// Numbas version: finer_feedback_settings
{"name": "Electrostatic potential and equipotentials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Electrostatic potential and equipotentials", "tags": ["coulomb", "elec", "potential", "tut1"], "metadata": {"description": "<p>The electrostatic potential due to a point charge is calculated at three points, who of which are at the same distance but different directions.&nbsp; This relates to the idea that the equipoptentials of a point charge are spheres centred on the charge, so all points at the same distance are at the same potential.</p><p>This question requires unit conversion, numerical calculations and some critical evaluation.</p>", "licence": "All rights reserved"}, "statement": "<p>A point charge is placed at the origin of a Cartesian co-ordinate system, in vacuum.&nbsp; In this problem we shall use the standard boundary condition for the electrostatic potential from a point charge, i.e. $V(\\infty)=0\\,\\text{V}$.</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>Quantiative errors may occur due to unit conversion errors.</p>\n<p>A common error is mis-remembering the Coulomb potential which is an inverse distance rule, and not an inverse-square (which is what we see for the electric field in Coulomb's Law).&nbsp; If you squared the distance, revise this section of the course.</p>\n<p>The correct formula for the electrostatic potential due to a point charge, taking the boudary condition that $V(\\infty)=0$, is&nbsp;</p>\n<p>$\\displaystyle V(r)=\\frac{q}{4\\pi\\varepsilon r}$</p>\n<p>where $r$ is the distance from the point charge.&nbsp; Note, the electrostatic potential is a scalar field - it does not have a direction.</p>\n<p>If you calculated the distance and potential at point $c$, then you may have not understood the symmetry of the system and the idea that a point charge has spherical equipotentials.&nbsp; Since the distance from the point charge to the points $a$ and $c$ are the same, so is the electrostatic potential.</p>\n<p>The potential difference between point $c$ and 'infinity' is obtainable directly from the definition of the potential difference, which is the energy per unit charge.&nbsp; Therefore the energy gained (they are both positive charges so therefore repel) is {q2}&times;10<sup>&minus;6</sup>&times;$V_c$.</p>", "rulesets": {}, "extensions": [], "variables": {"const": {"name": "const", "group": "Ungrouped variables", "definition": "(10^(-4))/(4*pi*eps0)", "description": "<p>$1/4\\pi\\varepsilon_0$ and a factor of 10,000 to convert cm and microC to get V in SI units.</p>", "templateType": "anything"}, "Vb": {"name": "Vb", "group": "Ungrouped variables", "definition": "q1*const/sqrt(x1^2+x1^2)", "description": "<p>Electrostatic potential at (x1,x1) in V.</p>", "templateType": "anything"}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "random(10..100#10)*1.0", "description": "<p>Randomised value of the charge at the origin in $mu$C.</p>", "templateType": "anything"}, "Va": {"name": "Va", "group": "Ungrouped variables", "definition": "q1*const/sqrt(x1^2+x2^2)", "description": "<p>Electrostatic potential at (x1,x2) in V.</p>", "templateType": "anything"}, "rc": {"name": "rc", "group": "Ungrouped variables", "definition": "ra", "description": "<p>Distance of point c from the charge in cm.</p>", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(5..20#1)*1.0", "description": "<p>First $x$-location in cm.</p>", "templateType": "anything"}, "Vc": {"name": "Vc", "group": "Ungrouped variables", "definition": "Va", "description": "<p>Electrostatic potential at (x2,x1) in V.</p>", "templateType": "anything"}, "eps0": {"name": "eps0", "group": "Ungrouped variables", "definition": "decimal(8.854*10^(-12))", "description": "<p>Permittivity of free space (F/m) to four significant figures.</p>", "templateType": "anything"}, "rb": {"name": "rb", "group": "Ungrouped variables", "definition": "sqrt(x1^2+x1^2)", "description": "<p>Distance of point b from the charge in cm.</p>", "templateType": "anything"}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(5..20#1 except x1)*1.0", "description": "<p>Second $x$-position in cm.</p>", "templateType": "anything"}, "ra": {"name": "ra", "group": "Ungrouped variables", "definition": "sqrt(x1^2+x2^2)", "description": "<p>Distance of point a from the charge in cm.</p>", "templateType": "anything"}, "work": {"name": "work", "group": "Ungrouped variables", "definition": "q2*Vc*10^-6", "description": "", "templateType": "anything"}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "random(1..100 except q1)*1.0", "description": "<p>Primitive charge</p>", "templateType": "anything"}, "test": {"name": "test", "group": "Ungrouped variables", "definition": "dec(1.6*10^-1.0)*1000000", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["const", "eps0", "q1", "ra", "rb", "rc", "Va", "Vb", "Vc", "x1", "x2", "work", "q2", "test"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Potential at $a$", "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 electrostitic potential at the point $a$, with co-ordinates $x=${x1} cm, $y=${x2} cm, $z=$0 cm for a point charge of {q1} $\\mu$C located at the origin?</p>\n<p style=\"text-align: center;\">$r_a=$ [[1]] cm.</p>\n<p style=\"text-align: center;\">$V(r_a)=$[[0]] Volts.</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "V_a", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Va*0.995", "maxValue": "Va*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": "r_a", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ra*0.995", "maxValue": "ra*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": "gapfill", "useCustomName": true, "customName": "Potential at $b$", "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 electrostitic potential at the point $b$, with co-ordinates $x=${x1} cm, $y=${x1} cm, $z=$0 cm for a point charge of {q1} $\\mu$C located at the origin?</p>\n<p style=\"text-align: center;\">$r_b=$&nbsp;[[1]]&nbsp;cm.</p>\n<p style=\"text-align: center;\">$V(r_b)=$[[0]]&nbsp;Volts.</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "V_b", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Vb*0.995", "maxValue": "Vb*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": "r_b", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "rb*0.95", "maxValue": "rb*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": "Potential at $c$", "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 electrostitic potential at the point $c$, with co-ordinates $x=${x2} cm, $y=${x1} cm, $z=$0 cm for the point charge of {q1} $\\mu$C located at the origin?</p>\n<p style=\"text-align: center;\">$r_c=$&nbsp;[[1]]&nbsp;cm.</p>\n<p style=\"text-align: center;\">$V(r_c)=$[[0]]&nbsp;Volts.</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "V_c", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Vc*0.995", "maxValue": "Vc*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": "r_c", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "rc*0.995", "maxValue": "rc*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": "1_n_2", "useCustomName": true, "customName": "Relationship of points", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Two of the three values of $V(r_a)$, $V(r_b)$ and $V(r_c)$ have the same value.&nbsp; This is because...</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["The points are in the same direction from the charge.", "The points are at the same distance from the charge.", "Random chance - there's no underlying reason.", "Conservation of energy", "Conservation of momentum", "The principle of superposition"], "matrix": [0, "1", 0, 0, 0, 0], "distractors": ["No - equipotenials of a point charge are spheres centred upon it.", "Yes - equipotenials of a point charge are spheres centred upon it.", "No - equipotenials of a point charge are spheres centred upon it.", "", "", ""]}, {"type": "gapfill", "useCustomName": true, "customName": "Energy", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A point charge of {q2} &mu;C point is initially placed at $c$.&nbsp; The repulsive force moves it far from the origin.&nbsp; Taking it's final position to be effectively infinitly far from the {q1} &mu;C point charge at the origin, how much energy does this second charge gain in this process?</p>\n<p style=\"text-align: center;\">Energy gained = [[0]]&nbsp;J</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Energy", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "work*0.999", "maxValue": "work*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"}], "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/"}]}