128 lines
4.6 KiB
XML
128 lines
4.6 KiB
XML
<problem><script type="loncapa/python">
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scale = float(random.randrange(1000,2000,10))/1000
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V = 10*scale
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R1 = 3*scale
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R2 = 2*scale
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vV = V
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R1R2S = R1+R2
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i1 = V/R1R2S
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i2 = i1
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iV = -i1
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v1 = i1*R1
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v2 = i2*R2
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Ps= vV*iV + v1*i1 + v2*i2
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I = V
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R3 = 1/R1
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R4 = 1/R2
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iI = -I
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R3R4P = (R3*R4)/(R3+R4)
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vI = I*R3R4P
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v3 = vI
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v4 = vI
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i3 = v3/R3
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i4 = v4/R4
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Pp = vI*iI + v3*i3 + v4*i4
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</script><startouttext/>
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In this problem we will investigate a fun idea called "duality."
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<br/>
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Consider the series circuit in the diagram shown.
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<center><img src="/static/circuits/duality.gif"/></center>
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We are given device parameters \(V=$V\)V, \(R_1=$R1\Omega\), and \(R_2=$R2\Omega\).
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All of the unknown voltages and currents are labeled in associated reference
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directions. Solve this circuit for the unknowns and enter them into
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the boxes given.
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<br/>
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The value (in Volts) of \(v_1\) is:
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<endouttext/>
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<numericalresponse answer="$v1"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Amperes) of \(i_1\) is:
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<endouttext/>
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<numericalresponse answer="$i1"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Volts) of \(v_2\) is:
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<endouttext/>
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<numericalresponse answer="$v2"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Amperes) of \(i_2\) is:
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<endouttext/>
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<numericalresponse answer="$i2"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Volts) of \(v_V\) is:
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<endouttext/>
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<numericalresponse answer="$vV"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Amperes) of \(i_V\) is:
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<endouttext/>
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<numericalresponse answer="$iV"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The sum of the powers (in Watts) entering all of the elements is:
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<endouttext/>
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<numericalresponse answer="$Ps"><responseparam type="tolerance" default="0.00001" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/> <br/> Now, let's turn our attentions to the parallel circuit.
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The numerical value of the strength of the current source is set to be
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the same as the numerical value of the strength of the voltage source:
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\(I=$I\)A. The resistances are set to be reciprocals of the
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resistances in the series circuit: \(R_3=$R3\Omega\) and
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\(R_4=$R4\Omega\).
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<br/>
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Solve this circuit for the unknowns and enter them into
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the boxes given.
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<br/>
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The value (in Volts) of \(v_3\) is:
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<endouttext/>
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<numericalresponse answer="$v3"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Amperes) of \(i_3\) is:
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<endouttext/>
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<numericalresponse answer="$i3"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Volts) of \(v_4\) is:
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<endouttext/>
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<numericalresponse answer="$v4"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Amperes) of \(i_4\) is:
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<endouttext/>
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<numericalresponse answer="$i4"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value (in Volts) of \(v_I\) is:
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<endouttext/>
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<numericalresponse answer="$vI"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The value of (in Amperes) \(i_I\) is:
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<endouttext/>
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<numericalresponse answer="$iI"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/>
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The sum of the powers (in Watts) entering all of the elements is:
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<endouttext/>
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<numericalresponse answer="$Pp"><responseparam type="tolerance" default="0.00001" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
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<startouttext/>
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<br/> <br/> Look carefully at the numbers you have derived.
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Compare the series and parallel circuit. Do you see the pattern?
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<endouttext/>
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</problem>
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