// Numbas version: exam_results_page_options
{"name": "PA2240 Unit 3 Three Core Exercises", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": "60", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "2240_U3_Seminar_Q1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Barstow", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5191/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A plane wave in free space is normally incident on a dielectric.</p>", "advice": "", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Part a)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Show that at the boundary, the components of the magnetic intensity along the surface in the 2 media are continuous.</p>\n<p>(hint:&nbsp;The result is an immediate consequence of the Maxwell equation involving the curl of H. Follow similar working to showing that the E field is continuous)</p>\n<p>Provide the missing part of the equation in the answer.</p>\n<p style=\"text-align: left;\">$\\rm \\nabla \\times H =$&nbsp;[[0]]</p>\n<p><strong>Notes for entering symbols in the test:</strong></p>\n<p style=\"text-align: left;\">subscripts - use _, superscripts - use ^ e.g., E_0T or e^k</p>\n<p style=\"text-align: left;\">multiplication - use *, division - use /</p>\n<p style=\"text-align: left;\">for a vectors either -</p>\n<p style=\"text-align: left;\">use m:v: followed by the relevant letter, e.g. m:v:H to get a bold symbol - or -</p>\n<p style=\"text-align: left;\">use vec: followed by the relevant letter, e.g., vec:E to get ~ over an italic letter</p>\n<p style=\"text-align: left;\">for greek letters - use their names, e.g., theta, omega</p>\n<p style=\"text-align: left;\">if using a greek letter in a derivative, leave a space, e.g., d theta</p>\n<p style=\"text-align: left;\">for the operator Del, just type Del</p>\n<p style=\"text-align: left;\">for cross product just type * or leave a space</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "dD/dt", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "dd", "value": ""}, {"name": "dt", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Let's consider a rectangular circuit $\\Gamma$ with two sides orthogonal to the surface and two sides parallel to it; by using Stokes's theorem, we get:</p>\n<p>$\\rm &nbsp;\\oint_\\Gamma $[[0]]&nbsp;$ . \\it d\\bf{l} = \\int_{\\cal S}$&nbsp;[[1]]&nbsp;$\\cdot d {\\bf s}=\\int_{\\cal S} \\frac{\\partial {\\bf D}}{\\partial t}\\cdot d{\\bf s} $</p>\n<p>Again, fill in the missing parts of the equation.</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 1", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "m:v:H", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "m:v:h", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 2", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": " Del*m:v:H", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "del", "value": ""}, {"name": "m:v:h", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If we shrink the circuit so that the two sides perpendicular to the surface become infinitesimal we get:</p>\n<p>$({H}_{\\parallel 1} - {H}_{\\parallel 2})&nbsp;\\Delta l=\\int_{\\cal S} \\nabla \\times {\\bf H}\\cdot d {\\bf s}=$[[0]]&nbsp;</p>\n<p>where ${H}_{\\parallel 1}$ and ${H}_{\\parallel 2}$ are the components&nbsp;of the H-field parallel inside and outside of the dielectric.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "2240_U3_Seminar_Q2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Tim Yeoman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4835/"}, {"name": "Martin Barstow", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5191/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A laser beam, having a power of 100 MW and a diameter of 1 mm, passes through a</p>\n<p>glass window of refractive index 1.59.</p>", "advice": "", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the amplitude of the electric and magnetic fields of the incident laser beam in the air.</p>\n<p>Fill the numerical answers in the gaps</p>\n<p></p>\n<p>The amplitude of the incident Poynting vector in air is:</p>\n<p>(Note: to enter scientific notation such as $4.6 \\times 10^{-5}$, write 4.5e-5)</p>\n<p><br/>$\\left|\\langle {\\bf S}_{air} \\rangle\\right| =\\frac{P_t}{Area_{laser}}=\\frac{10^8 W}<br/>{\\pi 10^{-6}/4 m^2}\\approx $ [[0]]$\\; W/m^2=\\frac{E_{air}^2}{2Z_0}$<br/><br/>and therefore:<br/><br/>$E_{0_I}[air]=\\sqrt{2\\;Z_0\\;\\langle {\\rm \\bf S}_{air}\\rangle }= $[[1]]$\\:V/m$<br/><br/>and the magnetic field will be<br/><br/>$B_{0_I}[air]=E_{0_I}[air]/c= $[[2]]$T$</p>\n<p></p>\n<p><strong>Notes for entering symbols in the test:</strong></p>\n<p style=\"text-align: left;\">subscripts - use _, superscripts - use ^ e.g., E_0T or e^k</p>\n<p style=\"text-align: left;\">multiplication - use *, division - use /</p>\n<p style=\"text-align: left;\">for a vectors either -</p>\n<p style=\"text-align: left;\">use m:v: followed by the relevant letter, e.g. m:v:H to get a bold symbol - or -</p>\n<p style=\"text-align: left;\">use vec: followed by the relevant letter, e.g., vec:E to get ~ over an italic letter</p>\n<p style=\"text-align: left;\">for greek letters - use their names, e.g., theta, omega</p>\n<p style=\"text-align: left;\">if using a greek letter in a derivative, leave a space, e.g., d theta</p>\n<p style=\"text-align: left;\">for the operator Del, just type Del</p>\n<p style=\"text-align: left;\">for cross product just type * or leave a space</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 1", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1.0*10^14", "maxValue": "1.6*10^14", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap 2", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "28.3*10^7", "maxValue": "34.3*10^7", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap 3", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1.0", "maxValue": "1.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the amplitude of the electric and magnetic fields of the reflected laser beam in the air:</p>\n<p>Fill the numerical answers in the gaps.</p>\n<p>At normal incidence for an air-glass interface:</p>\n<p>$\\rm&nbsp;\\frac{E_{0_R}[air]}{E_{0_I}[air]}=\\frac{1-n}{1+n}=$[[0]]&nbsp;</p>\n<p>and therefore:</p>\n<p>$\\rm E_{0_R}[air]=-0.2278 \\:E_{0_I}[air]=$[[1]]&nbsp;V/m</p>\n<p>whereas the amplitude of the magnetic field is:</p>\n<p>$\\rm B_{0_R}[air]=\\frac{E_{0_R}[air]}{c}=$[[2]]&nbsp;T&nbsp;</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 4", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-0.2", "maxValue": "-0.25", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap 5", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-6.0*10^7", "maxValue": "-8.0*10^7", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap 6", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-0.22", "maxValue": "-0.26", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the amplitude of the electric and magnetic fields &nbsp;of the transmitted laser beam in the glass.&nbsp;</p>\n<p>Fill the numerical answers in the gaps.</p>\n<p>At normal incidence for an air-glass interface:<br/><br/>$\\rm \\frac{E_{0_T}[glass]}{E_{0_I}[air]}=\\frac{2}{1+n}=$[[0]]&nbsp;<br/><br/>and therefore</p>\n<p>$\\rm E_{0_T}[glass]=$[[1]]&nbsp;V/m</p>\n<p>whereas the amplitude of the magnetic field is:<br/><br/>$\\rm B_{0_T}[glass]=\\frac{E_{0_T}[glass]}{v_{glass}}=\\frac{E_{0_T}[glass]}{c}n_{glass}=$ [[2]]&nbsp;T</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 7", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.7", "maxValue": "0.9", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap 8", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "20.0*10^7", "maxValue": "30.0*10^7", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap 9", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1.1", "maxValue": "1.4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Compute the reflectance and the transmittance of the beam at the glass interface.</p>\n<p>The reflectance is:<br/><br/>$\\rm R=\\left(\\frac{E_{0_R}[air]}{E_{0_I}[air]}\\right)^2=(-0.2278)^2=$[[0]]&nbsp;%&nbsp;<br/><br/></p>\n<p>and the transmittance is:</p>\n<p>$\\rm T=1-R=$[[1]]&nbsp;%.</p>\n<p></p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 10", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "4.7", "maxValue": "5.7", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap 11", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "90", "maxValue": "100", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "2240_U3_Seminar_Q3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Barstow", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5191/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>An electromagnetic wave travelling in a vacuum is normally incident on a slab of glass of refractive index. $n$, and of relative permeability, $\\mu = 1$.</p>", "advice": "", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "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 expressions for the incident, transmitted and reflected<strong> E</strong>&nbsp;for angular frequency $\\omega $. What is the wavenumber $k$ in each medium in terms of $\\omega $ and $n$.</p>\n<p></p>\n<p>Incident - $E_I(r,t) = E_{0I}e^{i(k_I.r-\\omega _Rt)}$,&nbsp;$k_I=$[[0]]</p>\n<p>Reflected - $E_R(r,t) = E_{0R}$ [[1]] ,&nbsp;$k_R=n_1{\\omega _I/c}$</p>\n<p>Transmitted - $E_T(r,t) = E_{0T}e^{i(k_T.r-\\omega _Rt)}$,&nbsp;$k_T=$[[2]]</p>\n<p><strong>Notes for entering symbols in the test:</strong></p>\n<p style=\"text-align: left;\">subscripts - use _, superscripts - use ^ e.g., E_0T or e^k</p>\n<p style=\"text-align: left;\">multiplication - use *, division - use /</p>\n<p style=\"text-align: left;\">for a vectors either -</p>\n<p style=\"text-align: left;\">use m:v: followed by the relevant letter, e.g. m:v:H to get a bold symbol - or -</p>\n<p style=\"text-align: left;\">use vec: followed by the relevant letter, e.g., vec:E to get ~ over an italic letter</p>\n<p style=\"text-align: left;\">for greek letters - use their names, e.g., theta, omega</p>\n<p style=\"text-align: left;\">if using a greek letter in a derivative, leave a space, e.g., d theta</p>\n<p style=\"text-align: left;\">for the operator Del, just type Del</p>\n<p style=\"text-align: left;\">for cross product just type * or leave a space</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "n_1*omega_R/c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "n_1", "value": ""}, {"name": "omega_r", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 2", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "e^(k_R*r-omega_R*t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "k_r", "value": ""}, {"name": "omega_r", "value": ""}, {"name": "r", "value": ""}, {"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "n_2*(omega_T/c)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "n_2", "value": ""}, {"name": "omega_t", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Using the two boundary conditions for tangential fields, give the ratio of the reflected electric field, $E_{0R}$, to the incident electric field, $E_{0I}$ in terms of the impedances of the vacuum and glass, $Z_0$ and $Z_g$, respectively.&nbsp;</p>\n<p>x-component of E is continuous:</p>\n<p>$(\\tilde E_{0I} + \\tilde E_{0R})cos\\theta _I = \\tilde E_{0T}$[[0]]</p>\n<p>At normal incidence $cos\\theta_I=cos\\theta_T=1$</p>\n<p>Therefore:</p>\n<p>$(\\tilde E_{0I} + \\tilde E_{0R}) = \\tilde E_{0T}$&nbsp;</p>\n<p>y-component of H is continuous:</p>\n<p>$\\tilde H_{0I}-$[[1]] $=\\tilde H_{0T}$</p>\n<p>Impedance $|Z| = $[[2]]</p>\n<p>$|\\tilde E_{0R}|/|\\tilde E_{0I}|=$[[3]]$/(Z_g+Z_0)$</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 4", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos(theta_T)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "theta_t", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 5", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "vec:H_0R", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "vec:h_0r", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 6", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "vec:E/vec:H", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "vec:e", "value": ""}, {"name": "vec:h", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 7", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "Z_g-Z_0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "z_0", "value": ""}, {"name": "z_g", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "2240_U3_Seminar_Q4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Barstow", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5191/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A uniform plane wave travelling in a medium with refractive index $n_1$, is incident on a second medium of refractive index $n_2$ at angle $\\theta_I$.</p>", "advice": "", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Show that<br/><br/>$r =\\left( {\\frac{{\\tilde{E}_{0_R} }}{{\\tilde{E}_{0_I} }}} \\right) = \\frac{{Z_2 \\cos \\theta_I - Z_1 \\cos \\theta _T }}{{Z_2 \\cos \\theta _I + Z_1 \\cos \\theta _T }}$</p>\n<p>where $\\tilde{E}_{0_R}$ and $\\tilde{E}_{0_I}$ are the complex magnitudes of the reflected and incident electric fields, $Z_1$ and $Z_2$ are the characteristic impedances of medium 1 and 2, and $\\theta_T$ is the angle of the transmitted wave to the normal.</p>\n<p></p>\n<p>By imposing the continuity of the tangential components of the <strong>E</strong> and <strong>H</strong> fields we get:</p>\n<p>(enter the missing parts of the equations)</p>\n<p><br/>$\\tilde{E}_{0_I}+\\tilde{E}_{0_R} = $ [[0]]</p>\n<p><br/>$ \\tilde{H}_{0_I}\\cos\\theta_I- $ [[1]]$&nbsp;= &nbsp;\\tilde{H}_{0_T} \\cos\\theta_T $</p>\n<p>By using the relation between the ${\\bf E}$ and ${\\bf H}$ fields via the impedances of the two media we get:<br/><br/>$\\frac{\\tilde{E}_{0_I}}{Z_1}\\cos\\theta_I- $ [[2]]$\\ cos\\theta_I= \\frac{\\tilde{E}_{0_T}}{Z_2} $ [[3]]<br/><br/>and by solving the system of two equations in two unknowns we get:<br/><br/>$ r =\\left( {\\frac{{\\tilde{E}_{0_R} }}{{\\tilde{E}_{0_I} }}} \\right) = \\frac{{Z_2 \\cos \\theta _I - Z_1 \\cos \\theta _T }}{{Z_2 \\cos \\theta _I + Z_1 \\cos \\theta _T }} $</p>\n<p></p>\n<p><strong>Notes for entering symbols in the test:</strong></p>\n<p style=\"text-align: left;\">subscripts - use _, superscripts - use ^ e.g., E_0T or e^k</p>\n<p style=\"text-align: left;\">multiplication - use *, division - use /</p>\n<p style=\"text-align: left;\">for a vectors either -</p>\n<p style=\"text-align: left;\">use m:v: followed by the relevant letter, e.g. m:v:H to get a bold symbol - or -</p>\n<p style=\"text-align: left;\">use vec: followed by the relevant letter, e.g., vec:E to get ~ over an italic letter</p>\n<p style=\"text-align: left;\">for greek letters - use their names, e.g., theta, omega</p>\n<p style=\"text-align: left;\">if using a greek letter in a derivative, leave a space, e.g., d theta</p>\n<p style=\"text-align: left;\">for the operator Del, just type Del</p>\n<p style=\"text-align: left;\">for cross product just type * or leave a space</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 1", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "vec:E_0T", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "vec:e_0t", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 2", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "vec:H_0R*cos(theta_I)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "theta_i", "value": ""}, {"name": "vec:h_0r", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 3", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "vec:E_0R/Z_1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "vec:e_0r", "value": ""}, {"name": "z_1", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Gap 4", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos(theta_T)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "theta_t", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Show that at normal incidence for dielectrics the previous formula reduces to:</p>\n<p>$\\frac{{\\tilde{E}_{0_R} }}{{\\tilde{E}_{0_I} }} = \\frac{{n_1-n_2}}{{n_1+n_2 }}$</p>\n<p>This comes immediately from the previous result with $\\theta _I = \\theta _R = \\theta _T = $[[0]], and $Z_1=Z_0/n_1$ plus<br/> $Z_2=Z_0/$[[1]], which are valid for dielectrics.</p>\n<p></p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 5", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": true, "customName": "Gap 6", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "n_2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "n_2", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "review", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": "", "allowAttemptDownload": false, "downloadEncryptionKey": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": false, "advicethreshold": 0, "intro": "", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Martin Barstow", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5191/"}], 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