When all the components in a circuit are linked through series connections, it's called a series circuit. If they are connected through parallel connections, it's known as a parallel circuit. A third type, called a compound or combination circuit, involves both series and parallel connections.
In the circuit described, the total resistance and the potential difference (voltage) across resistors, as well as the current, can be calculated based on how the resistors are connected in each mode. Let's go through each mode using the simulation below that allows students to practise calculating potential differences and currents of a simple circuit consisting of two resistors in series (in mode 1) and two combination circuits (in modes 2 and 3). These three different modes can be accessed by clicking on the pink-coloured switch.
When resistors \( R_1 \) and \( R_2 \) are connected in series, the total resistance is simply the sum of the individual resistances:
\[ R_{\text{total}} = R_1 + R_2 \]
The current \( I \) through the circuit is given by Ohm's Law:
\[ I = \frac{V_{\text{total}}}{R_{\text{total}}} = \frac{V_{\text{total}}}{R_1 + R_2} \]
where \( V_{\text{total}} \) is the total potential difference supplied by the source.
The potential difference across each resistor can be calculated using:
\[ V_1 = I \cdot R_1, \quad V_2 = I \cdot R_2 \]
In this mode, resistors \( R_1 \) and \( R_3 \) are in parallel, and \( R_2 \) is in series with the combination. First, calculate the equivalent resistance of the parallel combination:
\[ \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_3} \]
Thus, the total resistance is:
\[ R_{\text{total}} = R_{\text{parallel}} + R_2 \]
The current through the circuit is:
\[ I = \frac{V_{\text{total}}}{R_{\text{total}}} \]
The potential difference across \( R_2 \) is:
\[ V_2 = I \cdot R_2 \]
Since \( R_1 \) and \( R_3 \) are in parallel, they share the same potential difference:
\[ V_1 = V_3 = V_{\text{total}} - V_2 \]
The current through each parallel resistor can be found using Ohm’s Law:
\[ I_1 = \frac{V_1}{R_1}, \quad I_3 = \frac{V_1}{R_3} \]
Here, resistors \( R_1 \) and \( R_2 \) are connected in series, and the combination is in parallel with \( R_3 \). First, calculate the resistance of the series combination:
\[ R_{\text{series}} = R_1 + R_2 \]
Then, find the total resistance of the parallel combination:
\[ \frac{1}{R_{\text{total}}} = \frac{1}{R_{\text{series}}} + \frac{1}{R_3} \]
The total current is:
\[ I = \frac{V_{\text{total}}}{R_{\text{total}}} \]
The voltage across the parallel combination is the same for both branches:
\[ V_1 + V_2 = V_3 = V_{\text{total}} \]
The current through \( R_3 \) is:
\[ I_3 = \frac{V_3}{R_3} \]
The current through \( R_1 \) and \( R_2 \), which are in series, is the same:
\[ I_{\text{series}} = \frac{V_{\text{total}}}{R_1 + R_2} \]
The voltage across each series resistor is:
\[ V_1 = I_{\text{series}} \cdot R_1, \quad V_2 = I_{\text{series}} \cdot R_2 \]