$N$ sources of current with different emf's are connected as shown in the figure. The emf's of the sources are proportional to their internal resistances, i.e., $\xi =\alpha R$, where, $\alpha$ is an assigned constant. The lead wire resistance is negligible. Find, 

$\left(a\right)$ the current in the circuit,
$\left(b\right)$ the potential difference between points $A$ and $B$ dividing the circuit in $n$ and $N-n$ links.

 

In the circuit shown in the figure, the sources have emf's ${\mathcal{E}}_1=1.0 \mathrm{V}$ and ${\mathcal{E}}_2=2.5 \mathrm{V}$ and the resistances have the values $R_1=10 \mathrm{\Omega }$ and $R_2=20 \mathrm{\Omega }$. The internal resistances of the sources are negligible. Find a potential difference ${\phi }_{\mathrm{A}}-{\phi }_{\mathrm{B}}$ between the plates $A$ and $B$ of the capacitor $C$.

Figure illustrates a potentiometric circuit by means of which we can vary a voltage $V$  applied to a certain device possessing a resistance $R$. The potentiometer has a length $l$ and a resistance $R_0$, and voltage $V_0$  is applied to its terminals. Find the voltage $V$ fed to the device as a function of distance $x$. Analyze the $R\gg R_0$ case separately.

Find the magnitude and direction of the current flowing through the resistance $R$ in the circuit shown in figure if the emf's of the sources are equal to ${\mathcal{E}}_1=1.5 \mathrm{V}$ and ${\mathcal{E}}_2=3.7 \mathrm{V}$ and the resistances are equal to $R_1=10 \mathrm{\Omega }, R_2=20 \mathrm{\Omega }, R=5.0 \mathrm{\Omega }$. The internal resistances of the sources are negligible.

In the circuit shown in figure the sources have emf's ${\mathcal{E}}_1=1.5 \mathrm{V}, {\mathcal{E}}_2=2.0 \mathrm{V}, {\mathcal{E}}_3=2.5 \mathrm{V}$, and the resistances are equal to $R_1=10 \mathrm{\Omega }, R_2=20 \mathrm{\Omega }, R_3=30 \mathrm{\Omega }$. The internal resistances of the sources are negligible. Find,

$\left(a\right)$ the current flowing through the resistance $R_1$, 

$\left(b\right)$ a potential difference ${\phi }_A-{\phi }_B$ between the points $A$ and $B$.

Find the current flowing through the resistance $R$ in the circuit shown in the figure. The internal resistances of the batteries are negligible. 

Find a potential difference ${\phi }_A-{\phi }_B$ between the plates of a capacitor $C$in the circuit shown in figure. If the sources have emf's ${\mathcal{E}}_1=4.0 \mathrm{V}$ and ${\mathcal{E}}_2=1.0 \mathrm{V}$ and the resistances are equal to $R_1=10 \mathrm{\Omega }, R_2=20 \mathrm{\Omega },$ and $R_3=30 \mathrm{\Omega }$. The internal resistances of the sources are negligible.