// Numbas version: exam_results_page_options
{"name": "Time varying current - Faraday's law", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Time varying current - Faraday's law", "tags": ["faraday", "flux", "mag", "solenoid", "tut2"], "metadata": {"description": "Calculation of physical quantities, differentiation and recognition/recall of mathematical relationships.", "licence": "All rights reserved"}, "statement": "<p>A coil comprised from {turns} turns is wrapped around a ferrite core carrying a peak flux density of {Bfield} T varying sinusoidally at a frequency of {frequency} kHz. The cross sectional area of the core is&nbsp;{xsection} mm<sup>2</sup>.</p>\n<p>Use the values of constant provided in the course when performing calculations.&nbsp; Submit numerical answers to four significant figures and you may use scientific notation.&nbsp;&nbsp;</p>", "advice": "<p>The point to this question is that we need a time varying field to induce a current.&nbsp; We're told the flux density varies sinusoidally with a given frequency, $f$, and provided with a peak amplitude, $B_0$.&nbsp; We're also told that the flux is zero at $t=0$s.&nbsp; We can therefore express the $B$-field as&nbsp;</p>\n<p style=\"text-align: center;\">$B(t)=B_0 \\sin(\\omega t+\\gamma)$</p>\n<p>where $\\omega$ is the angular frequency $2 \\pi f$, and $\\gamma$ is the phase angle.&nbsp; Since the flux (and therefore the flux density) is zero at $t=0$s, the phase angle can be taken to be zero or $\\pi$ radians.&nbsp; We can choose zero for simplicity in this case.</p>\n<p>The flux is related to the flux density as $\\phi(t)=B(t)A$, and use the relationship that the flux linkage is $\\Psi=N\\phi$ to write $\\Psi(t)$.&nbsp;&nbsp;</p>\n<p>Applying Faraday's Law requires a derivitive of the flux linkage, which is easily obtained as&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle{{d\\Psi\\over dt}={d\\over dt}N\\phi={d\\over dt}NAB(t)=NA{dB(t)\\over dt} = N A B_0 \\omega \\cos(\\omega t)}$.&nbsp;</p>\n<p>Hence $|V|=NA\\omega B_0=2\\pi NAfB_0$.</p>", "rulesets": {}, "extensions": [], "variables": {"Bfield": {"name": "Bfield", "group": "Ungrouped variables", "definition": "random(0.05..0.25#0.05)", "description": "<p>Peak B-field amplitude in T.</p>", "templateType": "anything"}, "xsection": {"name": "xsection", "group": "Ungrouped variables", "definition": "random(25..50)", "description": "<p>Cross-sectional area of ferrite core in mm<sup>2</sup>.</p>", "templateType": "anything"}, "vmax": {"name": "vmax", "group": "Ungrouped variables", "definition": "omega*turns*flux", "description": "<p>Calculated amplitude of sinusoidally varying induced voltage, Volts.</p>", "templateType": "anything"}, "frequency": {"name": "frequency", "group": "Ungrouped variables", "definition": "random(5..20)", "description": "<p>Sinusoidal frequency in kHz.</p>", "templateType": "anything"}, "turns": {"name": "turns", "group": "Ungrouped variables", "definition": "random(200..300#10)", "description": "<p>Number of turns in the solenoid.</p>", "templateType": "anything"}, "omega": {"name": "omega", "group": "Ungrouped variables", "definition": "2*pi*frequency*1000", "description": "<p>Caclulated angular frequency in cycles/s.</p>", "templateType": "anything"}, "flux": {"name": "flux", "group": "Ungrouped variables", "definition": "Bfield*xsection*10^(-6)", "description": "<p>Calculated flux in Wb (note conversion of area from mm2 to m2)</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Bfield", "flux", "frequency", "omega", "turns", "vmax", "xsection"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Flux in the ferrite core", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What the mathematical expression for the maximum flux in the ferrite core in terms of one or more of the following variables: $B$ (peak magnetic flux density), $\\mu$ (permeability), $A$ (cross-sectional area), $N$ (number of turns) and $f$ (frequency).</p>\n<p style=\"text-align: center;\">$\\phi_m=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Flux", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "B A", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Flux value", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the value of the flux in this case?</p>\n<p style=\"text-align: center;\">$\\phi_m=$[[0]] Wb</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Flux value", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "flux*0.95", "maxValue": "flux*1.05", "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}, {"type": "gapfill", "useCustomName": true, "customName": "Time dependence", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the mathematical form of the magnetic flux as a function of time ($t$), frequency ($f$), and peak flux ($\\phi_m$)?&nbsp; You may assume that the flux is zero at $t=0$s.</p>\n<p style=\"text-align: center;\">$\\phi(t)=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Time dependence", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "phi_m sin(2 pi f t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "f", "value": ""}, {"name": "phi_m", "value": ""}, {"name": "t", "value": ""}]}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": true, "customName": "Flux linkage", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the relationship between the flux, $\\phi$ and flux-linkage, $\\Psi$ for the coil?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "2", "showCellAnswerState": true, "choices": ["$\\Psi = N\\phi$", "$\\phi=N\\Psi$", "$\\Psi=\\displaystyle{N^2\\over\\phi}$", "$\\Psi \\times \\phi = {\\rm constant}$", "$\\Psi=\\displaystyle{d\\over dt}\\phi(t)$", "$\\Psi=\\displaystyle{1\\over2}\\phi i^2$"], "matrix": ["1", 0, 0, 0, 0, 0], "distractors": ["", "", "", "", "", ""]}, {"type": "gapfill", "useCustomName": true, "customName": "Maximum voltage", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Using Faraday's Law,</p>\n<p style=\"text-align: center;\">$|V|=\\displaystyle{\\left|{{d\\Psi}\\over{dt}}\\right|},$</p>\n<p>that relates the voltage flux linkage, determine the amplitude of the induced voltage.</p>\n<p style=\"text-align: center;\">$V_{\\rm max}=$[[0]]Volts</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Maximum voltage", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "vmax*0.95", "maxValue": "vmax*1.05", "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", "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/"}]}