A battery consists of $12$ cells in series, each having an emf E and internal resistor r. Some of the cells in the battery are connected with wrong polarity. This battery is connected to another source of emf $2\mathrm{E}$ and internal resistance $2\mathrm{r}$. An ammeter in the circuit reads $3\mathrm{A}$ when battery and the source aid each other and $2\mathrm{A}$ in the same direction when they oppose each other. Find how many cells in the battery are connected with wrong polarity.

In the circuit shown in Fig. , each battery is of $5 \mathrm{V}$ and has an internal resistance of $0.2\mathrm{\Omega }$. What will be the reading of an ideal voltmeter connected across a battery?

In the circuit shown in Fig. , the heat produced in $5\mathrm{\Omega }$ resistor, due to the current flowing through it is $10$ calorie per second. Find the heat produced in $4\mathrm{\Omega }$ resistor.

A heater is designed to operate with a power of $1000 \mathrm{W}$ in a $100 \mathrm{V}$ line. It is connected in combination with a resistance of $10\mathrm{\Omega }$ and a resistance $\mathrm{R}$ to a $100 \mathrm{V}$ mains, as shown in Fig. . What should be the value of $\mathrm{R}$ so that the heater operates with a power of $62.5 \mathrm{W}$ ?

Three equal resistors, each of $\mathrm{R}$ ohm, are connected as shown in Fig. . A battery of $2$ volt and of internal resistance $0.1 \mathrm{ohm}$ is connected across the circuit. Calculate the value of $\mathrm{R}$ for which the heat generated in the circuit is maximum.

Determine the currents through the resistors ${\mathrm{R}}_1, {\mathrm{R}}_2$ and ${\mathrm{R}}_3$ shown in Fig.

Find the $emf\left({\mathcal{\epsilon }}_0\right)$ and internal resistance $\left(r_0\right)$ of a battery which is equivalent to a parallel combination of two batteries of emfs ${\mathrm{\epsilon }}_1$ and ${\epsilon }_2$ and internal resistances ${\mathrm{r}}_1$ and $r_2$ respectively, with polarities as shown in Fig.

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$\text{'} \mathrm{n}\text{'}$ cells, each of emf $\text{'}\mathrm{e}\text{'}$ and internal resistance $\text{'}\mathrm{r}\text{'}$ are connected in a closed circuit so that the positive terminal of a cell is joined to the negative terminal of the next, as shown in Fig. . Any two points of the circuit are connected by an external resistance $\mathrm{R}$. Find the current in $\mathrm{R}.$