Parallel circuits Knowledge-Based Study

Study notes

1. What is a Parallel Circuit?

   In a direct current (DC) circuit, components are connected in parallel when they are linked across the same two common electrical nodes. This arrangement creates multiple alternative paths through which electric charge can flow from the positive terminal of a power source to the negative terminal.

   Unlike a series circuit, where charge passes sequentially through one component after another along a single path, a parallel circuit provides each branch with its own independent conducting route. If one branch is disconnected or fails, current continues to flow through the remaining parallel branches unaffected.

   Uniform Potential Difference

   Because every parallel branch connects directly across the same two nodes, the electric potential difference (voltage) across each individual branch is identical to the voltage across the whole parallel combination:

   $V_{\text{total}} = V_1 = V_2 = V_3 = \dots = V_n$

   For example, if two resistors are connected in parallel across an ideal 12 V battery, both resistors experience a full potential difference of 12 V across their terminals, regardless of whether their resistance values are the same or different.

   Conservation of Charge at Junctions

   When the total current $I_{\text{total}}$ entering a junction splits into multiple branches, electric charge is conserved. According to Kirchhoff's Current Law (Junction Rule), the total current entering any junction must equal the sum of the currents leaving that junction:

   $I_{\text{total}} = I_1 + I_2 + I_3 + \dots + I_n$

   The current through each individual resistor can be determined using Ohm's law:

   $I_k = \frac{V}{R_k}$

   Branches with lower resistance carry larger currents, while branches with higher resistance carry smaller currents.

   Calculating Equivalent Resistance

   To find a single equivalent resistance $R_{\text{eq}}$ that draws the same total current from the source as the parallel combination, we combine Ohm's law with the junction rule:

   $\frac{V}{R_{\text{eq}}} = \frac{V}{R_1} + \frac{V}{R_2} + \dots + \frac{V}{R_n}$

   Dividing both sides by $V$ yields the reciprocal formula:

   $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$

   For the special case of two resistors in parallel, this relation simplifies to the product-over-sum rule:

   $R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}$

   Key Insight: The equivalent resistance of a parallel network is always less than the resistance of the smallest individual resistor in that network, because adding branches provides additional pathways for current flow.

   Why Household Circuits Use Parallel Wiring

   Parallel connections are standard in residential and commercial electrical distribution for two main reasons:

   • Independent Operation: Turning off or unplugging an appliance in one branch does not break the circuit for appliances in other branches.
   • Consistent Operating Voltage: Every socket and appliance receives the full rated mains voltage, ensuring predictable operation regardless of how many devices are switched on.

   However, adding more parallel loads increases the total current drawn from the supply, which necessitates safety devices such as fuses and circuit breakers to prevent overheating in the supply wires.