// Numbas version: exam_results_page_options
{"name": "Coulomb's Law for a pair of charges vs. the dipole approximation", "extensions": [], "custom_part_types": [], "resources": [["question-resources/CoulombForce1.png", "/srv/numbas/media/question-resources/CoulombForce1.png"], ["question-resources/CoulombForce2.png", "/srv/numbas/media/question-resources/CoulombForce2.png"], ["question-resources/CoulombForce3.png", "/srv/numbas/media/question-resources/CoulombForce3.png"], ["question-resources/CoulombForce4.png", "/srv/numbas/media/question-resources/CoulombForce4.png"], ["question-resources/DipoleArrangement.png", "/srv/numbas/media/question-resources/DipoleArrangement.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Coulomb's Law for a pair of charges vs. the dipole approximation", "tags": ["coulomb", "elec", "field", "mid1819"], "metadata": {"description": "Concepts and calculation involved in the use of Coulomb's Law for both the field and the force, including units and directions, and then a comparison with a point-dipole approximation.", "licence": "All rights reserved"}, "statement": "<p>This question assesses understanding of the names and units of symbols in the Coulomb equation, calculation of quantities and evaluation of force direction.&nbsp;&nbsp;</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 Coulomb field is for a point charge (the equation does not generally apply to all charge arrangements):</p>\n<table>\n<tbody>\n<tr>\n<td>Symbol</td>\n<td>Description</td>\n<td>Units</td>\n</tr>\n<tr>\n<td>$E$</td>\n<td>Electric field strength</td>\n<td>V/m (or N/C)</td>\n</tr>\n<tr>\n<td>$q$</td>\n<td>Point charge</td>\n<td>C</td>\n</tr>\n<tr>\n<td>$\\varepsilon_0$</td>\n<td>Permittivity of free space</td>\n<td>F/m</td>\n</tr>\n<tr>\n<td>$\\varepsilon_r$</td>\n<td>Relative permittivity of medium betweem the point charge and where the field is being established</td>\n<td>none</td>\n</tr>\n<tr>\n<td>$r$</td>\n<td>Distance from the point charge - it's definitely not a radius!</td>\n<td>m</td>\n</tr>\n<tr>\n<td>$\\hat{r}$</td>\n<td>This is the unit radial vector.&nbsp; It is not a displacement (which would have units of m).</td>\n<td>none</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>When we have two point charges, the principle of superposition means that we can determine the electric field strength from each charge separately, and then add these electric fields together, noting that they are vectors and therefore have both magnitude and direciton.&nbsp;&nbsp;</p>\n<p>The process required for the charges at points A and B, is first to determine at what distance each point charge lies from P.&nbsp; This value can be substituted in the part of the Coulomb expression that conveys the magnitude (we retain the information in the sign of the charge).&nbsp; We can then work out the unit vectors from each charge towards P.&nbsp; For example,&nbsp;the unit vectors pointing from A to P is&nbsp;</p>\n<p>$\\displaystyle {\\vec{BP}\\over \\left|\\vec{BP}\\right|}$</p>\n<p>Equivalently for the case we have here, the field from B can be expressed using trigonometric functions:&nbsp;</p>\n<p>$\\displaystyle \\vec{E}_B(P)=\\frac{Q_B}{4\\pi\\varepsilon r_B^2}\\left(\\cos(\\theta)\\hat{i}+\\sin(\\theta)\\right)\\hat{j}, \\tan(\\theta)=\\frac{y(P)-y(B)}{x(P)-x(B)}$.</p>\n<p>Combining the magnitudes and directions for the fields from both point charges gives you the total field at P.</p>\n<p>The condition for the point-dipole approximation to be valid is when the distance of the point at which the field is being approximated is far bigger than the distance between two charges of equal magnitude but opposite sign.&nbsp; This is not the case in this worked example, so the point dipole approximation would be a very poor approach.</p>", "rulesets": {}, "extensions": [], "variables": {"yamm": {"name": "yamm", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "anything"}, "xbmm": {"name": "xbmm", "group": "Ungrouped variables", "definition": "-xamm", "description": "", "templateType": "anything"}, "xp": {"name": "xp", "group": "Ungrouped variables", "definition": "xa", "description": "", "templateType": "anything"}, "qb": {"name": "qb", "group": "Ungrouped variables", "definition": "-qa", "description": "<p>Second point charge in C.</p>", "templateType": "anything"}, "xa": {"name": "xa", "group": "Ungrouped variables", "definition": "xamm/1000", "description": "", "templateType": "anything"}, "rhatbpy": {"name": "rhatbpy", "group": "Ungrouped variables", "definition": "yp / sqrt(4 xa^2 +yp^2)", "description": "", "templateType": 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"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Coulomb's Law is embodied in the equation&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle \\vec{E}= {q\\over 4 \\pi \\varepsilon_0 \\varepsilon_r r^2}\\hat{r}$.</p>"}, {"type": "gapfill", "useCustomName": true, "customName": "Symbols, their meanings and their units", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: center;\">$\\vec{E}$ is&nbsp;[[0]] with units&nbsp;[[1]]</p>\n<p style=\"text-align: center;\">$q$ is&nbsp;[[2]] with units&nbsp;[[3]]</p>\n<p style=\"text-align: center;\">$\\varepsilon_0$ is&nbsp;[[4]] with units&nbsp;[[5]]</p>\n<p style=\"text-align: center;\">$\\varepsilon_r$ is&nbsp;[[6]] with units&nbsp;[[7]]</p>\n<p style=\"text-align: center;\">$r$ is&nbsp;[[8]] with units&nbsp;[[9]]</p>\n<p style=\"text-align: center;\">$\\hat{r}$ is&nbsp;[[10]] with units&nbsp;[[11]]</p>", "gaps": [{"type": "1_n_2", "useCustomName": true, "customName": "DE", "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": 0, "showCellAnswerState": true, "choices": ["electric field strength", "electric flux", "charge", "point charge", "permeability", "permeability 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"choices": ["V/m", "N/C", "C", "F/m", "none", "m", "J", "A", "V", "N", "F", "K", "eV", "Pa", "kg", "Hz"], "matrix": ["1", "1", 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], "distractors": ["", "", "", "", "", "", "", "", "", "", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Dq", "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": 0, "showCellAnswerState": true, "choices": ["electric field strength", "electric flux", "charge", "point charge", "permeability", "permeability of free space", "permeability of the material", "relative permeability of the material", "permittivity 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"K", "eV", "Pa", "kg", "Hz"], "matrix": [0, 0, 0, "1", 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], "distractors": ["", "", "", "", "", "", "", "", "", "", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Depsr", "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": 0, "showCellAnswerState": true, "choices": ["electric field strength", "electric flux", "charge", "point charge", "permeability", "permeability of free space", "permeability of the material", "relative permeability of the material", "permittivity of the material", "permittivity of free space", "relative permittivity of 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"Pa", "kg", "Hz"], "matrix": [0, 0, 0, 0, "1", 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], "distractors": ["", "", "", "", "Yes - it simply scales $\\varepsilon_0$", "", "", "", "", "", "", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "Dr", "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": 0, "showCellAnswerState": true, "choices": ["electric field strength", "electric flux", "charge", "point charge", "permeability", "permeability of free space", "permeability of the material", "relative permeability of the material", "permittivity of the material", "permittivity of free space", 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"sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "E-field value", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: center;\"><img style=\"width: 250px;\" src=\"resources/question-resources/DipoleArrangement.png\"/></p>\n<p>Two point charges are placed in vacuum at the points labelled $A$ and $B$ in the diagram.&nbsp; The point $P$ is defined, but there is no charge at this point.&nbsp; The values of the charges, and the positions $x$ and $y$ are shown in the table</p>\n<table style=\"border: 1px solid black; margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr>\n<td><strong>Site</strong></td>\n<td><strong>Charge (nC)</strong></td>\n<td><strong>$x$ (mm)</strong></td>\n<td><strong>$y$ (mm)</strong></td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$A$</td>\n<td style=\"text-align: center;\">{qan}</td>\n<td style=\"text-align: center;\">{xamm}</td>\n<td style=\"text-align: center;\">{yamm}</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$B$</td>\n<td style=\"text-align: center;\">{qbn}</td>\n<td style=\"text-align: center;\">{xbmm}</td>\n<td style=\"text-align: center;\">{ybmm}</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$P$</td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">{xpmm}</td>\n<td style=\"text-align: center;\">{ypmm}</td>\n</tr>\n</tbody>\n</table>\n<p>Determine the electric field at $P$.</p>\n<p style=\"text-align: center;\">$E_x(P)=$&nbsp;[[0]]&nbsp;[[2]]</p>\n<p style=\"text-align: center;\">$E_y(P)=$&nbsp;[[1]]&nbsp;[[2]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Epx", "marks": 1, "scripts": {}, 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