PK [@{)mathcad/worksheet.xml (Child's lawDiode vacuum tubeIn the diode vacuum tube in the figure, electrons are emitted from a heated cathode and flow to a positively biased anode. Two observations have been made. 1) With the cathode heated a small amount, the current is equal to the current emitted from the cathode. It does not change with anode voltage because all of the emitted electrons are collected. 2) With large cathode heating, the current varies as the 3/2 power of voltage. Why does the current vary with voltage?At high levels of cathode emission, the cloud of electrons in front of the cathode generates a negative potential causing some of the electrons to return to the cathode. The number going to the anode is determined by the space charge of the electrons in the gap between the anode and cathode. The number of electrons can be so great that Debye shielding reduces the electric field at the cathode nearly to zero. The diode can be modeled by using Poisson's equation in the region between the electrodes.ObjectsThese are Poisson's equation changed to two first order equations so that Runge Kutta can be used.Objectswhere J = n q v is the current and we find the charge density from nq = J / v. Define some familiar variables:q1.610-19ε08.85410-12m9.1110-31The starting values for f and E at the sheath boundary will be f is y0E is y1TextVariantIsSubscripty0.01We have made E = 0.01 because we are looking for the space charge limited solution in which E is nearly shielded out at the cathode. Later we can change E to see if this value affects the answer.The starting value for J is J1amps / m2TextVariantIsSuperscriptThis exercise is in SI units. We assume that the cathode temperature is 2000 - 3000 K so that the mean electron energy is ~0.2 eV plus the additional energy from falling through the potential drop. The electron velocity as a function of position is thenwhere y0 is the potential f(x).TextVariantIsSubscript2qy00.2mThe differential equations are: The meaning is: df/dx = -EdE/dx = nq = J/vNumber of iterations:DYxyy1-12qy00.2mJε0npoints200i01npointsThe anode-cathode separation isd0.1metersStart and end pointsx10x2dThe Runge-KuttaYrkfixedyx1x2npointsDYThe final value for the potential is<legend /><traces><trace resultRef="28"><traceStyle color="#FFFF0000" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle></trace></traces><graph-size width="187.200005578995" height="249.60000743866" /><axes><xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="0.11"><axisLine position="bottom" positionticmark="-1" legendWidth="78.6466666666667" /><axisGrid><gridFrequency>12</gridFrequency><gridLabels display="true" /><gridLines /><tickMarks display="true" /></axisGrid><axisLabel /><markers /><numberFormat><general precision="3" show-trailing-zeros="false" radix="dec" 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/></numberFormat><plotEquations><plotEquation><math resultRef="31"><ml:apply><ml:indexer /><ml:id xml:space="preserve">Y</ml:id><ml:sequence><ml:id xml:space="preserve">i</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply></math><math resultRef="32"><ml:placeholder /></math></plotEquation></plotEquations><xyDomain scale-type="linear" auto-scale="true"><startValue><ml:placeholder /></startValue><endValue><ml:placeholder /></endValue></xyDomain></yAxis></axes></xyPlot></plot><mc14-conversion-loss-annotations><annotation>PlotLabelsNotSupported</annotation></mc14-conversion-loss-annotations></region><region region-id="42" top="1190.4" left="9.6000000000000014"><math resultRef="33"><ml:eval><ml:apply><ml:indexer /><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">Y</ml:id><ml:sequence><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">npoints</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply><ml:unitOverride><ml:placeholder /></ml:unitOverride></ml:eval></math></region><region region-id="43" top="1190.4" left="201.60000000000002" width="43.600002288818359"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>volts </Run></Paragraph></FlowDocument></text></region><region region-id="44" top="1190.4" left="393.60000000000008" width="190.37037037037021"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph>Potential as a function of x</Paragraph></FlowDocument></text></region><region region-id="45" top="1238.4" left="9.6000000000000014" width="244"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>It takes a voltage of about 250 V across a 0.1 m gap to create a current density of 1 A/m</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Superscript"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>2</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>.</Run></Paragraph></FlowDocument></text><mc14-conversion-loss-annotations><annotation>TextVariantIsSuperscript</annotation></mc14-conversion-loss-annotations></region><region region-id="46" top="1324.8000000000002" left="67.200000000000017" width="164.91999816894531"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>x,</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations> </Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>y</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Subscript"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>0</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations> or </Run><Run FontFamily="Symbol" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>f,</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations> </Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>y</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Subscript"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>1</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations> or E</Run></Paragraph></FlowDocument></text><mc14-conversion-loss-annotations><annotation>TextVariantIsSubscript</annotation></mc14-conversion-loss-annotations></region><region region-id="47" top="1497.6000000000001" left="0"><math resultRef="35"><ml:eval><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">Y</ml:id><ml:unitOverride><ml:placeholder /></ml:unitOverride></ml:eval><resultFormat><matrix size="16,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /></resultFormat></math></region><region region-id="48" top="1545.6000000000001" left="384.00000000000006" width="253"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Click the table and scroll down to see more.</Run></Paragraph></FlowDocument></text></region><region region-id="49" top="1603.2000000000003" left="364.80000000000007" width="244.62001037597656"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Bold" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Try it:</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations> What happens to the final voltage value for different values of J?</Run></Paragraph></FlowDocument></text></region><region region-id="50" top="1747.2000000000003" left="9.6000000000000014" width="582.760009765625"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Italic" FontWeight="Bold" FontSize="16" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Method of solving a differential equation for different values of coefficient</Run></Paragraph></FlowDocument></text></region><region region-id="51" top="1776.0000000000002" left="9.6000000000000014" width="568"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Suppose we want to solve Poisson's equation for the voltage required for different values of J. The Runge Kutta routine in Mathcad does not allow us to change the J used inside of the derivatives DY without typing DY again. We cannot make DY a function of J. </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Bold" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>There is a way to work around this!</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations> We can add an additional term in the vector y and put the value of J in it. The derivative of J (now called y</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Subscript"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>2</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>) is set equal to zero so that J does not change inside of the Runge Kutta routine. This makes our answers a function of J.</Run></Paragraph></FlowDocument></text><mc14-conversion-loss-annotations><annotation>TextVariantIsSubscript</annotation></mc14-conversion-loss-annotations></region><region region-id="52" top="1900.8000000000002" left="192.00000000000003" width="432.24002075195312"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>The current J is now the third component of the vector y. Our starting y can be made a function of J but we cannot make DY a function of J.</Run></Paragraph></FlowDocument></text></region><region region-id="53" top="1920.0000000000002" left="9.6000000000000014"><math resultRef="37"><ml:define><ml:function><ml:id labels="VARIABLE" xml:space="preserve">ystart</ml:id><ml:boundVars><ml:id labels="VARIABLE" xml:space="preserve">J</ml:id></ml:boundVars></ml:function><ml:matrix rows="3" cols="1"><ml:real>0</ml:real><ml:real>0.01</ml:real><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">J</ml:id></ml:matrix></ml:define></math></region><region region-id="54" top="1968.0000000000002" left="268.80000000000007" width="113.44000244140625"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>The meaning is: </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" /><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>d</Run><Run FontFamily="Symbol" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>f</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>/dx = -E</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" /><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>dE/dx = nq = J/v</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" /><Paragraph Margin="0,0,0,0" TextAlignment="Left" /><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>dJ/dx = 0</Run></Paragraph></FlowDocument></text></region><region region-id="55" top="2025.6000000000004" left="412.80000000000007" width="179.29482580114291"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>The variable y</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Subscript"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>2</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations> is the current J.</Run></Paragraph></FlowDocument></text><mc14-conversion-loss-annotations><annotation>TextVariantIsSubscript</annotation></mc14-conversion-loss-annotations></region><region region-id="56" top="2044.8000000000004" left="9.6000000000000014"><math resultRef="39"><ml:define><ml:function><ml:id labels="VARIABLE" xml:space="preserve">DY</ml:id><ml:boundVars><ml:id labels="VARIABLE" xml:space="preserve">x</ml:id><ml:id labels="VARIABLE" xml:space="preserve">y</ml:id></ml:boundVars></ml:function><ml:matrix rows="3" cols="1"><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">y</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:mult /><ml:apply><ml:div /><ml:real>-1</ml:real><ml:apply><ml:nthRoot /><ml:placeholder /><ml:apply><ml:div /><ml:apply><ml:mult /><ml:apply><ml:mult /><ml:real>2</ml:real><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">q</ml:id></ml:apply><ml:parens><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">y</ml:id><ml:real>0</ml:real></ml:apply><ml:real>0.2</ml:real></ml:apply></ml:parens></ml:apply><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:div /><ml:apply><ml:indexer /><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">y</ml:id><ml:real>2</ml:real></ml:apply><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve"><Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:pw="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">ε<pw:Subscript>0</pw:Subscript></Span></ml:id></ml:apply></ml:apply><ml:real>0</ml:real></ml:matrix></ml:define></math></region><region region-id="57" top="2102.4" left="355.20000000000005" width="258.39999389648437"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Put zero here so that J does not change.</Run></Paragraph></FlowDocument></text></region><region region-id="58" top="2160.0000000000005" left="9.6000000000000014" width="118.46000671386719"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>The Runge-Kutta</Run></Paragraph></FlowDocument></text></region><region region-id="59" top="2160.0000000000005" left="182.40000000000003"><math resultRef="41"><ml:define><ml:function><ml:id labels="VARIABLE" xml:space="preserve">Y</ml:id><ml:boundVars><ml:id labels="VARIABLE" xml:space="preserve">J</ml:id></ml:boundVars></ml:function><ml:apply><ml:id labels="FUNCTION" label-is-contextual="true" xml:space="preserve">rkfixed</ml:id><ml:sequence><ml:apply><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">ystart</ml:id><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">J</ml:id></ml:apply><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x1</ml:id><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">x2</ml:id><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">npoints</ml:id><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">DY</ml:id></ml:sequence></ml:apply></ml:define></math></region><region region-id="60" top="2208.0000000000005" left="403.20000000000005"><math resultRef="43"><ml:eval><ml:apply><ml:indexer /><ml:apply><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">Y</ml:id><ml:real>1</ml:real></ml:apply><ml:sequence><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">npoints</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply><ml:unitOverride><ml:placeholder /></ml:unitOverride></ml:eval></math></region><region region-id="61" top="2217.6000000000004" left="0" width="291.27999877929687"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>For J = 1 we have the same answer as before:</Run></Paragraph></FlowDocument></text></region><region region-id="62" top="2256.0000000000005" left="9.6000000000000014" width="567.67999267578125"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Below is an analytic solution to our problem that we can compare to the Runge Kutta answer. </Run></Paragraph></FlowDocument></text></region><region region-id="63" top="2323.2000000000003" left="268.80000000000007" width="361.22000122070312"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>This is called both Child's law and the Langmuir-Child law. </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Child found the answer first but Langmuir popularized it. </Run></Paragraph></FlowDocument></text></region><region region-id="64" top="2332.8" left="9.6000000000000014"><math resultRef="45"><ml:define><ml:function><ml:id labels="VARIABLE" xml:space="preserve">i</ml:id><ml:boundVars><ml:id labels="VARIABLE" xml:space="preserve">ϕ</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult /><ml:apply><ml:mult /><ml:apply><ml:mult /><ml:apply><ml:div /><ml:real>4</ml:real><ml:real>9</ml:real></ml:apply><ml:apply><ml:nthRoot /><ml:placeholder /><ml:apply><ml:div /><ml:apply><ml:mult /><ml:real>2</ml:real><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">q</ml:id></ml:apply><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">m</ml:id></ml:apply></ml:apply></ml:apply><ml:parens><ml:apply><ml:div /><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve"><Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:pw="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">ε<pw:Subscript>0</pw:Subscript></Span></ml:id><ml:apply><ml:pow /><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">d</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:parens></ml:apply><ml:apply><ml:pow /><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">ϕ</ml:id><ml:apply><ml:div /><ml:real>3</ml:real><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math></region><region region-id="65" top="2390.4000000000005" left="9.6000000000000014" width="591.60003662109375"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Since Y is now a function of J, we can make a plot of the final voltage as a function of the current. </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>For the theory, we make a plot of the current as a function of the voltage. </Run></Paragraph></FlowDocument></text></region><region region-id="66" top="2448.0000000000005" left="9.6000000000000014" width="239.40000915527344"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Create a range of J values for the plot</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>of </Run><Run FontFamily="Symbol" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>f</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>(J)</Run></Paragraph></FlowDocument></text></region><region region-id="67" top="2448.0000000000005" left="307.20000000000005"><math resultRef="47"><ml:define><ml:id labels="VARIABLE" xml:space="preserve">J</ml:id><ml:range><ml:sequence><ml:real>1</ml:real><ml:real>2</ml:real></ml:sequence><ml:real>10</ml:real></ml:range></ml:define></math></region><region region-id="68" top="2505.6000000000004" left="9.6000000000000014" width="216.30000305175781"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Create a range of </Run><Run FontFamily="Symbol" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>F</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations> for </Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>the theory</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>plot of i(</Run><Run FontFamily="Symbol" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>f</Run><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>)</Run></Paragraph></FlowDocument></text></region><region region-id="69" top="2505.6000000000004" left="307.20000000000005"><math resultRef="49"><ml:define><ml:id labels="VARIABLE" xml:space="preserve">Φ</ml:id><ml:range><ml:sequence><ml:real>100</ml:real><ml:real>150</ml:real></ml:sequence><ml:real>1200</ml:real></ml:range></ml:define></math></region><region region-id="70" top="2563.2000000000003" left="9.6000000000000014" width="204.20001220703125"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Create a simpler variable name </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>for the Runge Kutta answer</Run></Paragraph></FlowDocument></text></region><region region-id="71" top="2572.8" left="307.20000000000005"><math resultRef="51"><ml:define><ml:function><ml:id labels="VARIABLE" xml:space="preserve">V</ml:id><ml:boundVars><ml:id labels="VARIABLE" xml:space="preserve">J</ml:id></ml:boundVars></ml:function><ml:apply><ml:indexer /><ml:apply><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">Y</ml:id><ml:id labels="VARIABLE" label-is-contextual="true" xml:space="preserve">J</ml:id></ml:apply><ml:sequence><ml:id xml:space="preserve">npoints</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply></ml:define></math></region><region region-id="72" top="2601.6000000000004" left="9.6000000000000014" height="278.40000572204588" width="374.80000818967824"><plot><xyPlot><title /><legend /><traces><trace resultRef="53"><traceStyle color="#FFFF0000" symbol="circle" line-weight="1" line-style="None">lines</traceStyle></trace><trace resultRef="54"><traceStyle color="#FF0000FF" symbol="none" line-weight="1" line-style="Dot">lines</traceStyle></trace></traces><graph-size width="274.800008189678" height="192.000005722046" /><axes><xAxis rank="1" legend-position="PlotBoundaryBottom" start="100" end="1200"><axisLine position="bottom" positionticmark="-1" legendWidth="85.6166666666667" /><axisGrid><gridFrequency>12</gridFrequency><gridLabels display="true" /><gridLines /><tickMarks display="true" /></axisGrid><axisLabel /><markers /><numberFormat><general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" /></numberFormat><plotEquations><plotEquation><math resultRef="55"><ml:apply><ml:id xml:space="preserve">V</ml:id><ml:id xml:space="preserve">J</ml:id></ml:apply></math><math resultRef="56"><ml:placeholder /></math></plotEquation><plotEquation><math resultRef="57"><ml:id xml:space="preserve">Φ</ml:id></math><math resultRef="58"><ml:placeholder /></math></plotEquation></plotEquations><xyDomain scale-type="linear" auto-scale="true"><startValue><ml:placeholder /></startValue><endValue><ml:placeholder /></endValue></xyDomain></xAxis><yAxis rank="1" legend-position="PlotBoundaryLeft" start="0.2" end="10.2"><axisLine position="left" positionticmark="-1" legendWidth="82.54" /><axisGrid><gridFrequency>11</gridFrequency><gridLabels display="true" /><gridLines /><tickMarks display="true" /></axisGrid><axisLabel /><markers /><numberFormat><general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" /></numberFormat><plotEquations><plotEquation><math resultRef="59"><ml:id xml:space="preserve">J</ml:id></math><math resultRef="60"><ml:placeholder /></math></plotEquation><plotEquation><math resultRef="61"><ml:apply><ml:id xml:space="preserve">i</ml:id><ml:id xml:space="preserve">Φ</ml:id></ml:apply></math><math resultRef="62"><ml:placeholder /></math></plotEquation></plotEquations><xyDomain scale-type="linear" auto-scale="true"><startValue><ml:placeholder /></startValue><endValue><ml:placeholder /></endValue></xyDomain></yAxis></axes></xyPlot></plot><mc14-conversion-loss-annotations><annotation>PlotLabelsNotSupported</annotation></mc14-conversion-loss-annotations></region><region region-id="73" top="2611.2000000000003" left="115.20000000000002" width="227.40740740740739"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph>RK solution and analytic solution</Paragraph></FlowDocument></text></region><region region-id="74" top="2688.0000000000005" left="403.20000000000005" width="216.760009765625"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>The Runge Kutta solution and the analytic solution agree pretty well.</Run></Paragraph></FlowDocument></text></region><region region-id="75" top="2928.0000000000005" left="9.6000000000000014" width="606.08001708984375"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Bold" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Try it:</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Does the initial electron energy of 0.2 eV affect the answer? For example, is there a significant change if the 0.2 is changed to 0.5 or 0.1?</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Does the initial value for E that we assumed, 0.01, have an effect? For example, is the final potential changed if we change the initial E to 0.1? 0.001? </Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Is the answer sensitive to the number of Runge Kutta steps, npoints?</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left" /><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>The initial value of E and the initial energy do not appear in Child's law. This suggests that these values are not important. The analytic solution for Child's law is derived using E = 0 at the cathode. </Run></Paragraph></FlowDocument></text></region><region region-id="76" top="3100.8000000000006" left="9.6000000000000014" width="263.70001220703125"><text><FlowDocument FontFamily="Tahoma" FontStyle="Normal" FontWeight="Normal" FontSize="14.6666666666667" Foreground="#FF000000" Background="#00FFFFFF" TextAlignment="Left" PagePadding="5,0,5,0" AllowDrop="True" Typography.Variants="Normal" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Bold" FontSize="13.3333333333333" Typography.Variants="Normal"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Reference:</Run></Paragraph><Paragraph Margin="0,0,0,0" TextAlignment="Left"><Run FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="13.3333333333333" Typography.Variants="Normal" xml:space="preserve"><Run.TextDecorations><TextDecorationCollection /></Run.TextDecorations>Chen's textbook, Chapter 8 through 8.2.4. </Run></Paragraph></FlowDocument></text></region></regions></worksheet>PK ������\@5 �� �� ��_rels/.rels �(���������������������<?xml version="1.0" encoding="utf-8"?><Relationships xmlns="http://schemas.openxmlformats.org/package/2006/relationships"><Relationship Type="http://schemas.openxmlformats.org/officeDocument/2006/relationships/officeDocument" Target="/mathcad/worksheet.xml" Id="R218a289b29b1452f" /><Relationship Type="http://schemas.openxmlformats.org/officeDocument/2006/relationships/officeDocument" Target="/mathcad/header.xml" Id="Racef55e9e8e34f90" /><Relationship Type="http://schemas.openxmlformats.org/officeDocument/2006/relationships/officeDocument" Target="/mathcad/footer.xml" Id="Redaafdd091dd4213" /><Relationship Type="http://schemas.openxmlformats.org/package/2006/relationships/metadata/core-properties" Target="/docProps/core.xml" Id="R9d2cc8846cec4b25" /></Relationships>PK �����[@+������mathcad/media/Image2.png �(���������������������PNG  ��� IHDR�� ���c���nY>���sRGB����gAMA�� a��� pHYs����od��wIDATx^N3;C/@jUKYOrx37@�^/{^_|Cpm ,X!'xn7lC`E`\c$ `!Gw#@ZvvI*FgTnRe}\.0*Blj%Pϥ\P߅@%_s%@`ZW HnNK0LD?@c сe5ɾ'0nOZ M8%h&F2γ"TbC)�I7X1X@aH^F'T``9$!"x |Su}ømi_nRt<@*8&h=C%GWG`;{j uGp[_mhv#0TrM=n}yJXjI{W*9s_c ڭ/P-#8|{}/'0Trx{*MC%PǺڭR$0Tr @1mv7%J:nCߟ (DtWa*uY QUJnAGڭ/$*8gA;G=n}7 .(㖕uɭ'0Trե;4q뷲\G`6iWK`=x1ڿ{*wC%wxݬ{ W*9~ڭr'0Trp~u"q[ Iz[aK¢}p= 0Trשmtv} 0Trץ%3?CoePX74w[_;n=ͱ@>*CRۊߵ?jΆ]j<ANn*:Ǖ-Mhj C%'%]vlɓ<WJ`R0pGK~[_m{qpyڭ6h$wAmlpWD`&֭~uH<WG`&ԭ0mX=n}JnnY v%0TrBYIvM&0TruOce&t[_m6kKm|㖥tí6PQw4@*9&h<C%GWG`;{j uGp[_mVq : Su HlJ*cTsÐIN:rHR Cj'E 0:<! 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