<\/sub> is related to the speed of light.) Since the wire is very long, the magnitude of the field depends only on distance from the wire <em>r<\/em>, not on position along the wire.<\/p>\n<div title=\"Example 22.6. Calculating Current that Produces a Magnetic Field\">\n<div title=\"Example 22.6. Calculating Current that Produces a Magnetic Field\">\n<div class=\"textbox examples\">\n<h3>Example 1. Calculating Current that Produces a Magnetic Field<\/h3>\n<div>\n<p>Find the current in a long straight wire that would produce a magnetic field twice the strength of the Earth\u2019s at a distance of 5.0 cm from the wire.<\/p>\n<h4><strong>Strategy<\/strong><\/h4>\n<p>The Earth\u2019s field is about\u00a05.0 \u00d7 10<sup>\u22125\u00a0<\/sup>T, and so here <em>B<\/em> due to the wire is taken to be\u00a0<span id=\"MathJax-Span-91782\" class=\"mrow\"><span id=\"MathJax-Span-91783\" class=\"semantics\"><span id=\"MathJax-Span-91784\" class=\"mrow\"><span id=\"MathJax-Span-91785\" class=\"mrow\"><span id=\"MathJax-Span-91786\" class=\"mrow\"><span id=\"MathJax-Span-91787\" class=\"mrow\"><span id=\"MathJax-Span-91788\" class=\"mn\">1<\/span><span id=\"MathJax-Span-91789\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-91790\" class=\"mrow\"><span id=\"MathJax-Span-91791\" class=\"mn\">0\u00a0<\/span><span id=\"MathJax-Span-91792\" class=\"mo\">\u00d7\u00a0<\/span><span id=\"MathJax-Span-91793\" class=\"msup\"><span id=\"MathJax-Span-91794\" class=\"mtext\">10<\/span><sup><span id=\"MathJax-Span-91795\" class=\"mrow\"><span id=\"MathJax-Span-91796\" class=\"mrow\"><span id=\"MathJax-Span-91797\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-91798\" class=\"mn\">4\u00a0<\/span><\/span><\/span><\/sup><\/span><\/span><span id=\"MathJax-Span-91799\" class=\"mspace\"><\/span><span id=\"MathJax-Span-91800\" class=\"mn\">T<\/span><\/span><\/span><span id=\"MathJax-Span-91801\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span>. The equation [latex]B=\\frac{\\mu_{0}I}{2\\pi r}\\\\[\/latex]\u00a0can be used to find <em>I<\/em>, since all other quantities are known.<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Solving for <em>I<\/em> and entering known values gives<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}I& =& \\frac{2\\pi rB}{\\mu _{0}}=\\frac{2\\pi\\left(5.0\\times 10^{-2}\\text{ m}\\right)\\left(1.0\\times 10^{-4}\\text{ T}\\right)}{4\\pi \\times 10^{-7}\\text{ T}\\cdot\\text{m\/A}}\\\\ & =& 25\\text{ A}\\end{array}\\\\[\/latex]<\/p>\n<h4><strong>Discussion<\/strong><\/h4>\n<p>So a moderately large current produces a significant magnetic field at a distance of 5.0 cm from a long straight wire. Note that the answer is stated to only two digits, since the Earth\u2019s field is specified to only two digits in this example.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Ampere\u2019s Law and Others\">\n<div>\n<div>\n<div>\n<h2>Ampere\u2019s Law and Others<\/h2>\n<\/div>\n<\/div>\n<\/div>\n<p>The magnetic field of a long straight wire has more implications than you might at first suspect. <em>Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment.<\/em> The formal statement of the direction and magnitude of the field due to each segment is called the <em> Biot-Savart law<\/em>. Integral calculus is needed to sum the field for an arbitrary shape current. This results in a more complete law, called <em> Ampere\u2019s law<\/em>, which relates magnetic field and current in a general way. Ampere\u2019s law in turn is a part of <em> Maxwell\u2019s equations<\/em>, which give a complete theory of all electromagnetic phenomena. Considerations of how Maxwell\u2019s equations appear to different observers led to the modern theory of relativity, and the realization that electric and magnetic fields are different manifestations of the same thing. Most of this is beyond the scope of this text in both mathematical level, requiring calculus, and in the amount of space that can be devoted to it. But for the interested student, and particularly for those who continue in physics, engineering, or similar pursuits, delving into these matters further will reveal descriptions of nature that are elegant as well as profound. In this text, we shall keep the general features in mind, such as RHR-2 and the rules for magnetic field lines listed in <a title=\"22.3. Magnetic Fields and Magnetic Field Lines\" href=\".\/chapter\/22-3-magnetic-fields-and-magnetic-field-lines\/\" target=\"_blank\">Magnetic Fields and Magnetic Field Lines<\/a>, while concentrating on the fields created in certain important situations.<\/p>\n<div>\n<div class=\"textbox learning-objectives\">\n<h3><strong>Making Connections: Relativity<\/strong><\/h3>\n<div>Hearing all we do about Einstein, we sometimes get the impression that he invented relativity out of nothing. On the contrary, one of Einstein\u2019s motivations was to solve difficulties in knowing how different observers see magnetic and electric fields.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Magnetic Field Produced by a Current-Carrying Circular Loop\">\n<div>\n<div>\n<div>\n<h2>Magnetic Field Produced by a Current-Carrying Circular Loop<\/h2>\n<\/div>\n<\/div>\n<\/div>\n<p>The magnetic field near a current-carrying loop of wire is shown in Figure 2. Both the direction and the magnitude of the magnetic field produced by a current-carrying loop are complex. RHR-2 can be used to give the direction of the field near the loop, but mapping with compasses and the rules about field lines given in <a title=\"22.3. Magnetic Fields and Magnetic Field Lines\" href=\".\/chapter\/22-3-magnetic-fields-and-magnetic-field-lines\/\" target=\"_blank\">Magnetic Fields and Magnetic Field Lines<\/a> are needed for more detail. There is a simple formula for the <em> magnetic field strength at the center of a circular loop<\/em>. It is<\/p>\n<p style=\"text-align: center;\">[latex]B=\\frac{\\mu_{0}I}{2R}\\left(\\text{at center of loop}\\right)\\\\[\/latex],<\/p>\n<p>where <em>R<\/em> is the radius of the loop. This equation is very similar to that for a straight wire, but it is valid <em>only<\/em> at the center of a circular loop of wire. The similarity of the equations does indicate that similar field strength can be obtained at the center of a loop. One way to get a larger field is to have <em>N<\/em> loops; then, the field is <em>B\u00a0<\/em>=\u00a0<em>N\u03bc<\/em><sub>0<\/sub><em>I<\/em>\/(2<em>R<\/em>). Note that the larger the loop, the smaller the field at its center, because the current is farther away.<\/p>\n<div title=\"Figure 22.39.\">\n<div>\n<div>\n<div style=\"width: 460px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20110618\/Figure_23_09_02a.jpg\" alt=\"Figure a illustrates use of the right hand rule 2 to determine the direction of the magnetic field around a current-carrying loop. The right hand thumb points in the direction of I while the fingers curl around in the direction of B. Figure b shows the magnetic field lines circling the wire, as viewed from the side.\" width=\"450\" height=\"505\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. (a) RHR-2 gives the direction of the magnetic field inside and outside a current-carrying loop. (b) More detailed mapping with compasses or with a Hall probe completes the picture. The field is similar to that of a bar magnet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Magnetic Field Produced by a Current-Carrying Solenoid\">\n<div>\n<div>\n<div>\n<h2>Magnetic Field Produced by a Current-Carrying Solenoid<\/h2>\n<\/div>\n<\/div>\n<\/div>\n<p>A <em> solenoid<\/em> is a long coil of wire (with many turns or loops, as opposed to a flat loop). Because of its shape, the field inside a solenoid can be very uniform, and also very strong. The field just outside the coils is nearly zero. Figure 3\u00a0shows how the field looks and how its direction is given by RHR-2.<\/p>\n<div title=\"Figure 22.40.\">\n<div>\n<div>\n<div style=\"width: 560px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20110619\/Figure_23_09_03a.jpg\" alt=\"A diagram of a solenoid. The current runs up from the battery on the left side and spirals around with the solenoid wire such that the current runs upward in the front sections of the solenoid and then down the back. An illustration of the right hand rule 2 shows the thumb pointing up in the direction of the current and the fingers curling around in the direction of the magnetic field. A length wise cutaway of the solenoid shows magnetic field lines densely packed and running from the south pole to the north pole, through the solenoid. Lines outside the solenoid are spaced much farther apart and run from the north pole out around the solenoid to the south pole.\" width=\"550\" height=\"479\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. (a) Because of its shape, the field inside a solenoid of length l is remarkably uniform in magnitude and direction, as indicated by the straight and uniformly spaced field lines. The field outside the coils is nearly zero. (b) This cutaway shows the magnetic field generated by the current in the solenoid.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the <em> magnetic field strength inside a solenoid<\/em> is simply<\/p>\n<p style=\"text-align: center;\">[latex]B={\\mu }_{0}nI\\left(\\text{inside a solenoid}\\right)\\\\[\/latex],<\/p>\n<p>where <em>n<\/em> is the number of loops per unit length of the solenoid (<em>n\u00a0<\/em>=\u00a0<em>N<\/em>\/<em>l<\/em>, with <em>N<\/em> being the number of loops and <em>l<\/em> the length). Note that <em>B<\/em> is the field strength anywhere in the uniform region of the interior and not just at the center. Large uniform fields spread over a large volume are possible with solenoids, as Example 2\u00a0implies.<\/p>\n<div title=\"Example 22.7. Calculating Field Strength inside a Solenoid\">\n<div title=\"Example 22.7. Calculating Field Strength inside a Solenoid\">\n<div class=\"textbox examples\">\n<h3>Example 2. Calculating Field Strength inside a Solenoid<\/h3>\n<div>\n<p>What is the field inside a 2.00-m-long solenoid that has 2000 loops and carries a 1600-A current?<\/p>\n<h4><strong>Strategy<\/strong><\/h4>\n<p>To find the field strength inside a solenoid, we use [latex]B={\\mu }_{0}nI\\\\[\/latex]. First, we note the number of loops per unit length is<\/p>\n<p style=\"text-align: center;\">[latex]n=\\frac{N}{l}=\\frac{2000}{2.00\\text{ m}}=1000\\text{ m}^{-1}=10{\\text{ cm}}^{-1}\\\\[\/latex].<\/p>\n<p><strong>Solution<\/strong> Substituting known values gives<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}B & =& {\\mu}_{0}nI=\\left(4\\pi \\times 10^{-7}\\text{ T}\\cdot\\text{m\/A}\\right)\\left(1000\\text{ m}^{-1}\\right)\\left(1600\\text{ A}\\right)\\\\ & =& 2.01\\text{ T}\\end{array}\\\\[\/latex]<\/p>\n<h4><strong>Discussion<\/strong><\/h4>\n<p>This is a large field strength that could be established over a large-diameter solenoid, such as in medical uses of magnetic resonance imaging (MRI). The very large current is an indication that the fields of this strength are not easily achieved, however. Such a large current through 1000 loops squeezed into a meter\u2019s length would produce significant heating. Higher currents can be achieved by using superconducting wires, although this is expensive. There is an upper limit to the current, since the superconducting state is disrupted by very large magnetic fields.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>There are interesting variations of the flat coil and solenoid. For example, the toroidal coil used to confine the reactive particles in tokamaks is much like a solenoid bent into a circle. The field inside a toroid is very strong but circular. Charged particles travel in circles, following the field lines, and collide with one another, perhaps inducing fusion. But the charged particles do not cross field lines and escape the toroid. A whole range of coil shapes are used to produce all sorts of magnetic field shapes. Adding ferromagnetic materials produces greater field strengths and can have a significant effect on the shape of the field. Ferromagnetic materials tend to trap magnetic fields (the field lines bend into the ferromagnetic material, leaving weaker fields outside it) and are used as shields for devices that are adversely affected by magnetic fields, including the Earth\u2019s magnetic field.<\/p>\n<\/div>\n<div>\n<div class=\"textbox\">\n<div>\n<h2><strong>PhET Explorations: Generator<\/strong><\/h2>\n<div>Generate electricity with a bar magnet! Discover the physics behind the phenomena by exploring magnets and how you can use them to make a bulb light.<\/div>\n<\/div>\n<div>\n<div style=\"width: 310px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/phet.colorado.edu\/sims\/faraday\/generator_en.jnlp\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/phet.colorado.edu\/sims\/faraday\/generator-600.png\" alt=\"Generator screenshot\" width=\"300\" height=\"197\" \/><\/a><\/p>\n<p class=\"wp-caption-text\">Click to download the simulation. Run using Java.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Section Summary<\/h2>\n<ul class=\"itemizedlist\">\n<li class=\"listitem\">The strength of the magnetic field created by current in a long straight wire is given by\n<div id=\"m42382-eip-807\" class=\"equation\" title=\"Equation 22.36.\">\n<div class=\"mediaobject\" style=\"text-align: center;\">[latex]B=\\frac{{\\mu}_{0}I}{2\\pi r}\\left(\\text{long straight wire}\\right)\\\\[\/latex]<\/div>\n<\/div>\n<p>where <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>I<\/em><\/span><\/span> is the current, <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>r<\/em><\/span><\/span> is the shortest distance to the wire, and the constant[latex]{\\mu}_{0}=4\\pi \\times 10^{-7}\\text{ T}\\cdot\\text{ m\/A}\\\\[\/latex]\u00a0is the permeability of free space.<\/li>\n<li class=\"listitem\">The direction of the magnetic field created by a long straight wire is given by right hand rule 2 (RHR-2): <span class=\"emphasis\"><em>Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops<\/em><\/span> created by it.<\/li>\n<li class=\"listitem\">The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampere\u2019s law.<\/li>\n<li class=\"listitem\">The magnetic field strength at the center of a circular loop is given by\n<div id=\"m42382-eip-430\" class=\"equation\" title=\"Equation 22.37.\">\n<div class=\"mediaobject\" style=\"text-align: center;\">[latex]B=\\frac{\\mu_{0}I}{2R}\\left(\\text{at center of loop}\\right)\\\\[\/latex]<\/div>\n<\/div>\n<p>where <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>R<\/em><\/span><\/span> is the radius of the loop. This equation becomes\u00a0<span id=\"MathJax-Span-94193\" class=\"mrow\"><span id=\"MathJax-Span-94194\" class=\"mi\">B\u00a0<\/span><span id=\"MathJax-Span-94195\" class=\"mo\">=<em>\u00a0<\/em><\/span><span id=\"MathJax-Span-94196\" class=\"msub\"><em><span id=\"MathJax-Span-94197\" class=\"mi\">\u03bc<\/span><\/em><sub><span id=\"MathJax-Span-94198\" class=\"mrow\"><span id=\"MathJax-Span-94199\" class=\"mn\">0<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-94200\" class=\"mrow\"><em><span id=\"MathJax-Span-94201\" class=\"mstyle\"><span id=\"MathJax-Span-94202\" class=\"mrow\"><span id=\"MathJax-Span-94203\" class=\"mrow\"><span id=\"MathJax-Span-94204\" class=\"mtext\">nI<\/span><\/span><\/span><\/span><\/em><span id=\"MathJax-Span-94205\" class=\"mo\">\/<\/span><span id=\"MathJax-Span-94206\" class=\"mo\">(<\/span><\/span><span id=\"MathJax-Span-94207\" class=\"mn\">2<\/span><em><span id=\"MathJax-Span-94208\" class=\"mi\">R<\/span><\/em><span id=\"MathJax-Span-94209\" class=\"mo\">)<\/span>\u00a0for a flat coil of <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>N<\/em><\/span><\/span> loops. RHR-2 gives the direction of the field about the loop. A long coil is called a solenoid.<\/li>\n<li class=\"listitem\">The magnetic field strength inside a solenoid is\n<div id=\"m42382-eip-942\" class=\"equation\" title=\"Equation 22.38.\">\n<div class=\"mediaobject\" style=\"text-align: center;\">[latex]B={\\mu }_{0}\\text{nI}\\left(\\text{inside a solenoid}\\right)\\\\[\/latex]<\/div>\n<\/div>\n<p>where <span class=\"token\"><span class=\"emphasis mathml-mi\"><em>n<\/em><\/span><\/span> is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<section>\n<div class=\"textbox key-takeaways\">\n<h3>Conceptual Questions<\/h3>\n<div>\n<div>\n<p>1. Make a drawing and use RHR-2 to find the direction of the magnetic field of a current loop in a motor (such as in Figure 1 from <a href=\".\/chapter\/22-8-torque-on-a-current-loop-motors-and-meters\/\" target=\"_blank\">Torque on a Current Loop<\/a>). Then show that the direction of the torque on the loop is the same as produced by like poles repelling and unlike poles attracting.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<dl>\n<dt>right hand rule 2 (RHR-2):<\/dt>\n<dd>a rule to determine the direction of the magnetic field induced by a current-carrying wire: Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops<\/dd>\n<\/dl>\n<dl>\n<dt>magnetic field strength (magnitude) produced by a long straight current-carrying wire:<\/dt>\n<dd>defined as [latex]B=\\frac{\\mu_{0}I}{2\\pi r}\\\\[\/latex], where <em>I\u00a0<\/em>is the current,<em> r<\/em>\u00a0is the shortest distance to the wire, and\u00a0<em><span id=\"MathJax-Span-94293\" class=\"mi\">\u03bc<\/span><\/em><sub><span id=\"MathJax-Span-94294\" class=\"mrow\"><span id=\"MathJax-Span-94295\" class=\"mn\">0<\/span><\/span><\/sub> is the permeability of free space<\/dd>\n<\/dl>\n<dl>\n<dt>permeability of free space:<\/dt>\n<dd>the measure of the ability of a material, in this case free space, to support a magnetic field; the constant [latex]\\mu_{0}=4\\pi \\times 10^{-7}T\\cdot \\text{m\/A}\\\\[\/latex]<\/dd>\n<\/dl>\n<dl>\n<dt>magnetic field strength at the center of a circular loop:<\/dt>\n<dd>defined as [latex]B=\\frac{{\\mu }_{0}I}{2R}\\\\[\/latex] where <em>R<\/em>\u00a0is the radius of the loop<\/dd>\n<\/dl>\n<dl>\n<dt>solenoid:<\/dt>\n<dd>a thin wire wound into a coil that produces a magnetic field when an electric current is passed through it<\/dd>\n<\/dl>\n<dl>\n<dt>magnetic field strength inside a solenoid:<\/dt>\n<dd>defined as [latex]B={\\mu }_{0}\\text{nI}\\\\[\/latex] where <em>n<\/em>\u00a0is the number of loops per unit length of the solenoid\u00a0<em><span id=\"MathJax-Element-5847-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-94377\" class=\"math\"><span id=\"MathJax-Span-94378\" class=\"mrow\"><span id=\"MathJax-Span-94379\" class=\"semantics\"><span id=\"MathJax-Span-94380\" class=\"mrow\"><span id=\"MathJax-Span-94381\" class=\"mrow\"><span id=\"MathJax-Span-94382\" class=\"mrow\"><span id=\"MathJax-Span-94383\" class=\"mrow\"><span id=\"MathJax-Span-94385\" class=\"mrow\"><span id=\"MathJax-Span-94386\" class=\"mi\">n\u00a0<\/span><span id=\"MathJax-Span-94387\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-94388\" class=\"mrow\"><span id=\"MathJax-Span-94389\" class=\"mi\">N<\/span><span id=\"MathJax-Span-94390\" class=\"mo\">\/<\/span><span id=\"MathJax-Span-94391\" class=\"mi\">l<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/em>, with <em><span id=\"MathJax-Element-5848-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-94392\" class=\"math\"><span id=\"MathJax-Span-94393\" class=\"mrow\"><span id=\"MathJax-Span-94394\" class=\"semantics\"><span id=\"MathJax-Span-94395\" class=\"mrow\"><span id=\"MathJax-Span-94396\" class=\"mrow\"><span id=\"MathJax-Span-94397\" class=\"mrow\"><span id=\"MathJax-Span-94398\" class=\"mi\">N<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/em> being the number of loops and<em> l\u00a0<\/em>the length)<\/dd>\n<\/dl>\n<dl>\n<dt>Biot-Savart law:<\/dt>\n<dd>a physical law that describes the magnetic field generated by an electric current in terms of a specific equation<\/dd>\n<\/dl>\n<dl>\n<dt>Ampere\u2019s law:<\/dt>\n<dd>the physical law that states that the magnetic field around an electric current is proportional to the current; each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment<\/dd>\n<\/dl>\n<dl>\n<dt>Maxwell\u2019s equations:<\/dt>\n<dd>a set of four equations that describe electromagnetic phenomena<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4816\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Physics. <strong>Authored by<\/strong>: OpenStax College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics\">http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Located at License<\/li><li>PhET Interactive Simulations . <strong>Provided by<\/strong>: University of Colorado Boulder . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/phet.colorado.edu\">http:\/\/phet.colorado.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":1,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Physics\",\"author\":\"OpenStax College\",\"organization\":\"\",\"url\":\"http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Located at License\"},{\"type\":\"cc\",\"description\":\"PhET Interactive Simulations \",\"author\":\"\",\"organization\":\"University of Colorado Boulder \",\"url\":\"http:\/\/phet.colorado.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4816","chapter","type-chapter","status-publish","hentry"],"part":7661,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/4816","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/4816\/revisions"}],"predecessor-version":[{"id":12088,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/4816\/revisions\/12088"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/parts\/7661"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/4816\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/media?parent=4816"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapter-type?post=4816"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/contributor?post=4816"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/license?post=4816"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}