1.
The resistance of Rb when the cursor position C in the center of the variable resistance:
[tex]RBC=Rb'=\frac{Rb}{2}=\frac{48}{2}=24\Omega[/tex]
The resistance of the bulb:
[tex]R'=\frac{U^{2}}{P}=\frac{12^{2}}{6}=24\Omega[/tex]
a. The equipvalent resistance of RAC:
RAC=[tex]\frac{Rb.R'}{Rb+R'}=\frac{24.24}{24+24}=12\Omega[/tex]
The equipvalent resistance of the circuit:
Req=RAC+RBC=12+24=36 [tex]\Omega[/tex]
b) The current of the circuit:
[tex]IAB=IBC=IAC=\frac{UAB}{Req}=\frac{18}{36}[/tex]=0,5 (A)
The voltage of RCB:
UCB=ICB.RBC=0,5.24=12 (V)
The voltage of RAC:
UAC=Ub'=U'=UAB-UBC=18-12=6 (V)
The capacity consumption of the bulb:
P'=[tex]\frac{U'^{2}}{R'}=\frac{6^{2}}{24}[/tex]=1,5 (W)
2.
We have IBC=IAC
IBC=[tex]\frac{UAB-U'}{48-Rb'}=\frac{6}{48-Rb'}[/tex] (1)
IAC=[tex]\frac{P'}{U'}+\frac{U'}{Rb'}=\frac{6}{12}+\frac{12}{Rb'}=0,5+\frac{12}{Rb'}[/tex] (2)
From (1) and (2), we have:
[tex]\frac{6}{48-Rb'}=0,5+\frac{12}{Rb'}[/tex]
=> Rb'[tex]\approx 40,5\Omega[/tex]
So the position cursor C is at the point that Rb'[tex]\approx 40,5\Omega[/tex] and RBC''[tex]\approx 7,5\Omega[/tex] to make the bulb works normally
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