{"id":1233,"date":"2017-10-27T16:31:43","date_gmt":"2017-10-27T16:31:43","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/magnetic-field-strength-force-on-a-moving-charge-in-a-magnetic-field\/"},"modified":"2017-11-08T03:26:38","modified_gmt":"2017-11-08T03:26:38","slug":"magnetic-field-strength-force-on-a-moving-charge-in-a-magnetic-field","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/magnetic-field-strength-force-on-a-moving-charge-in-a-magnetic-field\/","title":{"raw":"Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field","rendered":"Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe the effects of magnetic fields on moving charges.<\/li>\n<li>Use the right hand rule 1 to determine the velocity of a charge, the direction of the magnetic field, and the direction of the magnetic force on a moving charge.<\/li>\n<li>Calculate the magnetic force on a moving charge.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2921850\">What is the mechanism by which one magnet exerts a force on another? The answer is related to the fact that all magnetism is caused by current, the flow of charge. <em data-effect=\"italics\">Magnetic fields exert forces on moving charges<\/em>, and so they exert forces on other magnets, all of which have moving charges.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"import-auto-id2052696\">\n<h1 data-type=\"title\">Right Hand Rule 1<\/h1>\n<p id=\"import-auto-id1969358\">The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force. The magnitude of the <span data-type=\"term\" id=\"import-auto-id2086468\">magnetic force<\/span> [latex]F[\/latex] on a charge [latex]q[\/latex] moving at a speed [latex]v[\/latex] in a magnetic field of strength [latex]B[\/latex] is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1848216\">[latex]F=\\text{qvB}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1527820\">where [latex]\\theta [\/latex] is the angle between the directions of [latex]\\mathbf{\\text{v}}[\/latex] and [latex]\\mathbf{\\text{B}}.[\/latex] This force is often called the <span data-type=\"term\">Lorentz force<\/span>. In fact, this is how we define the magnetic field strength [latex]B[\/latex]\u2014in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength [latex]B[\/latex] is called the <span data-type=\"term\">tesla<\/span> (T) after the eccentric but brilliant inventor Nikola Tesla (1856\u20131943). To determine how the tesla relates to other SI units, we solve [latex]F=\\text{qvB}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] for [latex]B[\/latex].<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1536271\">[latex]B=\\frac{F}{\\text{qv}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta }[\/latex]<\/div>\n<p id=\"import-auto-id1990649\">Because<br>\n        [latex]\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex]<br>\n     is unitless, the tesla is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1593576\">[latex]\\text{1 T}=\\frac{\\text{1 N}}{C\\cdot \\text{m\/s}}=\\frac{\\text{1 N}}{A\\cdot m}[\/latex]<\/div>\n<p id=\"import-auto-id1582580\">(note that C\/s = A).<\/p>\n<p id=\"import-auto-id1956130\">Another smaller unit, called the <span data-type=\"term\" id=\"import-auto-id1803673\">gauss<\/span> (G), where [latex]1 G={\\text{10}}^{-4}\\phantom{\\rule{0.25em}{0ex}}T[\/latex], is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. The Earth\u2019s magnetic field on its surface is only about [latex]5\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}T[\/latex], or 0.5 G.<\/p>\n<p id=\"import-auto-id2165273\">The <em data-effect=\"italics\">direction<\/em> of the magnetic force [latex]\\mathbf{\\text{F}}[\/latex] is perpendicular to the plane formed by [latex]\\mathbf{\\text{v}}[\/latex] and [latex]\\mathbf{\\text{B}}[\/latex], as determined by the <span data-type=\"term\">right hand rule 1<\/span> (or RHR-1), which is illustrated in <a href=\"#import-auto-id1473446\" class=\"autogenerated-content\">(Figure)<\/a>. RHR-1 states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of [latex]\\mathbf{\\text{v}}[\/latex], the fingers in the direction of [latex]\\mathbf{\\text{B}}[\/latex], and a perpendicular to the palm points in the direction of [latex]\\mathbf{\\text{F}}[\/latex]. One way to remember this is that there is one velocity, and so the thumb represents it. There are many field lines, and so the fingers represent them. The force is in the direction you would push with your palm. The force on a negative charge is in exactly the opposite direction to that on a positive charge.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1473446\">\n<div class=\"bc-figcaption figcaption\">Magnetic fields exert forces on moving charges. This force is one of the most basic known. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by [latex]\\mathbf{\\text{v}}[\/latex] and [latex]\\mathbf{\\text{B}}[\/latex] and follows right hand rule\u20131 (RHR-1) as shown. The magnitude of the force is proportional to [latex]q[\/latex], [latex]v[\/latex], [latex]B[\/latex], and the sine of the angle between [latex]\\mathbf{\\text{v}}[\/latex] and [latex]\\mathbf{\\text{B}}[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1336624\" data-alt=\"The right hand rule 1. An outstretched right hand rests palm up on a piece of paper on which a vector arrow v points to the right and a vector arrow B points toward the top of the paper. The thumb points to the right, in the direction of the v vector arrow. The fingers point in the direction of the B vector. B and v are in the same plane. The F vector points straight up, perpendicular to the plane of the paper, which is the plane made by B and v. The angle between B and v is theta. The magnitude of the magnetic force F equals q v B sine theta.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_23_04_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The right hand rule 1. An outstretched right hand rests palm up on a piece of paper on which a vector arrow v points to the right and a vector arrow B points toward the top of the paper. The thumb points to the right, in the direction of the v vector arrow. The fingers point in the direction of the B vector. B and v are in the same plane. The F vector points straight up, perpendicular to the plane of the paper, which is the plane made by B and v. The angle between B and v is theta. The magnitude of the magnetic force F equals q v B sine theta.\" width=\"275\"><\/span><\/p><\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1891449\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Charges and Magnets<\/div>\n<p id=\"import-auto-id1102772\">There is no magnetic force on static charges. However, there is a magnetic force on moving charges. When charges are stationary, their electric fields do not affect magnets. But, when charges move, they produce magnetic fields that exert forces on other magnets. When there is relative motion, a connection between electric and magnetic fields emerges\u2014each affects the other.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1549068\">\n<div data-type=\"title\" class=\"title\">Calculating Magnetic Force: Earth\u2019s Magnetic Field on a Charged Glass Rod<\/div>\n<p id=\"import-auto-id2770918\">With the exception of compasses, you seldom see or personally experience forces due to the Earth\u2019s small magnetic field. To illustrate this, suppose that in a physics lab you rub a glass rod with silk, placing a 20-nC positive charge on it. Calculate the force on the rod due to the Earth\u2019s magnetic field, if you throw it with a horizontal velocity of 10 m\/s due west in a place where the Earth\u2019s field is due north parallel to the ground. (The direction of the force is determined with right hand rule 1 as shown in <a href=\"#import-auto-id1698149\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1698149\">\n<div class=\"bc-figcaption figcaption\">A positively charged object moving due west in a region where the Earth\u2019s magnetic field is due north experiences a force that is straight down as shown. A negative charge moving in the same direction would feel a force straight up.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1544337\" data-alt=\"The effects of the Earth\u2019s magnetic field on moving charges. Figure a shows a positive charge with a velocity vector due west, a magnetic field line B oriented due north, and a magnetic force vector F straight down. Figure b shows the right hand facing down, with the fingers pointing north with B, the thumb pointing west with v, and force down away from the hand.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_23_04_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"The effects of the Earth\u2019s magnetic field on moving charges. Figure a shows a positive charge with a velocity vector due west, a magnetic field line B oriented due north, and a magnetic force vector F straight down. Figure b shows the right hand facing down, with the fingers pointing north with B, the thumb pointing west with v, and force down away from the hand.\" width=\"400\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2013069\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1844254\">We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation [latex]F=\\text{qvB}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] to find the force.<\/p>\n<p id=\"import-auto-id1466325\"><strong>Solution<\/strong><\/p>\n<p id=\"fs-id2090550\">The magnetic force is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1535424\">[latex]F=\\text{qvb}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta .[\/latex]<\/div>\n<p>We see that [latex]\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta =1[\/latex], since the angle between the velocity and the direction of the field is [latex]\\text{90\u00ba}[\/latex]. Entering the other given quantities yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2093723\">[latex]\\begin{array}{lll}F&amp; =&amp; \\left(\\text{20}\u00d7{\\text{10}}^{\u20139}\\phantom{\\rule{0.25em}{0ex}}C\\right)\\left(\\text{10 m\/s}\\right)\\left(5\u00d7{\\text{10}}^{\u20135}\\phantom{\\rule{0.25em}{0ex}}T\\right)\\\\ &amp; =&amp; 1\u00d7{\\text{10}}^{\\text{\u201311}}\\phantom{\\rule{0.25em}{0ex}}\\left(C\\cdot \\text{m\/s}\\right)\\left(\\frac{N}{C\\cdot \\text{m\/s}}\\right)=1\u00d7{\\text{10}}^{\\text{\u201311}}\\phantom{\\rule{0.25em}{0ex}}N.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1752970\"><strong>Discussion<\/strong><\/p>\n<p id=\"fs-id2551470\">This force is completely negligible on any macroscopic object, consistent with experience. (It is calculated to only one digit, since the Earth\u2019s field varies with location and is given to only one digit.) The Earth\u2019s magnetic field, however, does produce very important effects, particularly on submicroscopic particles. Some of these are explored in <a href=\"\/contents\/a66a7a2a-4d91-4887-8c5f-3757971a23b5@2\">Force on a Moving Charge in a Magnetic Field: Examples and Applications<\/a>.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1907447\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"import-auto-id2165913\">\n<li>Magnetic fields exert a force on a moving charge <em data-effect=\"italics\">q<\/em>, the magnitude of which is\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1758727\">[latex]F=\\text{qvB}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta ,[\/latex]<\/div>\n<p>where [latex]\\theta [\/latex] is the angle between the directions of [latex]v[\/latex] and [latex]B[\/latex].\n<\/p><\/li>\n<li>The SI unit for magnetic field strength [latex]B[\/latex] is the tesla (T), which is related to other units by\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2091790\">[latex]1 T=\\frac{\\text{1 N}}{C\\cdot \\text{m\/s}}=\\frac{\\text{1 N}}{A\\cdot m}.[\/latex]<\/div>\n<\/li>\n<li>The <em data-effect=\"italics\">direction<\/em> of the force on a moving charge is given by right hand rule 1 (RHR-1): Point the thumb of the right hand in the direction of [latex]v[\/latex], the fingers in the direction of [latex]B[\/latex], and a perpendicular to the palm points in the direction of [latex]F[\/latex].<\/li>\n<li>The force is perpendicular to the plane formed by [latex]\\mathbf{\\text{v}}[\/latex] and [latex]\\mathbf{\\text{B}}[\/latex]. Since the force is zero if [latex]\\mathbf{\\text{v}}[\/latex] is parallel to [latex]\\mathbf{\\text{B}}[\/latex], charged particles often follow magnetic field lines rather than cross them.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1751385\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2869610\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1416913\">\n<p id=\"import-auto-id1314412\">If a charged particle moves in a straight line through some region of space, can you say that the magnetic field in that region is necessarily zero?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1399302\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1796004\">\n<p id=\"import-auto-id1844190\">What is the direction of the magnetic force on a positive charge that moves as shown in each of the six cases shown  in <a href=\"#import-auto-id1755657\" class=\"autogenerated-content\">(Figure)<\/a>?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1755657\"><span data-type=\"media\" id=\"import-auto-id1403435\" data-alt=\"figure a shows magnetic field line direction symbols with solid circles labeled B out; a velocity vector points down; figure b shows B vectors pointing right and v vector pointing up; figure c shows B in and v to the right; figure d shows B vector pointing right and v vector pointing left; figure e shows B vectors up and v vector into the page; figure f shows B vectors pointing left and v vectors out of the page\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/FIgure_23_04_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"figure a shows magnetic field line direction symbols with solid circles labeled B out; a velocity vector points down; figure b shows B vectors pointing right and v vector pointing up; figure c shows B in and v to the right; figure d shows B vector pointing right and v vector pointing left; figure e shows B vectors up and v vector into the page; figure f shows B vectors pointing left and v vectors out of the page\" height=\"425\"><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2047765\">\n<p id=\"import-auto-id1649236\">(a) Left (West)<\/p>\n<p id=\"import-auto-id2150173\">(b) Into the page<\/p>\n<p id=\"import-auto-id1545470\">(c) Up (North)<\/p>\n<p id=\"import-auto-id1298931\">(d) No force<\/p>\n<p id=\"import-auto-id2746047\">(e) Right (East)<\/p>\n<p>(f) Down (South)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1116147\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1495514\">\n<p id=\"import-auto-id1426633\">Repeat <a href=\"#fs-id1399302\" class=\"autogenerated-content\">(Figure)<\/a> for a negative charge.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1327231\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1535392\">\n<p id=\"import-auto-id1033962\">What is the direction of the velocity of a negative charge that experiences the magnetic force shown in each of the three cases in <a href=\"#import-auto-id1396305\" class=\"autogenerated-content\">(Figure)<\/a>, assuming it moves perpendicular to [latex]\\mathbf{\\text{B}}?[\/latex]<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1396305\"><span data-type=\"media\" id=\"import-auto-id1886637\" data-alt=\"Figure a shows the force vector pointing up and B out of the page. Figure b shows the F vector pointing up and the B vector pointing to the right. Figure c shows the F vector pointing to the left and the B vector pointing into the page.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_23_04_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure a shows the force vector pointing up and B out of the page. Figure b shows the F vector pointing up and the B vector pointing to the right. Figure c shows the F vector pointing to the left and the B vector pointing into the page.\" width=\"350\"><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1847923\">\n<p>(a) East (right)<\/p>\n<p id=\"import-auto-id1798800\">(b) Into page<\/p>\n<p id=\"import-auto-id1947568\">(c) South (down)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1708858\">Repeat <a href=\"#fs-id1327231\" class=\"autogenerated-content\">(Figure)<\/a> for a positive charge.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2031113\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1993559\">\n<p id=\"import-auto-id1798927\">What is the direction of the magnetic field that produces the magnetic force on a positive charge as shown in each of the three cases in the figure below, assuming [latex]\\mathbf{\\text{B}}[\/latex] is perpendicular to [latex]\\mathbf{\\text{v}}[\/latex]?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1612786\"><span data-type=\"media\" data-alt=\"Figure a shows a force vector pointing toward the left and a velocity vector pointing up. Figure b shows the force vector pointing into the page and the velocity vector pointing down. Figure c shows the force vector pointing up and the velocity vector pointing to the left.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_23_04_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure a shows a force vector pointing toward the left and a velocity vector pointing up. Figure b shows the force vector pointing into the page and the velocity vector pointing down. Figure c shows the force vector pointing up and the velocity vector pointing to the left.\" width=\"300\"><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2490682\">\n<p id=\"import-auto-id2603345\">(a) Into page<\/p>\n<p id=\"import-auto-id1758251\">(b) West (left)<\/p>\n<p id=\"import-auto-id1915782\">(c) Out of page<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1572551\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1897259\">\n<p id=\"import-auto-id2864827\">Repeat <a href=\"#fs-id2031113\" class=\"autogenerated-content\">(Figure)<\/a> for a negative charge.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1726851\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1395400\">\n<p id=\"import-auto-id2092090\">What is the maximum force on an aluminum rod with a [latex]0\\text{.}\\text{100}\\text{-\u03bcC}[\/latex] charge that you pass between the poles of a 1.50-T permanent magnet at a speed of 5.00 m\/s? In what direction is the force?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1796961\">\n<p id=\"eip-id2230778\">[latex]7\\text{.}\\text{50}\u00d7{\\text{10}}^{-7}\\phantom{\\rule{0.25em}{0ex}}\\text{N}[\/latex] perpendicular to both the magnetic field lines and the velocity<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1911116\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2495625\">\n<p id=\"import-auto-id2620248\">(a) Aircraft sometimes acquire small static charges. Suppose a supersonic jet has a [latex]0\\text{.}\\text{500}\\text{-\u03bcC}[\/latex] charge and flies due west at a speed of 660 m\/s over the Earth\u2019s south magnetic pole, where the [latex]8\\text{.}\\text{00}\u00d7{\\text{10}}^{-5}\\text{-T}[\/latex] magnetic field points straight up. What are the direction and the magnitude of the magnetic force on the plane? (b) Discuss whether the value obtained in part (a) implies this is a significant or negligible effect.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2092579\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2166152\">\n<p id=\"import-auto-id2028436\">(a) A cosmic ray proton moving toward the Earth at [latex]\\text{5.00}\u00d7{\\text{10}}^{7}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex] experiences a magnetic force of [latex]1\\text{.}\\text{70}\u00d7{\\text{10}}^{-\\text{16}}\\phantom{\\rule{0.25em}{0ex}}\\text{N}[\/latex]. What is the strength of the magnetic field if there is a [latex]\\text{45\u00ba}[\/latex] angle between it and the proton\u2019s velocity? (b) Is the value obtained in part (a) consistent with the known strength of the Earth\u2019s magnetic field on its surface? Discuss.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1332784\">\n<p id=\"eip-id1164037422717\">(a) [latex]3\\text{.}\\text{01}\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}\\text{T}[\/latex]<\/p>\n<p id=\"eip-id1164037435356\">(b) This is slightly less then the magnetic field strength of [latex]5\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}\\text{T}[\/latex] at the surface of the Earth, so it is consistent.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1809133\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2877727\">\n<p id=\"import-auto-id1942132\">An electron moving at [latex]4\\text{.}\\text{00}\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex] in a 1.25-T magnetic field experiences a magnetic force of [latex]1\\text{.}\\text{40}\u00d7{\\text{10}}^{-\\text{16}}\\phantom{\\rule{0.25em}{0ex}}\\text{N}[\/latex]. What angle does the velocity of the electron make with the magnetic field? There are two answers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1645778\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id2025134\">(a) A physicist performing a sensitive measurement wants to limit the magnetic force on a moving charge in her equipment to less than [latex]1\\text{.}\\text{00}\u00d7{\\text{10}}^{-\\text{12}}\\phantom{\\rule{0.25em}{0ex}}N[\/latex]. What is the greatest the charge can be if it moves at a maximum speed of 30.0 m\/s in the Earth\u2019s field? (b) Discuss whether it would be difficult to limit the charge to less than the value found in (a) by comparing it with typical static electricity and noting that static is often absent.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2376778\">\n<p id=\"import-auto-id1261791\">(a) [latex]6\\text{.}\\text{67}\u00d7{\\text{10}}^{-\\text{10}}\\phantom{\\rule{0.25em}{0ex}}\\text{C}[\/latex] (taking the Earth\u2019s field to be [latex]5\\text{.}\\text{00}\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}\\text{T}[\/latex])<\/p>\n<p>(b) Less than typical static, therefore difficult<\/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-id1536928\">\n<dt>right hand rule 1 (RHR-1)<\/dt>\n<dd id=\"fs-id1703934\"> the rule to determine the direction of the magnetic force on a positive moving charge: when the thumb of the right hand points in the direction of the charge\u2019s velocity [latex]\\mathbf{\\text{v}}[\/latex] and the fingers point in the direction of the magnetic field [latex]\\mathbf{\\text{B}}[\/latex], then the force on the charge is perpendicular and away from the palm; the force on a negative charge is perpendicular and into the palm<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2045976\">\n<dt>Lorentz force<\/dt>\n<dd id=\"fs-id1208355\"> the force on a charge moving in a magnetic field<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1547164\">\n<dt>tesla<\/dt>\n<dd>T, the SI unit of the magnetic field strength; [latex]\\text{1 T}=\\frac{\\text{1 N}}{A\\cdot m}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1461809\">\n<dt>magnetic force<\/dt>\n<dd id=\"fs-id1420080\">the force on a charge produced by its motion through a magnetic field; the Lorentz force <\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1779442\">\n<dt>gauss<\/dt>\n<dd id=\"fs-id2095086\">G, the unit of the magnetic field strength; [latex]\\text{1 G}={\\text{10}}^{\u20134}\\phantom{\\rule{0.25em}{0ex}}T[\/latex]<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe the effects of magnetic fields on moving charges.<\/li>\n<li>Use the right hand rule 1 to determine the velocity of a charge, the direction of the magnetic field, and the direction of the magnetic force on a moving charge.<\/li>\n<li>Calculate the magnetic force on a moving charge.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2921850\">What is the mechanism by which one magnet exerts a force on another? The answer is related to the fact that all magnetism is caused by current, the flow of charge. <em data-effect=\"italics\">Magnetic fields exert forces on moving charges<\/em>, and so they exert forces on other magnets, all of which have moving charges.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"import-auto-id2052696\">\n<h1 data-type=\"title\">Right Hand Rule 1<\/h1>\n<p id=\"import-auto-id1969358\">The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force. The magnitude of the <span data-type=\"term\" id=\"import-auto-id2086468\">magnetic force<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> on a charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> moving at a speed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> in a magnetic field of strength <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1848216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-551e8ac2293e6b5b5c1ef94e987018e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#118;&#66;&#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;&#105;&#110;&#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;&#104;&#101;&#116;&#97;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"112\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1527820\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the angle between the directions of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-598b841609930773255f7b8964582095_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> This force is often called the <span data-type=\"term\">Lorentz force<\/span>. In fact, this is how we define the magnetic field strength <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>\u2014in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is called the <span data-type=\"term\">tesla<\/span> (T) after the eccentric but brilliant inventor Nikola Tesla (1856\u20131943). To determine how the tesla relates to other SI units, we solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0acdfaf3e7a33547151abc8119af4cc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#118;&#66;&#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;&#105;&#110;&#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;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"108\" style=\"vertical-align: -3px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1536271\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c0b6a6cc26c78c2b45173311cf6d32c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#118;&#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;&#105;&#110;&#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;&#104;&#101;&#116;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1990649\">Because<br \/>\n        <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b691404dafbb2033aee29a4307ec732d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#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;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\" \/><br \/>\n     is unitless, the tesla is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1593576\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-080a103b24acc4b1a4ff650361b86377_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#84;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#78;&#125;&#125;&#123;&#67;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#78;&#125;&#125;&#123;&#65;&#92;&#99;&#100;&#111;&#116;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"145\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1582580\">(note that C\/s = A).<\/p>\n<p id=\"import-auto-id1956130\">Another smaller unit, called the <span data-type=\"term\" id=\"import-auto-id1803673\">gauss<\/span> (G), where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e7a3321334fc389d5847f038926d8275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#32;&#71;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#52;&#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;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -1px;\" \/>, is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. The Earth\u2019s magnetic field on its surface is only about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7636516df208ffe9da9c815c6fc7c993_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#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;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"62\" style=\"vertical-align: -1px;\" \/>, or 0.5 G.<\/p>\n<p id=\"import-auto-id2165273\">The <em data-effect=\"italics\">direction<\/em> of the magnetic force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5cd16916557096d431ff3a1af0c9119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: -1px;\" \/> is perpendicular to the plane formed by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, as determined by the <span data-type=\"term\">right hand rule 1<\/span> (or RHR-1), which is illustrated in <a href=\"#import-auto-id1473446\" class=\"autogenerated-content\">(Figure)<\/a>. RHR-1 states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/>, the fingers in the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, and a perpendicular to the palm points in the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5cd16916557096d431ff3a1af0c9119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: -1px;\" \/>. One way to remember this is that there is one velocity, and so the thumb represents it. There are many field lines, and so the fingers represent them. The force is in the direction you would push with your palm. The force on a negative charge is in exactly the opposite direction to that on a positive charge.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1473446\">\n<div class=\"bc-figcaption figcaption\">Magnetic fields exert forces on moving charges. This force is one of the most basic known. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> and follows right hand rule\u20131 (RHR-1) as shown. The magnitude of the force is proportional to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, and the sine of the angle between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1336624\" data-alt=\"The right hand rule 1. An outstretched right hand rests palm up on a piece of paper on which a vector arrow v points to the right and a vector arrow B points toward the top of the paper. The thumb points to the right, in the direction of the v vector arrow. The fingers point in the direction of the B vector. B and v are in the same plane. The F vector points straight up, perpendicular to the plane of the paper, which is the plane made by B and v. The angle between B and v is theta. The magnitude of the magnetic force F equals q v B sine theta.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_23_04_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The right hand rule 1. An outstretched right hand rests palm up on a piece of paper on which a vector arrow v points to the right and a vector arrow B points toward the top of the paper. The thumb points to the right, in the direction of the v vector arrow. The fingers point in the direction of the B vector. B and v are in the same plane. The F vector points straight up, perpendicular to the plane of the paper, which is the plane made by B and v. The angle between B and v is theta. The magnitude of the magnetic force F equals q v B sine theta.\" width=\"275\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1891449\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Charges and Magnets<\/div>\n<p id=\"import-auto-id1102772\">There is no magnetic force on static charges. However, there is a magnetic force on moving charges. When charges are stationary, their electric fields do not affect magnets. But, when charges move, they produce magnetic fields that exert forces on other magnets. When there is relative motion, a connection between electric and magnetic fields emerges\u2014each affects the other.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1549068\">\n<div data-type=\"title\" class=\"title\">Calculating Magnetic Force: Earth\u2019s Magnetic Field on a Charged Glass Rod<\/div>\n<p id=\"import-auto-id2770918\">With the exception of compasses, you seldom see or personally experience forces due to the Earth\u2019s small magnetic field. To illustrate this, suppose that in a physics lab you rub a glass rod with silk, placing a 20-nC positive charge on it. Calculate the force on the rod due to the Earth\u2019s magnetic field, if you throw it with a horizontal velocity of 10 m\/s due west in a place where the Earth\u2019s field is due north parallel to the ground. (The direction of the force is determined with right hand rule 1 as shown in <a href=\"#import-auto-id1698149\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1698149\">\n<div class=\"bc-figcaption figcaption\">A positively charged object moving due west in a region where the Earth\u2019s magnetic field is due north experiences a force that is straight down as shown. A negative charge moving in the same direction would feel a force straight up.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1544337\" data-alt=\"The effects of the Earth\u2019s magnetic field on moving charges. Figure a shows a positive charge with a velocity vector due west, a magnetic field line B oriented due north, and a magnetic force vector F straight down. Figure b shows the right hand facing down, with the fingers pointing north with B, the thumb pointing west with v, and force down away from the hand.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_23_04_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"The effects of the Earth\u2019s magnetic field on moving charges. Figure a shows a positive charge with a velocity vector due west, a magnetic field line B oriented due north, and a magnetic force vector F straight down. Figure b shows the right hand facing down, with the fingers pointing north with B, the thumb pointing west with v, and force down away from the hand.\" width=\"400\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2013069\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1844254\">We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0acdfaf3e7a33547151abc8119af4cc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#118;&#66;&#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;&#105;&#110;&#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;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"108\" style=\"vertical-align: -3px;\" \/> to find the force.<\/p>\n<p id=\"import-auto-id1466325\"><strong>Solution<\/strong><\/p>\n<p id=\"fs-id2090550\">The magnetic force is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1535424\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef8ebd49a9abbf6bd1ae5c40a0f671f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#118;&#98;&#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;&#105;&#110;&#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;&#104;&#101;&#116;&#97;&#32;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/div>\n<p>We see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-70dcb8b9397fc9fe2382d66e248d1eae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#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;&#104;&#101;&#116;&#97;&#32;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: -1px;\" \/>, since the angle between the velocity and the direction of the field is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>. Entering the other given quantities yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2093723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-561b5aadf41a675b4158c89159d3a343_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;&#70;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#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;&#67;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#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;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#49;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#49;&#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;&#108;&#101;&#102;&#116;&#40;&#67;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#78;&#125;&#123;&#67;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#49;&#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;&#78;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"344\" style=\"vertical-align: -23px;\" \/><\/div>\n<p id=\"import-auto-id1752970\"><strong>Discussion<\/strong><\/p>\n<p id=\"fs-id2551470\">This force is completely negligible on any macroscopic object, consistent with experience. (It is calculated to only one digit, since the Earth\u2019s field varies with location and is given to only one digit.) The Earth\u2019s magnetic field, however, does produce very important effects, particularly on submicroscopic particles. Some of these are explored in <a href=\"\/contents\/a66a7a2a-4d91-4887-8c5f-3757971a23b5@2\">Force on a Moving Charge in a Magnetic Field: Examples and Applications<\/a>.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1907447\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"import-auto-id2165913\">\n<li>Magnetic fields exert a force on a moving charge <em data-effect=\"italics\">q<\/em>, the magnitude of which is\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1758727\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-eba9c85e2772fb1bef8d10e419a13bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#118;&#66;&#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;&#105;&#110;&#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;&#104;&#101;&#116;&#97;&#32;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the angle between the directions of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>.\n<\/p>\n<\/li>\n<li>The SI unit for magnetic field strength <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the tesla (T), which is related to other units by\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2091790\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc1c781aebddc276f6624c41f0e6e959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#32;&#84;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#78;&#125;&#125;&#123;&#67;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#78;&#125;&#125;&#123;&#65;&#92;&#99;&#100;&#111;&#116;&#32;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"144\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/li>\n<li>The <em data-effect=\"italics\">direction<\/em> of the force on a moving charge is given by right hand rule 1 (RHR-1): Point the thumb of the right hand in the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>, the fingers in the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, and a perpendicular to the palm points in the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>.<\/li>\n<li>The force is perpendicular to the plane formed by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>. Since the force is zero if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/> is parallel to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, charged particles often follow magnetic field lines rather than cross them.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1751385\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2869610\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1416913\">\n<p id=\"import-auto-id1314412\">If a charged particle moves in a straight line through some region of space, can you say that the magnetic field in that region is necessarily zero?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1399302\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1796004\">\n<p id=\"import-auto-id1844190\">What is the direction of the magnetic force on a positive charge that moves as shown in each of the six cases shown  in <a href=\"#import-auto-id1755657\" class=\"autogenerated-content\">(Figure)<\/a>?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1755657\"><span data-type=\"media\" id=\"import-auto-id1403435\" data-alt=\"figure a shows magnetic field line direction symbols with solid circles labeled B out; a velocity vector points down; figure b shows B vectors pointing right and v vector pointing up; figure c shows B in and v to the right; figure d shows B vector pointing right and v vector pointing left; figure e shows B vectors up and v vector into the page; figure f shows B vectors pointing left and v vectors out of the page\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/FIgure_23_04_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"figure a shows magnetic field line direction symbols with solid circles labeled B out; a velocity vector points down; figure b shows B vectors pointing right and v vector pointing up; figure c shows B in and v to the right; figure d shows B vector pointing right and v vector pointing left; figure e shows B vectors up and v vector into the page; figure f shows B vectors pointing left and v vectors out of the page\" height=\"425\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2047765\">\n<p id=\"import-auto-id1649236\">(a) Left (West)<\/p>\n<p id=\"import-auto-id2150173\">(b) Into the page<\/p>\n<p id=\"import-auto-id1545470\">(c) Up (North)<\/p>\n<p id=\"import-auto-id1298931\">(d) No force<\/p>\n<p id=\"import-auto-id2746047\">(e) Right (East)<\/p>\n<p>(f) Down (South)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1116147\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1495514\">\n<p id=\"import-auto-id1426633\">Repeat <a href=\"#fs-id1399302\" class=\"autogenerated-content\">(Figure)<\/a> for a negative charge.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1327231\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1535392\">\n<p id=\"import-auto-id1033962\">What is the direction of the velocity of a negative charge that experiences the magnetic force shown in each of the three cases in <a href=\"#import-auto-id1396305\" class=\"autogenerated-content\">(Figure)<\/a>, assuming it moves perpendicular to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-220fa3233858b80d7c18b5dc6886af03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1396305\"><span data-type=\"media\" id=\"import-auto-id1886637\" data-alt=\"Figure a shows the force vector pointing up and B out of the page. Figure b shows the F vector pointing up and the B vector pointing to the right. Figure c shows the F vector pointing to the left and the B vector pointing into the page.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_23_04_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure a shows the force vector pointing up and B out of the page. Figure b shows the F vector pointing up and the B vector pointing to the right. Figure c shows the F vector pointing to the left and the B vector pointing into the page.\" width=\"350\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1847923\">\n<p>(a) East (right)<\/p>\n<p id=\"import-auto-id1798800\">(b) Into page<\/p>\n<p id=\"import-auto-id1947568\">(c) South (down)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1708858\">Repeat <a href=\"#fs-id1327231\" class=\"autogenerated-content\">(Figure)<\/a> for a positive charge.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2031113\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1993559\">\n<p id=\"import-auto-id1798927\">What is the direction of the magnetic field that produces the magnetic force on a positive charge as shown in each of the three cases in the figure below, assuming <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is perpendicular to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/>?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1612786\"><span data-type=\"media\" data-alt=\"Figure a shows a force vector pointing toward the left and a velocity vector pointing up. Figure b shows the force vector pointing into the page and the velocity vector pointing down. Figure c shows the force vector pointing up and the velocity vector pointing to the left.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_23_04_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure a shows a force vector pointing toward the left and a velocity vector pointing up. Figure b shows the force vector pointing into the page and the velocity vector pointing down. Figure c shows the force vector pointing up and the velocity vector pointing to the left.\" width=\"300\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2490682\">\n<p id=\"import-auto-id2603345\">(a) Into page<\/p>\n<p id=\"import-auto-id1758251\">(b) West (left)<\/p>\n<p id=\"import-auto-id1915782\">(c) Out of page<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1572551\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1897259\">\n<p id=\"import-auto-id2864827\">Repeat <a href=\"#fs-id2031113\" class=\"autogenerated-content\">(Figure)<\/a> for a negative charge.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1726851\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1395400\">\n<p id=\"import-auto-id2092090\">What is the maximum force on an aluminum rod with a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7bf98dfcd62a77cc4d0f2556df1724ec_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;&#49;&#48;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&mu;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: -1px;\" \/> charge that you pass between the poles of a 1.50-T permanent magnet at a speed of 5.00 m\/s? In what direction is the force?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1796961\">\n<p id=\"eip-id2230778\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ece163350dec2c60dc90f4241ebfd353_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#55;&#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;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/> perpendicular to both the magnetic field lines and the velocity<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1911116\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2495625\">\n<p id=\"import-auto-id2620248\">(a) Aircraft sometimes acquire small static charges. Suppose a supersonic jet has a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a2ebec529db31b116d03b39acce114a6_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;&#53;&#48;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&mu;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: 0px;\" \/> charge and flies due west at a speed of 660 m\/s over the Earth\u2019s south magnetic pole, where the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-811f79a7ea759887fc07dbd9bc6d380e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#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;&#45;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -1px;\" \/> magnetic field points straight up. What are the direction and the magnitude of the magnetic force on the plane? (b) Discuss whether the value obtained in part (a) implies this is a significant or negligible effect.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2092579\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2166152\">\n<p id=\"import-auto-id2028436\">(a) A cosmic ray proton moving toward the Earth at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-282130dd57b13ac78276bf0e7854891c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#55;&#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;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -4px;\" \/> experiences a magnetic force of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-591714a99c1a57611b1ce6d42cb52f41_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;&#55;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#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;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -1px;\" \/>. What is the strength of the magnetic field if there is a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1142d1c44cfaf3459c45a3d6cc399899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/> angle between it and the proton\u2019s velocity? (b) Is the value obtained in part (a) consistent with the known strength of the Earth\u2019s magnetic field on its surface? Discuss.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1332784\">\n<p id=\"eip-id1164037422717\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b198fb12d33967484fde2a010039a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#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;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"eip-id1164037435356\">(b) This is slightly less then the magnetic field strength of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-60fc96a69731620c528afbceacee88d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#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;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"62\" style=\"vertical-align: -1px;\" \/> at the surface of the Earth, so it is consistent.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1809133\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2877727\">\n<p id=\"import-auto-id1942132\">An electron moving at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b1f264f094cfc3cdc7e54c69d0bb2b71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#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;&#51;&#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;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -4px;\" \/> in a 1.25-T magnetic field experiences a magnetic force of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03d106b53615f87a61eb8a63117a1027_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;&#52;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#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;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -1px;\" \/>. What angle does the velocity of the electron make with the magnetic field? There are two answers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1645778\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id2025134\">(a) A physicist performing a sensitive measurement wants to limit the magnetic force on a moving charge in her equipment to less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-21a94e4237c90af9047270cd08ba436e_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;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#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;&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"94\" style=\"vertical-align: -1px;\" \/>. What is the greatest the charge can be if it moves at a maximum speed of 30.0 m\/s in the Earth\u2019s field? (b) Discuss whether it would be difficult to limit the charge to less than the value found in (a) by comparing it with typical static electricity and noting that static is often absent.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2376778\">\n<p id=\"import-auto-id1261791\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d00f18a99ae79cf3523986554249c7a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#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;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -1px;\" \/> (taking the Earth\u2019s field to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3a6dffba4a2eadca364ec96729e45198_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#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;&#45;&#53;&#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;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/>)<\/p>\n<p>(b) Less than typical static, therefore difficult<\/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-id1536928\">\n<dt>right hand rule 1 (RHR-1)<\/dt>\n<dd id=\"fs-id1703934\"> the rule to determine the direction of the magnetic force on a positive moving charge: when the thumb of the right hand points in the direction of the charge\u2019s velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/> and the fingers point in the direction of the magnetic field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d9a84faf50d37982cb42228e5af4419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, then the force on the charge is perpendicular and away from the palm; the force on a negative charge is perpendicular and into the palm<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2045976\">\n<dt>Lorentz force<\/dt>\n<dd id=\"fs-id1208355\"> the force on a charge moving in a magnetic field<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1547164\">\n<dt>tesla<\/dt>\n<dd>T, the SI unit of the magnetic field strength; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3fb8c8df1135a51f9e0ed47d63da12b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#84;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#78;&#125;&#125;&#123;&#65;&#92;&#99;&#100;&#111;&#116;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"79\" style=\"vertical-align: -6px;\" \/><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1461809\">\n<dt>magnetic force<\/dt>\n<dd id=\"fs-id1420080\">the force on a charge produced by its motion through a magnetic field; the Lorentz force <\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1779442\">\n<dt>gauss<\/dt>\n<dd id=\"fs-id2095086\">G, the unit of the magnetic field strength; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c48f11df0479196f558e5d29e62e5303_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#71;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#52;&#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;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -1px;\" \/><\/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-1233","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1204,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1233","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\/1233\/revisions"}],"predecessor-version":[{"id":1234,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1233\/revisions\/1234"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1204"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1233\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1233"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1233"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1233"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}