Both batteries in the following figure (a) are ideal. Emf ${\epsilon }_1$ of battery $1$ has a fixed value, but emf ${\epsilon }_2$ of battery $2$ can be varied between$1.0 \mathrm{V}$ and $10 \mathrm{V}$. The plots in figure (b) give the currents through the two batteries as a function of ${\epsilon }_2.$ The vertical scale is set by ${\mathrm{i}}_{\mathrm{s}}=0.40 \mathrm{A}$. You must decide which plot corresponds to which battery, but for both plots, a negative current occurs when the direction of the current through the battery is opposite the direction of that battery's emf. What are (a) emf ${\epsilon }_1,$ (b) resistance $R_1,$ and (c) resistance $R_2?$

$\left(\mathrm{a}\right)$ 

$\left(\mathrm{b}\right)$ 

In the following figure, two batteries with emf $\epsilon =12.0 \mathrm{V}$ and an internal resistance $r=0.500 \mathrm{\Omega }$ are connected in parallel across a resistance $R$. (a) For what value of $R$ is the dissipation rate in the resistor a maximum? (b) What is that maximum?

Two identical batteries of emf $\epsilon =10.0 \mathrm{V}$ and internal resistance $r=0.200 \mathrm{\Omega }$ are to be connected to an external resistance $R,$ either in parallel (figure a) or in series (figure b). If $R=2.00 r$ , what is the current $i$ in the external resistance in the (a) parallel and (b) series arrangements? (c) For which arrangement is $i$ greater? If $R=r/2.00,$ what is $i$ in the external resistance in the (d) parallel arrangement and (e) series arrangement? (f) For which arrangement is $i$ greater now?

(a)

(b)

In figure a voltmeter of resistance $R_{\mathrm{V}}=300 \mathrm{\Omega }$ and an ammeter of resistance $R_{\mathrm{A}}=3.00 \mathrm{\Omega }$ are being used to measure a resistance $R$ in a circuit that also contains a resistance $R_0=100 \mathrm{\Omega }$ and an ideal battery with an emf of $\epsilon =18.0 \mathrm{V}.$ Resistance $R$ is given by $R=V/i,$ where $V$ is the potential across $R$ and $i$ is the ammeter reading. The voltmeter reading is $V^{\text{'}},$ which is $V$ plus the potential difference across the ammeter. Thus, the ratio of the two meter readings is not $R$ but only an apparent resistance $R^{\text{'}}=V^{\text{'}}/i.$ If $R=85.0 \mathrm{\Omega }$. What are (a) the ammeter reading, (b) the voltmeter reading, and (c) $R^{\text{'}}$? (d) If $R_{\mathrm{A}}$ is decreased, does the difference between $R^{\text{'}}$ and $R$ increase, decrease, or remain the same?

In the following figure, $R_1=100 \mathrm{\Omega }, R_2=R_3=50.0 \mathrm{\Omega }, R_4=75.0 \mathrm{\Omega },$ and the ideal battery has emf $\epsilon =12.0 \mathrm{V}$ (a) What is the equivalent resistance? What is $i$ in (b) resistance $1,$ (c) resistance $2,$ (d) resistance $3,$ and (e) resistance $4$?

In the following figure, $R_1=10.0 \mathrm{k\Omega }, R_2=15.0 \mathrm{k\Omega }, C=0.400 \mathrm{\mu F}$ , and the ideal battery has emf $\epsilon =20.0 \mathrm{V}$ . First, the switch is closed a long time so that the steady state is reached. Then the switch is opened at time $t=0.$ For resistor $2$ and at time $t=4.00 \mathrm{ms},$ what are (a) the current, (b) the rate at which the current is changing, and (c) the rate at which the dissipation rate is changing?

In the following figure, the resistors have the values $R_1=7.00 \mathrm{\Omega }$, $R_2=12.0 \mathrm{\Omega }$ and $R_3=4.00 \mathrm{\Omega }$ and the ideal battery's emf is $\epsilon =22.0 \mathrm{V}$. For what value of $R_4$ will the rate at which the battery transfers energy to the resistors equal (a) $60.0 \mathrm{W},$ (b) the maximum possible rate  $P_{\text{max }},$ and (c) the minimum possible rate $P_{\min }?$ What are (d) $P_{\max }$ and (e) $P_{\min }?$

In the following figure, the ideal batteries have emfs ${\epsilon }_1=5.0 \mathrm{V}$ and ${\epsilon }_2=12 \mathrm{V}$, the resistances are each $2.0 \mathrm{\Omega },$ and the potential is defined to be zero at the grounded point of the circuit. What are potentials (a) $V_1$and (b)$V_2$ at the indicated points? (c) What is the power of battery $2$?

Figure shows a battery connected across a uniform resistor ${\mathrm{R}}_0$. A sliding contact can move across the resistor from $x=0 \mathrm{cm}$ at the left to $x=10 \mathrm{cm}$ at the right. Moving the contact changes how much resistance is to the left of the contact and how much is to the right. Find the rate at which energy is dissipated in resistor $R$ as a function of $x$. Plot the function for $\epsilon =50 \mathrm{V}, R=2000 \mathrm{\Omega },$ and $R_0=100 \mathrm{\Omega }$.