Two resistors with resistances $R_1$ and $R_2$ are connected in series, and so are two capacitors with capacitances $C_1$ and $C_2$. The two systems are connected in parallel and an external voltage is applied to the new system (see the figure accompanying the problem). What must be the relationship between $R_1, R_2, C_1$, and $C_2$ for the potential difference between the points $a$ and $b$ to be zero?

         

All the resistances and emf's shown in the figure accompanying the problem are assumed known. How many values of current can exist for such a circuit? How many equations for finding these values must we construct on the basis of Kirchhoff's first law and how many must we construct on the basis of Kirchhoff's second law?

Twelve conductors are connected in such a way that they form a cube, and an emf source is connected into an edge of the cube. All the resistances and the emfs are known. There are eight junctions (eight vertices of the cube) and six loops (six faces of the cube) in the circuit. Construct the equations for determining all the currents in the circuit.

In two circuits, each of which contains a DC source and an external resistance, the maximal currents are the same, while the maximum power output in the external resistance of one circuit is twice that in the other. In what parameters do these circuits differ?

ADC source is connected to a rheostat. When the sliding contact is $x$ distant from either end of the rheostat (the length of the rheostat is set at unity), the power output in the rheostat is the same in both cases. Determine the internal resistance of the DC source if the resistance of the theostat is $R$.

A DC source with known emf $\mathcal{E}$ is charging a capacitor $C$. After the charging process has been completed, the capacitor is disconnected, via a key $K$, from the DC source and is connected to a resistor $R$, through which the capacitor discharges. The capacitance of the capacitor and the resistance of the resistor are selected in such a way that the charging process takes several minutes, so that the discharge current can be registered by a measuring device, $G$. The results of measurements are used to draw a rough curve on a diagram in which the time of discharge is laid off on the horizontal axis and the logarithm of the current, or the vertical axis. Determine the law by which the current varies and the curve representing the dependence of the logarithm of the current on the time of discharge. How can the curve help in determining the parameters of the discharge circuit, $R$ and $C$ ?