Ohm's Law

As shown in the figure, a source with EMF $\varepsilon = 10$ V has negligible internal resistance. The resistors are $R_1 = 20\ \Omega$, $R_2 = 10\ \Omega$, $R_3 = 10\ \Omega$, and the capacitance is $C = 1\ \mu\text{F}$. The source is in series with $R_2$, after which the circuit branches between node $A$ and the bottom rail: $R_4$ connects $A$ directly to the bottom rail, while $R_1$ leads from $A$ to node $B$, from which $R_3$ connects to the bottom rail. The capacitor $C$ is connected between node $B$ and the bottom rail, in parallel with $R_3$.

As shown in the figure, four resistors have values $R_1 = 2\ \Omega$, $R_2 = 3\ \Omega$, $R_3 = 4\ \Omega$, and $R_4 = 6\ \Omega$. The network is connected between two external terminals: $R_1$ lies along the upper branch and $R_2$ along the lower branch, both meeting at a pair of nodes between which $R_3$ and $R_4$ are connected in parallel. During the same time interval, the heat dissipated by each resistor is $Q_1, Q_2, Q_3, Q_4$, respectively.

A power source with internal resistance $r$ is sequentially connected to two different external resistors, $R_1$ and $R_2$. During the same time interval, $R_1$ and $R_2$ are measured to dissipate equal amounts of heat.

A supply of batteries is available, each with EMF $1.5 V$ and internal resistance $1 \Omega$, with a maximum allowable output current of $0.05 A$ per battery. Resistors of various values can be used as series voltage-dividing resistors. A load is to be operated at its rated values of $6 V$ and $0.1 A$.

A power source with EMF $30 V$ and internal resistance $1 \Omega$ is connected in a closed series loop with a light bulb rated $6 V$, $12 W$ and a motor whose coil resistance is $2 \Omega$. The bulb operates exactly at its rated values.

A galvanometer has internal resistance $R_g = 100 \Omega$ and full-scale deflection current $I_g = 0.1 mA$. It is converted into a multi-range voltmeter by connecting three multiplier resistors $R_1$, $R_2$, $R_3$ in series with the galvanometer $G$. Three tap terminals are provided, measured against the galvanometer's far terminal: the tap after $R_1$ corresponds to the $1 V$ range, the tap after $R_1 + R_2$ corresponds to the $10 V$ range, and the tap after $R_1 + R_2 + R_3$ corresponds to the $100 V$ range.

A potentiometer is labeled $1 k\Omega$, $40 W$.

The same galvanometer ($R_g = 100 \Omega$, $I_g = 0.1 mA$) is converted into a multi-range ammeter using an Ayrton (universal) shunt: three resistors $R_1$, $R_2$, $R_3$ are connected in series, and the galvanometer is placed in parallel across the entire string (from the leftmost node to the rightmost node). The leftmost end is the common (input) terminal. Three output taps are taken at: the node after $R_1$ ($10 A$ range), the node after $R_1 + R_2$ ($1 A$ range), and the far end after $R_3$ ($0.1 A$ range).

A ladder network connects terminals $A$ (top-left) and $B$ (bottom-left). Each cell of the ladder consists of a top horizontal resistor of value $R$, a bottom horizontal resistor of value $R$, and a vertical resistor (between the top and bottom rails) of value $R$. The network has an arbitrary number of cells. The rightmost vertical resistor is $R_x$ (unknown) rather than $R$.

A control circuit has input voltage $U = 24 V$ across input terminals $AA'$. A resistor $R$ connects $A$ to a junction node $X$, which serves as the upper output terminal $B$. From $X$, two parallel branches drop to the lower rail ($A' = B'$): one through $R_1$ in series with switch $S_1$, the other through $R_2$ in series with switch $S_2$. The output voltage at $BB'$ is to be selectable among $24 V$, $12 V$, $8 V$, and $6 V$. The current through any resistor must not exceed $12 mA$.