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{"name": "Parallel wires and superposition", "extensions": [], "custom_part_types": [], "resources": [["question-resources/wires.png", "/srv/numbas/media/question-resources/wires.png"], ["question-resources/current_angled_anti_qHgtK8m.png", "/srv/numbas/media/question-resources/current_angled_anti_qHgtK8m.png"], ["question-resources/current_angled_clock_dd9WdQN.png", "/srv/numbas/media/question-resources/current_angled_clock_dd9WdQN.png"], ["question-resources/current_anti_kwFIMFr.png", "/srv/numbas/media/question-resources/current_anti_kwFIMFr.png"], ["question-resources/current_clock_iH5WgUI.png", "/srv/numbas/media/question-resources/current_clock_iH5WgUI.png"], ["question-resources/current_mixed_xEHmLa0.png", "/srv/numbas/media/question-resources/current_mixed_xEHmLa0.png"], ["question-resources/current_no_direction_IQWaXTk.png", "/srv/numbas/media/question-resources/current_no_direction_IQWaXTk.png"], ["question-resources/current_spiral_Y13ML67.png", "/srv/numbas/media/question-resources/current_spiral_Y13ML67.png"], ["question-resources/Hcomp.png", "/srv/numbas/media/question-resources/Hcomp.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Parallel wires and superposition", "tags": [], "metadata": {"description": "<p>Practice at superimposing magnetic fields.</p>", "licence": "All rights reserved"}, "statement": "<p>This question focuses upon the quantities relating to the fields arising from currents in straight wires, and their units, and the superposition of fields from two wires separated in space.</p>\n<p>When providing numerical answers you may express them using scientific notation.&nbsp; Unless stated otherwise, express values to four significant figures and use the values of physical constants as provided in the course notes.</p>", "advice": "<p>From a Biot-Savart law, the magnetic fields from an infinitely long, straight wire can be derived to be&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle H={B\\over\\mu}={i\\over 2\\pi r}$</p>\n<p>where $H$ is the magnetizing field strength (A/m), $B$ is the magnetic flux denstity (T), $\\mu$ is the permeability (N/A$^2$), $i$ is the current (A), and $r$ is the distance from the wire (m).&nbsp; The units options provided in the question are not necessarily simply the same as those listed here, but there are acceptable or equivalent units.&nbsp; For example, an Amp is equivalent to a Coulomb per second, and &Aring; is the &Aring;ngstrom unit which is $10^{-10}$&nbsp;m.</p>\n<p>For the direction, we can employ the right-hand screw rule and note that both $H$ and $B$ circulate about the wire in a clockwise direction as viewed along the current.</p>\n<p>For the pair of wires, we can use simple geometry to determine the magnitudes and directions for the field arising at each point from each wire.&nbsp; At A the field from the right hand wire is directly to the right and has a magnitude of $\\var{current}/(2\\pi\\times\\var{pos})$ in units of A/cm. The field at A due to the left-hand wire is the most complicated of the four components.&nbsp; The distance of A from the wire using Pythagorus is $\\sqrt{(3\\times\\var{pos}^2+\\var{pos}^2}$ in cm and the direction can be obtained.</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/Hcomp.png\"/></p>\n<p>In the diagram we can see that the angle, $\\theta$ can be obtained since we know the two distances involved to be $3\\times\\var{pos}$ and $\\var{pos}$ in the horizontal and vertical directions, respectively (cm).&nbsp; Once we know the magnitude of $H$, we can determine the horizontal and vertical components from the diagram to be $-|H|\\sin(\\theta)$ and $|H|\\cos(\\theta)$, respectively.</p>\n<p>Adding the $x$-components from each wire together gives the $x$-component of the $H$ field at A, and so on.</p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"pos": {"name": "pos", "group": "Ungrouped variables", "definition": "random(5..25)*0.1", "description": "<p>Position variable that defines all parts of the system (cm).</p>", "templateType": "anything"}, "w2x": {"name": "w2x", "group": "Ungrouped variables", "definition": "pos", "description": "<p>x-co-ordinate of right-hand wire, cm.</p>", "templateType": "anything"}, "w2y": {"name": "w2y", "group": "Ungrouped variables", "definition": "0", "description": "<p>y-co-ordinate of right-hand wire, cm.</p>", "templateType": "anything"}, "w1x": {"name": "w1x", "group": "Ungrouped variables", "definition": "-2*pos", "description": "<p>x-co-ordinate of left-hand wire, cm.</p>", "templateType": "anything"}, "w1y": {"name": "w1y", "group": "Ungrouped variables", "definition": "0", "description": "<p>y-co-ordinate of left-hand wire, cm.</p>", "templateType": "anything"}, "ax": {"name": "ax", "group": "Ungrouped variables", "definition": "pos", "description": "<p>x-co-ordinate of point A, cm.</p>", "templateType": "anything"}, "ay": {"name": "ay", "group": "Ungrouped variables", "definition": "pos", "description": "<p>y-co-ordinate ofpoint A, cm.</p>", "templateType": "anything"}, "bx": {"name": "bx", "group": "Ungrouped variables", "definition": "0", "description": "<p>x-co-ordinate ofpoint B, cm.</p>", "templateType": "anything"}, "by": {"name": "by", "group": "Ungrouped variables", "definition": "0", "description": "<p>y-co-ordinate ofpoint B, cm.</p>", "templateType": "anything"}, "mu0": {"name": "mu0", "group": "Ungrouped variables", "definition": "4*10^-7 pi ", "description": "<p>Permeability of free space, N^2/A</p>", "templateType": "anything"}, "mur": {"name": "mur", "group": "Ungrouped variables", "definition": "random(30..250)", "description": "<p>Relative permeability of embedding material.</p>", "templateType": "anything"}, "ha2": {"name": "ha2", "group": "Ungrouped variables", "definition": "vector(current / 2 / pi / ay,0)", "description": "<p>H from right-wire at A, A/cm</p>", "templateType": "anything"}, "current": {"name": "current", "group": "Ungrouped variables", "definition": "random(1..50)*0.1", "description": "<p>Current in each wire, Amps.</p>", "templateType": "anything"}, "ha1": {"name": "ha1", "group": "Ungrouped variables", "definition": "vector(- current / 2 / pi / pos /10, 3 current / 2 / pi / pos /10)", "description": "<p>H from left-wire at A, A/cm</p>", "templateType": "anything"}, "ha": {"name": "ha", "group": "Ungrouped variables", "definition": "ha1 + ha2", "description": "<p>Field at A, A/cm.</p>", "templateType": "anything"}, "modha": {"name": "modha", "group": "Ungrouped variables", "definition": "sqrt(dot(ha,ha))", "description": "<p>Magnitude of H at A in A/cm.</p>", "templateType": "anything"}, "hb1": {"name": "hb1", "group": "Ungrouped variables", "definition": "vector(0,current/2/pi/2/pos)", "description": "<p>H from left-wire at B, A/cm</p>", "templateType": "anything"}, "hb2": {"name": "hb2", "group": "Ungrouped variables", "definition": "vector(0,current/2/pi/pos)", "description": "<p>H from 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"scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>As derived using the Biot-Savart or Ampere's Law, the magntiude of a magnetic field due to a long, straight wire is given by the formula</p>\n<p style=\"text-align: center;\">$\\displaystyle \\left|\\vec{H}\\right|={i\\over 2\\pi r}$.</p>\n<p>For each of the quantities in the equation, select the correct names and&nbsp;viable units.</p>\n<p>$H$ [[0]]&nbsp;[[1]]</p>\n<p>$i$ [[2]]&nbsp;[[3]]</p>\n<p>$r$ [[4]]&nbsp;[[5]]</p>", "gaps": [{"type": "1_n_2", "useCustomName": true, "customName": "H quantity", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, 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"A.J", "H", "C.m<sup>&minus;1</sup>.s<sup>&minus;1</sup>", "A/s", "H<sup>&minus;1</sup>", "&Aring;", "m<sup>&minus;1</sup>", "C/s"], "matrix": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, "2"], "distractors": ["", "", "", "", "", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "r quantity", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "3", "showCellAnswerState": true, "choices": ["magnetic flux density", "inductance", "reluctance", "flux linkage", "magnetising field strength", "current", "voltage", "capacitance", "radius of the field", "distance along the wire", "distance from the wire", "radius of the wire"], "matrix": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, "2", 0], "distractors": ["", "", "", "", "", "", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "r unit", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "4", "showCellAnswerState": true, "choices": ["V", "V/m", "V/A", "A.J", "H", "C.m<sup>&minus;1</sup>.s<sup>&minus;1</sup>", "A/s", "H<sup>&minus;1</sup>", "&Aring;", "m<sup>&minus;1</sup>", "C/s"], "matrix": [0, 0, 0, 0, 0, 0, 0, 0, "2", 0, 0], "distractors": ["", "", "", "", "", "", "", "", "", "", ""]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Pair of wires", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>In the diagram, we see the case of two, parallel wires carrying currents in opposite directions.</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/wires.png\"/></p>\n<p style=\"text-align: left;\">Two points in the $xy$-plane are also shown, labelled A and B.&nbsp; For each point select the direction of the field arisising from each of the wires.</p>\n<p style=\"text-align: left;\">[[0]]</p>", "gaps": [{"type": "m_n_x", "useCustomName": true, "customName": "Directions", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": 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"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>An experiment is set up so that the left-hand wire (as depicted in the diagram) is at $(\\var{w1x},\\var{w1y})$ and the right-hand wire is at $(\\var{w2x},\\var{w2y})$.&nbsp; The points in the $xy$-plane are also fixed so that A is at $(\\var{ax},\\var{ay})$ and B is at $(\\var{bx},\\var{by})$.&nbsp; All values for the co-ordinates in the $xy$-plane are in cm.&nbsp; Determine the total magnitudes of $\\vec{H}$ at A and B if the magnitude of the current in each wire is {current}&nbsp;A.</p>\n<p style=\"text-align: center;\">$\\left|\\vec{H}(A)\\right|=$[[0]] [[2]]</p>\n<p style=\"text-align: center;\">$\\left|\\vec{H}(B)\\right|=$[[1]] [[2]]</p>\n<p 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