Understanding Current Flow in Parallel Circuit with Identical Resistors

When two identical resistors are connected in a parallel circuit, the behavior of the current through each resistor and the overall circuit is of significant interest for electrical engineers and enthusiasts alike. Let's delve into the concept and explore some fundamental principles that underpin this scenario.

Electrical Principles in Parallel Circuits

In an electrical circuit, there are two primary ways to connect resistors: series and parallel. In a parallel circuit, the resistors are connected in such a way that the voltage across each resistor is the same, while the current divides at each branch.

Current Flow in Parallel Circuits

When two identical resistors are connected in a parallel circuit, the current flowing through each resistor can be analyzed as follows:

Current in Each Resistor

When two resistors of the same resistance (R1 R2 R) are connected in parallel, the current through each resistor will be the same as if they were connected individually. This is due to the fact that the voltage across each resistor remains constant and equal to the source voltage. Therefore, for an identical resistor, the current flowing through it in a parallel configuration will be identical to the current it would carry if it were an individual resistor.

Let's denote the individual resistance of each resistor as R, and the source voltage as V. The current through a single resistor (I) can be calculated using Ohm's Law:

[ I frac{V}{R} ]

Since the voltage across each resistor (V) is the same in a parallel circuit, the same current (I) will flow through each resistor in the parallel configuration.

Effect on Total Current

A key principle in parallel circuits is that the total current from the source increases. Instead of a single current I flowing, the current splits between the two branches. The total current (I_{total}) that flows from the source is the sum of the currents through each branch:

[ I_{total} I_1 I_2 ]

With ( I_1 I_2 frac{V}{R} ), the total current becomes:

[ I_{total} frac{V}{R} frac{V}{R} 2 times frac{V}{R} ]

This means that in a parallel circuit with two identical resistors, the total current drawn from the source is double the current drawn by a single resistor.

Effective Resistance of the Parallel Network

The effective resistance (R_{eff}) of resistors connected in parallel is calculated using the formula:

[ frac{1}{R_{eff}} frac{1}{R_1} frac{1}{R_2} ]

For identical resistors (R1 R2 R), the effective resistance simplifies to:

[ frac{1}{R_{eff}} frac{1}{R} frac{1}{R} frac{2}{R} ]

Hence:

[ R_{eff} frac{R}{2} ]

The effective resistance of the parallel network is thus halved, meaning that the total current drawn from the source is doubled.

Assumptions and Implications

It is important to note a few assumptions and implications for the analysis:

Low Source Impedance

When the source impedance is low compared to the resistance of the resistors, the current through each resistor remains the same as in an individual configuration. However, in a real-world scenario, the source impedance (Z) may affect the current distribution.

For a low source impedance, the current through each resistor does not change significantly, but the total current from the source is doubled. This is because the source can supply more current due to the low impedance, leading to the same current in each branch.

Conclusion

When two identical resistors are connected in parallel, the current flowing through each resistor remains identical to the individual currents. However, the total current from the source is doubled due to the reduced effective resistance. This is a crucial principle in circuit analysis and design, regardless of the source impedance. Understanding these concepts can help in designing more efficient and effective electrical systems.