<problem><script type="loncapa/python">
scale = float(random.randrange(1000,2000,10))/1000

V  = 10*scale
R1 = 3*scale
R2 = 2*scale

vV = V
R1R2S = R1+R2
i1 = V/R1R2S
i2 = i1
iV = -i1
v1 = i1*R1
v2 = i2*R2

Ps= vV*iV + v1*i1 + v2*i2


I  = V
R3 = 1/R1
R4 = 1/R2

iI = -I
R3R4P = (R3*R4)/(R3+R4)
vI = I*R3R4P
v3 = vI
v4 = vI
i3 = v3/R3
i4 = v4/R4

Pp = vI*iI + v3*i3 + v4*i4
</script><startouttext/>
In this problem we will investigate a fun idea called "duality."
<br/>
Consider the series circuit in the diagram shown.
<center><img src="/static/circuits/duality.gif"/></center>
We are given device parameters \(V=$V\)V, \(R_1=$R1\Omega\), and \(R_2=$R2\Omega\).
All of the unknown voltages and currents are labeled in associated reference
directions.  Solve this circuit for the unknowns and enter them into
the boxes given.
<br/>
The value (in Volts) of \(v_1\) is: 
<endouttext/>
<numericalresponse answer="$v1"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Amperes) of \(i_1\) is: 
<endouttext/>
<numericalresponse answer="$i1"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Volts) of \(v_2\) is: 
<endouttext/>
<numericalresponse answer="$v2"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Amperes) of \(i_2\) is: 
<endouttext/>
<numericalresponse answer="$i2"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Volts) of \(v_V\) is: 
<endouttext/>
<numericalresponse answer="$vV"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Amperes) of \(i_V\) is: 
<endouttext/>
<numericalresponse answer="$iV"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The sum of the powers (in Watts) entering all of the elements is:
<endouttext/>
<numericalresponse answer="$Ps"><responseparam type="tolerance" default="0.00001" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>

<startouttext/>
<br/> <br/> Now, let's turn our attentions to the parallel circuit.  
The numerical value of the strength of the current source is set to be
the same as the numerical value of the strength of the voltage source: 
\(I=$I\)A.  The resistances are set to be reciprocals of the
resistances in the series circuit:  \(R_3=$R3\Omega\) and
\(R_4=$R4\Omega\). 
<br/>
Solve this circuit for the unknowns and enter them into
the boxes given.
<br/>
The value (in Volts) of \(v_3\) is: 
<endouttext/>
<numericalresponse answer="$v3"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Amperes) of \(i_3\) is: 
<endouttext/>
<numericalresponse answer="$i3"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Volts) of \(v_4\) is: 
<endouttext/>
<numericalresponse answer="$v4"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Amperes) of \(i_4\) is: 
<endouttext/>
<numericalresponse answer="$i4"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value (in Volts) of \(v_I\) is: 
<endouttext/>
<numericalresponse answer="$vI"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The value of (in Amperes) \(i_I\) is: 
<endouttext/>
<numericalresponse answer="$iI"><responseparam type="tolerance" default="5%" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>
<startouttext/>
<br/>
The sum of the powers (in Watts) entering all of the elements is:
<endouttext/>
<numericalresponse answer="$Pp"><responseparam type="tolerance" default="0.00001" name="tol" description="Numerical Tolerance"/><textline/></numericalresponse>

<startouttext/>
<br/> <br/>  Look carefully at the numbers you have derived.
Compare the series and parallel circuit.  Do you see the pattern?
<endouttext/>


</problem>
