Source: High school physics (Chinese)
Problem Sets:
Problem
Six identical resistors, each of resistance $R$, are connected in the circuit shown in Figure. The battery has EMF $U$ and negligible internal resistance, and the resistance values do not change.
- With all three switches closed, which resistor dissipates the greatest power, and which dissipates the least?
- When $S_1$ is opened, how does the power dissipated by each resistor change?
- When $S_2$ is then also opened, how does the power dissipated by each resistor change?
All closed: greatest $P_1=\dfrac{36U^2}{121R}$; smallest $P_4=P_5=P_6=\dfrac{4U^2}{121R}$.
$S_1$ open: $P_1=\dfrac{U^2}{4R}$, $P_2=P_3=P_5=P_6=\dfrac{U^2}{16R}$, $P_4=0$. Changes: $P_1,P_2,P_3$ down; $P_5,P_6$ up; $P_4\to 0$. $S_1$ and $S_2$ both open: $P_1=P_3=\dfrac{4U^2}{25R}$, $P_5=P_6=\dfrac{U^2}{25R}$, $P_2=P_4=0$. Changes (vs. previous): $P_1,P_5,P_6$ down; $P_3$ up; $P_2\to 0$.With all switches closed: $R_4,R_5,R_6$ in parallel give $R/3$; $R_2,R_3$ in parallel give $R/2$; with $R_1$ in series, $R_{\text{tot}}=R+\frac{R}{2}+\frac{R}{3}=\frac{11R}{6}$ and the main current is $I=\frac{6U}{11R}$. Using $P=I^2R$ on each branch:
$$P_1=\frac{36U^2}{121R},\quad P_2=P_3=\frac{9U^2}{121R},\quad P_4=P_5=P_6=\frac{4U^2}{121R}.$$So $P_1$ is the largest and $P_4=P_5=P_6$ are the smallest.
When $S_1$ is opened, $R_4$ carries no current. The left group becomes $R_5\|R_6=R/2$, so $R_{\text{tot}}=2R$, $I=\frac{U}{2R}$, and the voltage across each parallel group is $\frac{U}{4}$:
$$P_1=\frac{U^2}{4R},\quad P_2=P_3=P_5=P_6=\frac{U^2}{16R},\quad P_4=0.$$Compared with the previous state, $P_1,P_2,P_3$ decrease; $P_5,P_6$ increase; $P_4\to 0$.
When $S_2$ is also opened, $R_2$ carries no current. The middle group reduces to $R_3$ alone, so $R_{\text{tot}}=R+R+\frac{R}{2}=\frac{5R}{2}$ and $I=\frac{2U}{5R}$. The voltage across the middle is $\frac{2U}{5}$; across the left, $\frac{U}{5}$:
$$P_1=P_3=\frac{4U^2}{25R},\quad P_5=P_6=\frac{U^2}{25R},\quad P_2=P_4=0.$$Compared with the previous state, $P_1,P_5,P_6$ decrease; $P_3$ increases; $P_2\to 0$.