// Numbas version: finer_feedback_settings
{"name": "Field and force on a charge in the centre of of a wire loop", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Picture_1_KeCQXYx.png", "/srv/numbas/media/question-resources/Picture_1_KeCQXYx.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Field and force on a charge in the centre of of a wire loop", "tags": [], "metadata": {"description": "<p>Field and force (no prior knowledge required for the force)</p>", "licence": "All rights reserved"}, "statement": "<p>For this question we will look at the case of a single loop of wire carring a constant current. Althought diagrams show the wires entering and leaving the loop at different points, assume that this gap is effectively zero in magnitude for the purposes of the analysis.</p>\n<p>When providing numerical answers you may express them using scientific notation.&nbsp; Express values to four significant figures and use the values of physical constants as provided in the course notes.</p>", "advice": "<p>The field at the centre of a circular current loop has a magnitude $H=j/2R$, where $j$ is the current and $R$ is the radius of the loop.&nbsp; The direction can be found using the right-hand screw rule: grab the wire with your right hand so that the thumb points along the direction of the current.&nbsp; The field circulates around the wire in the direction that your fingers are pointing.</p>\n<p>The expression for the force provided in the question is the magnetic term in the Lorentz force.&nbsp; You're told that it's an electron moving which means $q=-1.6\\times10^{-19}$C, and the formula for $B$ is what's required for the preceding step, i.e.&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle |\\vec{B}|=\\mu_0|\\vec{H}|={\\mu_0j \\over 2R}\\Rightarrow|\\vec{F}|={eu\\mu_0j \\over 2R}$</p>\n<p>You're also given the direction of the force in terms of the cross-product and order of the variables.&nbsp; The force is in the direction perpendicular to both the field (which is along $z$, as given in the question), and the velocity.&nbsp; The cross-product is a right handed rule so we can determine the direction of $\\vec{u}\\times\\vec{B}$ in a straight forward way.&nbsp; We must also note that it's an electron, which is negatively charged, so the direction is then reversed.</p>\n<p>The simplest approach is to write down the&nbsp;componentof $q\\vec{u}$ and $\\vec{B}$ in a determinant form and this will keep the orientations correctly specified.</p>\n<p>As a note, the force of {precround(forcef,4)} fN determined in this case seems rediculously small and essentially zero, but we have to note that the electron also has a very small mass (order $10^{-30}$kg), so it will result in a very high acceleration!</p>", "rulesets": {}, "extensions": [], "variables": {"radiuscm": {"name": "radiuscm", "group": "Ungrouped variables", "definition": "random(1..5#0.1)", "description": "", "templateType": "anything"}, "radius": {"name": "radius", "group": "Ungrouped variables", "definition": "radiuscm*0.01", "description": "", "templateType": "anything"}, "xspeed": {"name": "xspeed", "group": "Ungrouped variables", "definition": "random(-1001..1001#2)*10^5", "description": "", "templateType": "anything"}, "current": {"name": "current", "group": "Ungrouped variables", "definition": "random(0..2#0.01)+2", "description": "", "templateType": "anything"}, "hfield": {"name": "hfield", "group": "Ungrouped variables", "definition": "current*0.5/radius", "description": "", "templateType": "anything"}, "mu": {"name": "mu", "group": "Ungrouped variables", "definition": "4*10^-7*pi", "description": "", "templateType": "anything"}, "bfield": {"name": "bfield", "group": "Ungrouped variables", "definition": "mu*hfield", "description": "", "templateType": "anything"}, "xforcef": {"name": "xforcef", "group": "Ungrouped variables", "definition": "decimal(yspeed*bfield*q)", "description": "", "templateType": "anything"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "decimal(-1.6*10^-4)", "description": "<p>Charge on a proton in femto coulombs</p>", "templateType": "anything"}, "yspeed": {"name": "yspeed", "group": "Ungrouped variables", "definition": "random(-1001..1001#2)*10^5", "description": "", "templateType": "anything"}, "yforcef": {"name": "yforcef", "group": "Ungrouped variables", "definition": "decimal(-1*xspeed*bfield*q)", "description": "", "templateType": "anything"}, "forcef": {"name": "forcef", "group": "Unnamed group", "definition": "sqrt(xforcef^2+yforcef^2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["bfield", "current", "hfield", "mu", "q", "radius", "radiuscm", "xforcef", "xspeed", "yspeed", "yforcef"], "variable_groups": [{"name": "Unnamed group", "variables": ["forcef"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": true, "customName": "Field from a circular current", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Which of the following relative orientations of a current and the magnetic flux density its generated are consistent?&nbsp; More than one case may apply, but note that incorrect selections will reduce the credit for this part (the minimum score is zero).</p>\n<p><img src=\"resources/question-resources/Picture_1_KeCQXYx.png\" alt=\"Image showing current and field in different relative directions\"/></p>\n<p></p>", "minMarks": 0, "maxMarks": "4", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": "4", "minAnswers": "1", "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["A", "B", "C", "D"], "matrix": ["2", "-2", "-2", "2"], "distractors": ["", "", "", ""]}, {"type": "gapfill", "useCustomName": true, "customName": "Formula", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The high-symmetry and time-independence of this case renders the formula for the mangetic fields generated at the centre of the loop straight foward to obtain.&nbsp; Enter in the box below a formula for the magnitude of the magnetic flux density at the centre of the loop.&nbsp; Express your answer in terms of the radius of the loop, $R$, the current in the wire, $j$, and the permeability of free space $\\mu_0$ (mu_0).&nbsp; Take care to separate your variables with spaces and use brackets as appropriate to ensure that your formula is interpretted as you intend.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\left|\\vec{B}\\right|=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Formula", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "mu_0 j/(2R)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "j", "value": ""}, {"name": "mu_0", "value": ""}, {"name": "r", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Force", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>In an experiement, a current is flowing through a circular wire loop in the $xy$-plane with the centre of the loop at the origin.&nbsp;</p>\n<p>An electron is introduced to the system at the centre of the loop with a velocity $\\vec{u}=(\\var{xspeed/1000000})\\hat{i}+(\\var{yspeed/1000000})\\hat{j}$&nbsp;m/&mu;s. The loop of wire carries a current of $j=\\var{current}$&nbsp;Amps and has a radius of $R=\\var{radiuscm}$&nbsp;cm. The direction of the current is such that the magnetic flux at the origin is directed along the positive $z$-direction.</p>\n<p>The force on a moving charge in a magnetic field is given by&nbsp;</p>\n<p style=\"text-align: center;\">$\\vec{F}=q\\vec{u}\\times\\vec{B}$</p>\n<p style=\"text-align: left;\">where $q$ is the amount of charge, and $u$ is the velocity with which it is moving.</p>\n<p style=\"text-align: left;\">Determine $x$, $y$ and $z$ components of the force that the electron experiences, expressed in femto-newtons.</p>\n<p style=\"text-align: center;\">$\\vec{F}=$[[0]]$\\hat{i}+$[[1]]$\\hat{j}+$[[2]]$\\hat{k}$</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "F_x", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "xforcef*0.999", "maxValue": "xforcef*1.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "F_y", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "yforcef*0.999", "maxValue": "yforcef*1.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "F_z", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Jon Goss", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3925/"}]}]}], "contributors": [{"name": "Jon Goss", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3925/"}]}