// Numbas version: finer_feedback_settings
{"name": "E-field superposition for an electron inside a parallel plate capacitor", "extensions": [], "custom_part_types": [], "resources": [["question-resources/cap.png", "/srv/numbas/media/question-resources/cap.png"], ["question-resources/neg.png", "/srv/numbas/media/question-resources/neg.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "E-field superposition for an electron inside a parallel plate capacitor", "tags": ["coulomb", "elec", "force", "PPC", "Vector", "vector"], "metadata": {"description": "Evaluation of conceptual understanding and precise inclusion of direction when combining vectors.", "licence": "All rights reserved"}, "statement": "<p style=\"text-align: center;\"><img src=\"resources/question-resources/cap.png\" width=\"300pt\"/></p>\n<p>An electron is placed at the centre of an ideal parallel plate capacitor where the plates are separated by $d$ metres, and the potential difference between the plates is $V$&nbsp;Volts. The orientation and arrangement of the component parts including the electron is indicated in the diagram above.&nbsp; The upper electrode is at the more positive electrostatic potential.</p>\n<p>In your solution you should replace</p>\n<p style=\"text-align: center;\">$\\displaystyle{1\\over{4\\pi\\varepsilon}}$</p>\n<p>with $k$, and denote the charge on an electron as $-q$.&nbsp;&nbsp;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>From Gauss' Law for an infinite parallel plate capacitor we know that the electric field, $E=V/d$, is uniform in both magnitude and direction between the plates, and points from the positive electrode to the negative electrode.&nbsp; The electron generates a spherically symmetric field (Coulomb's Law), which can be viewed as&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle{-e\\over 4 \\pi \\varepsilon_0 r^2}$&nbsp;</p>\n<p>radially outwards or, equivalently,&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle{e\\over 4 \\pi \\varepsilon_0 r^2}$&nbsp;</p>\n<p>radially towards the electron:</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/neg.png\"/></p>\n<p>The field from the capacitor and the field from the electron add vectorially everywhere, so that midway between the electron and the positive electrode, where both fields are 'down', they combine to increase the field, giving&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle{{V\\over d}+{e\\over 4 \\pi \\varepsilon_0 (d/4)^2}}$ downwards.</p>\n<p>The $d/4$ arises since this point is 1/4 of the way between the electrodes, the electron being located half way between them.&nbsp; For halfway between the electron and the negative electrode the fields point in opposite directions and tend to cancel, yielding</p>\n<p style=\"text-align: center;\">$\\displaystyle{{V\\over d}-{e\\over 4 \\pi \\varepsilon_0 (d/4)^2}}$ downwards.</p>\n<p style=\"text-align: left;\">In practice the field from the capacitor will be generally be vastly bigger than the contribution from the electron.&nbsp;</p>\n<p style=\"text-align: left;\">Since the electron is negatively charged it will experience a force $\\vec{F}=q\\vec{E}=-e\\vec{E}$ which is upwards.&nbsp; You were asked to calculate the magnitude in femto-Newtons, so you need to remember that 'femto' means '$10^{-15}$'.&nbsp; Note that the area is given in the question but is not needed to perform any calculation - in real life you often have lots of information that does not help you solve the problem, and often you don't have some of the information that you really want.</p>", "rulesets": {}, "extensions": [], "variables": {"energy": {"name": "energy", "group": "Ungrouped variables", "definition": "0.5*pd*charge", "description": "<p>Energy stored in J</p>", "templateType": "anything"}, "force": {"name": "force", "group": "Ungrouped variables", "definition": "Efield*qe", "description": "<p>Force on the electron in N</p>", "templateType": "anything"}, "sigma": {"name": "sigma", "group": "Ungrouped variables", "definition": "eps0*Efield", "description": "<p>Charge denisity on each plate in C/m^2</p>", "templateType": "anything"}, "areamm": {"name": "areamm", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "<p>PPC area in mm^2.</p>", "templateType": "anything"}, "charge": {"name": "charge", "group": "Ungrouped variables", "definition": "sigma*area", "description": "<p>Charge on each plate</p>", "templateType": "anything"}, "pd": {"name": "pd", "group": "Ungrouped variables", "definition": "random(1..50)", "description": "<p>Potential difference across plates in Volts.</p>", "templateType": "anything"}, "forcefN": {"name": "forcefN", "group": "Ungrouped variables", "definition": "force*10^15", "description": "<p>Force on the electron in femto Newtons</p>", "templateType": "anything"}, "wmicron": {"name": "wmicron", "group": "Ungrouped variables", "definition": "random(10..50#5)", "description": "<p>Plate separation in micrometres.</p>", "templateType": "anything"}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "wmicron*10^-6", "description": "<p>Separation in metres.</p>", "templateType": "anything"}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "areamm*10^-6", "description": "<p>area in m^2</p>", "templateType": "anything"}, "cppc": {"name": "cppc", "group": "Ungrouped variables", "definition": "charge/pd", "description": "<p>Capacitance of device in F</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"}, "Efield": {"name": "Efield", "group": "Ungrouped variables", "definition": "pd/w", "description": "<p>E field (uniform) in V/m</p>", "templateType": "anything"}, "qe": {"name": "qe", "group": "Ungrouped variables", "definition": "1.6*10^-19", "description": "<p>Magnitude of the charge on a electron in C.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["area", "areamm", "charge", "cppc", "Efield", "energy", "eps0", "pd", "sigma", "w", "wmicron", "qe", "force", "forcefN"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": true, "customName": "Combining fields", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>When combining two or more sources of electric field we should use...</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "3", "showCellAnswerState": true, "choices": ["The principle of superposition", "The Law of Mass Action", "Coulomb's Law", "The second law of thermodynamics", "E=mc<sup>2</sup>", "Faraday's Law"], "matrix": ["2", 0, 0, "1", 0, 0], "distractors": ["This is exactly what I was looking for...", "This law is the principle that the rate of a chemical reaction is proportional to the concentrations of the reacting substances - this is not even in this module!", "This is the specific source of the electric field from a point charge, not a general principle of combining fields.", "This important law tells us the direction a force will point, but not really what we're after here...", "This is not even a complete law or principle.  ", "No - this tells you about induction of electrical currents"]}, {"type": "gapfill", "useCustomName": true, "customName": "Field between electron and positive plate", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write down the formula for the field midway between the <strong>positive plate</strong> and the <strong>electron</strong>.&nbsp; Express your answer in terms of $V$, $d$, $k$ and $q$.&nbsp; Do not use 'e' to represent the charge on a proton, as this will be interpretted as exp(1).</p>\n<p style=\"text-align: center;\">[[0]] in the downward&nbsp;direction on the diagram</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "formula", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "16 k q /d^2+V/d", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "d", "value": ""}, {"name": "k", "value": ""}, {"name": "q", "value": ""}, {"name": "v", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Field between electron and negative plate", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write down the formula for the field midway between the negative<strong>&nbsp;plate</strong> and the <strong>electron</strong>.&nbsp; Express your answer in terms of $V$, $d$, $k$ and $q$.&nbsp; &nbsp;Again, do not use 'e' for the charge on a proton as this will be interpretted as exp(1).</p>\n<p style=\"text-align: center;\">[[0]] in the downward&nbsp;direction on the diagram</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "formula", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-16 k q / d^2 + V / d", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "d", "value": ""}, {"name": "k", "value": ""}, {"name": "q", "value": ""}, {"name": "v", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Force on the electron", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The capacitor is biased so that the potential difference across the plates is {pd}&nbsp;Volts. The distance between the plates is {wmicron}&nbsp;&mu;m and the area of each plate is {areamm}&nbsp;mm$^2$. What force does the electron experience?</p>\n<p style=\"text-align: center;\">$|\\vec{F}|=$&nbsp;[[0]]&nbsp;femto-Newtons in the&nbsp;[[1]] direction.</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Force", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "forcefN*0.995", "maxValue": "forcefN*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": "1_n_2", "useCustomName": true, "customName": "Direction", "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": "dropdownlist", "displayColumns": "4", "showCellAnswerState": true, "choices": ["Up", "Down", "Left", "Right"], "matrix": ["1", 0, "0", 0], "distractors": ["Yes - the E-field is pointing down (from plus to minus) and the force on the electron (a negative charge) is therefore up.", "No - the E-field is pointing down (from plus to minus) and the force on the electron (a negative charge) is therefore up.", "No - the E-field is pointing down (from plus to minus) and the force on the electron (a negative charge) is therefore up.", "No - the E-field is pointing down (from plus to minus) and the force on the electron (a negative charge) is therefore up."]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Relationship of force to total field", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>This means that the electron is moving in the direction of&nbsp;[[0]]&nbsp;$E$-field.</p>", "gaps": [{"type": "1_n_2", "useCustomName": true, "customName": "Relationship", "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": "dropdownlist", "displayColumns": "2", "showCellAnswerState": true, "choices": ["...deminishing...", "..increasing..."], "matrix": [0, "1"], "distractors": ["No - the field above the electron is greater than below it", "Yes - the electron is moving towards a. region of higher $E$-field."]}], "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/"}]}