Problem 79.
What is the equivalent resistance between points A and B?
Solution
The voltage on the capacitor is equal to the voltage across 20 ohm resistor, which is connected in parallel to the capacitor. Thus in this problem, to find the voltage and correspondingly the charge on the capacitor, we need to calculate the voltage across the resistor. In a stationary state, there is no current through the capacitor. In this case we can consider an equivalent circuit without capacitor (see figure below). In this circuit the voltage across the resistor can be found if we know the current through 20 ohm resistor. In our case, the current though the resistor is equal to the current through the battery. Thus, we need to find the current through the battery. To find the current through the battery, we need to know the equivalent resistance of the circuit. Therefore, the first step is to find the equivalent resistance of the circuit.
In the circuit without the capacitor, two 20 ohm resistors (see figure) are connected in parallel. We replace them with an equivalent resistor with the resistance of
Finally, we have two resistors (10 ohm and 20 ohm) connected in series. The equivalent resistance of these two resistors can be found from the following expression (see figure below)
Now we can find the current through the battery. It is
This is also the current through 20 ohm resistor. Then the voltage across the resistor can be found from the Ohm’s law:
This is the voltage across the capacitor. Then the charge on the capacitor is
