// Numbas version: exam_results_page_options
{"name": "Parallel-plate capacitor characteristics practice", "extensions": [], "custom_part_types": [], "resources": [["question-resources/PPC_wVHgAAr.png", "/srv/numbas/media/question-resources/PPC_wVHgAAr.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Parallel-plate capacitor characteristics practice", "tags": ["elec", "energy", "field", "mid1718", "PPC"], "metadata": {"description": "Mathematical relationships and their evaluation for parallel plate capacitors.", "licence": "All rights reserved"}, "statement": "<p><img style=\"float: left; width: 300px;\" src=\"resources/question-resources/PPC_wVHgAAr.png\"/> A capacitor is constructed from circular plates of diameter {diametermm}&nbsp;mm separated by a layer of electrically insulating material with a uniform thickness of {wmu}&nbsp;&mu;m. When a potential difference of {pdmv}&nbsp;mV is applied, charges of &minus;{qn}&nbsp;nC and {qn}&nbsp;nC are stored on the two plates.<br/> 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 equations required to anwer this question are:</p>\n<p style=\"text-align: center;\">$\\displaystyle C={Q\\over V}={\\varepsilon A\\over d}$</p>\n<p style=\"text-align: center;\">$\\displaystyle E={V\\over d}={\\sigma \\over \\varepsilon}$</p>\n<p style=\"text-align: center;\">$\\displaystyle \\sigma = {Q\\over A}$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: center;\">Stored energy $\\displaystyle = {1\\over 2}QV$</p>\n<p style=\"text-align: left;\">where $C$ is the capacitance, $V$ is the potential difference between the plates, $A$ is the plate area, $d$ is the plate separation, $sigma$ is the charge density on a plate, $Q$ is the total charge on a plate and $\\varepsilon=\\varepsilon_0\\varepsilon_r$ is the permittivity of the material between the plates.&nbsp;</p>\n<p style=\"text-align: left;\">The formula for the electric field in terms of the charge density and the permittivity is derived from the application of Gauss' Law to an infinite sheet of uniform charge density, so is only an approximation.&nbsp; It is pretty accurate between the plates away from the edge if the gap is small compared to the length-scale of the plate.&nbsp; At the edge of the PPC the field is dipolar (not uniform).&nbsp; If the plates are far apart compared to the diameter of the plate, then the approximation is poor.</p>\n<p style=\"text-align: left;\">Finally, there are requirements to convert between units, so if you got the right numbers except for where the decimal point is, then check your unit conversion.&nbsp;</p>", "rulesets": {}, "extensions": [], "variables": {"den": {"name": "den", "group": "Ungrouped variables", "definition": "deltaenergy*10^9", "description": "<p>Increase in energy, nJ.</p>", "templateType": "anything"}, "energy1": {"name": "energy1", "group": "Ungrouped variables", "definition": "0.5*q*pd", "description": "<p>Energy stored initially, J.</p>", "templateType": "anything"}, "emax": {"name": "emax", "group": "Ungrouped variables", "definition": "siground(efield,4)*random(2..5)", "description": "<p>maximum E-field (breakdown), V/m.</p>", "templateType": "anything"}, "eps0": {"name": "eps0", "group": "Ungrouped variables", "definition": "8.854*10^-12", "description": "<p>Permittivity of free space in F/m.</p>", "templateType": "anything"}, "deltaenergy": {"name": "deltaenergy", "group": "Ungrouped variables", "definition": "energymax-energy1", "description": "<p>Energy required to increase the charge to the max, J.</p>", "templateType": "anything"}, "pd": {"name": "pd", "group": "Ungrouped variables", "definition": "pdmv/1000", "description": "<p>Potential difference in V.</p>", "templateType": "anything"}, "qn": {"name": "qn", "group": "Ungrouped variables", "definition": "random(1..10)*0.1", "description": "<p>Charge on each plate in nC.</p>", "templateType": "anything"}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "wmu*10^-6", "description": "<p>Separation of plates in m.</p>", "templateType": "anything"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "qn*10^-9", "description": "<p>Charge in Coulombs.</p>", "templateType": "anything"}, "energymax": {"name": "energymax", "group": "Ungrouped variables", "definition": "0.5*qmax*emax*w", "description": "<p>Max energy stored, J.</p>", "templateType": "anything"}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "pi*(diameter/2)^2", "description": "<p>Plate area in m^2.</p>", "templateType": "anything"}, "diametermm": {"name": "diametermm", "group": "Ungrouped variables", "definition": "random(2..5#0.5)", "description": "<p>Diameter of plates in mm.</p>", "templateType": "anything"}, "qmax": {"name": "qmax", "group": "Ungrouped variables", "definition": "capacitance * emax * w", "description": "<p>Maximum stored charge, C.</p>", "templateType": "anything"}, "wmu": {"name": "wmu", "group": "Ungrouped variables", "definition": "random(6..20#2)", "description": "<p>Separation of plates in um.</p>", "templateType": "anything"}, "pdmv": {"name": "pdmv", "group": "Ungrouped variables", "definition": "random(100..500#100)", "description": "<p>Potential difference in mV</p>", "templateType": "anything"}, "diameter": {"name": "diameter", "group": "Ungrouped variables", "definition": "diametermm/1000", "description": "<p>Diameter in m</p>", "templateType": "anything"}, "sigma": {"name": "sigma", "group": "Ungrouped variables", "definition": "q/area", "description": "<p>Charge density in C/m^2.</p>", "templateType": "anything"}, "epsr": {"name": "epsr", "group": "Ungrouped variables", "definition": "Q w / (pd * area *eps0)", "description": "<p>Relative permittivity.</p>", "templateType": "anything"}, "capacitance": {"name": "capacitance", "group": "Ungrouped variables", "definition": "q/pd", "description": "<p>Capacitance in F.</p>", "templateType": "anything"}, "efield": {"name": "efield", "group": "Ungrouped variables", "definition": "pd/w", "description": "<p>Electric field in V/m.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["area", "capacitance", "deltaenergy", "den", "diameter", "diametermm", "efield", "emax", "energy1", "energymax", "eps0", "epsr", "pd", "pdmv", "q", "qmax", "qn", "sigma", "w", "wmu"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Electric field strength", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>What is the formula for the electric field strength in terms of the potential difference, $V$, and the separation of the plates, $d$?</p>\n<p style=\"text-align: center;\">$|\\vec{E}|=$&nbsp;[[1]]</p>\n<p>Calculate the magnitude of the electric field.</p>\n<p style=\"text-align: center;\">$|\\vec{E}|=$&nbsp;[[0]] V/m</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "efield", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "efield*0.995", "maxValue": "efield*1.005", "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": "jme", "useCustomName": true, "customName": "Formula", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "V/d", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "d", "value": ""}, {"name": "v", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "The charge density", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>What is the formula for the&nbsp;charge density&nbsp;in terms of the charge, $Q$, and the&nbsp;radius of the plate, $r$.</p>\n<p style=\"text-align: center;\">$\\sigma=$&nbsp;[[1]]</p>\n<p>Calculate the charge density on the positive plate.</p>\n<p style=\"text-align: center;\">$\\sigma=$&nbsp;[[0]]&nbsp;C.m$^{-2}$</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "sigma", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "sigma*0.995", "maxValue": "sigma*1.005", "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": "jme", "useCustomName": true, "customName": "formula", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "Q/(pi r^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "q", "value": ""}, {"name": "r", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Capacitance", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>What is the formula for the capacitance&nbsp;in terms of the charge, $Q$, and the potential difference, $V$.</p>\n<p style=\"text-align: center;\">$C=$&nbsp;[[1]]</p>\n<p>Calculate the capacitance.</p>\n<p style=\"text-align: center;\">$C=$&nbsp;[[0]]&nbsp;Farads</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Capacitance", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "capacitance*0.995", "maxValue": "capacitance*1.005", "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": "jme", "useCustomName": true, "customName": "formula", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "Q/V", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "q", "value": ""}, {"name": "v", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Relative permittivity", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>What is the formula for the&nbsp;relative permittivity in terms of the charge, $Q$, the separation of the plates, $d$, the potential difference, $V$, the plate area, $A$ and the permittivity of free space, $\\varepsilon_0$ (epsilon_0).</p>\n<p style=\"text-align: center;\">$\\varepsilon_r=$&nbsp;[[1]]</p>\n<p>Calculate the value of the relative permittivity.</p>\n<p style=\"text-align: center;\">$\\varepsilon_r=$&nbsp;[[0]]&nbsp;</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Epsilon", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "epsr*0.995", "maxValue": "epsr*1.005", "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": "jme", "useCustomName": true, "customName": "formula", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "Q D/(V A epsilon_0)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "valuegenerators": [{"name": "a", "value": ""}, {"name": "d", "value": ""}, {"name": "epsilon_0", "value": ""}, {"name": "q", "value": ""}, {"name": "v", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Maximum charge", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>The insulating material between the plates will fail (start to conduct electricity) if the magnitude of the electric field strength exceeds {emax} N/C.&nbsp; What is the maximum charge that can be stored?</p>\n<p style=\"text-align: center;\">$Q_{\\rm max}=$&nbsp;[[0]]&nbsp;Coulombs</p>\n<p style=\"text-align: left;\">How much energy is required to increase the charge from {qn} nC to this maximum value?</p>\n<p style=\"text-align: center;\">Increase in energy =&nbsp;[[1]]&nbsp;nano-Joules.</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Max charge", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [{"variable": "epsr", "part": "p3g0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "qmax*0.995", "maxValue": "qmax*1.005", "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": "Energy", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "den*0.995", "maxValue": "den*1.005", "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": "m_n_x", "useCustomName": true, "customName": "Match the ideas", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Match the parts that are true for the PPC in the question.&nbsp; Note, there may be more than one acceptable answer.&nbsp; You will get a mark deducted for each incorrect answer.</p>", "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "checkbox", "warningType": "none", "showCellAnswerState": true, "choices": ["The field as $E=\\sigma/\\varepsilon$ is best approximated", "The field is dipolar", "The electric field points", "The field as $E=\\sigma/\\varepsilon$ is a poor approximation"], "matrix": [["1", "-1", "1", "-1", "-1", "-1"], ["-1", "1", "-1", "-1", "-1", "1"], ["-1", "-1", "-1", "1", 0, "-1"], ["-1", "1", "-1", "-1", "-1", "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["in the middle of the capacitor", "at the edge of the capacitor", "if the diameter is much bigger than the separation", "from positive to negative", "from negative to positive", "if the diameter is&nbsp;similar to or smaller&nbsp;than the separation"]}], "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Jon Goss", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3925/"}]}]}], "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Jon Goss", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3925/"}]}