Consider a parallel combination of 3 resistors $ R_1 $, $ R_2 $ and $ R_3 $ with currents $ I_1 $, $ I_2 $ and

Question

Consider a parallel combination of 3 resistors $ R_1 $, $ R_2 $ and $ R_3 $ with currents $ I_1 $, $ I_2 $ and $ I_3 $, respectively. Which of the following is true? ($ I = 	ext{total current}$, $ R_{	ext{eq}} = 	ext{equivalent resistance of the circuit}$)

Examoogle · Accepted Answer

The correct answer is Option B: $I = I_1 + I_2 + I_3$. In a parallel electrical circuit, the total current supplied by the source is equal to the sum of the individual branch currents. This is a direct consequence of Kirchhoff's Current Law (KCL), which states that the sum of currents entering a junction must equal the sum of currents leaving it.

Step-by-Step Solution & Concepts:
In a parallel electrical circuit, the total current supplied by the source is equal to the sum of the individual branch currents. This is a direct consequence of Kirchhoff's Current Law (KCL), which states that the sum of currents entering a junction must equal the sum of currents leaving it.

Working Principle: According to Kirchhoff's Current Law (KCL) for any node or junction in an electric circuit, the algebraic sum of currents is zero. For a parallel connection, all resistors are connected across the same potential difference $V$. Consequently, the total current $I$ drawn from the source is the summation of individual currents $I_1, I_2,$ and $I_3$ flowing through the respective branches ($I = I_1 + I_2 + I_3$).

Key Formulas: $I = I_1 + I_2 + I_3$ — Kirchhoff's Current Law for parallel circuits; $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ — Equivalent resistance in parallel

Key Points: Potential difference remains constant across all parallel resistors.; The total current is distributed among branches based on their individual resistances.; Lower resistance branches carry higher current according to Ohm's Law ($I = \frac{V}{R}$).

Advantages: Independent operation of components; Constant voltage supply for each component

Applications: Household electrical wiring; Power distribution systems

Additional Info: Option A represents a specific case of calculating equivalent resistance using series-parallel reduction, not the general law for current.. Option C is dimensionally incorrect as it implies current behaves like resistance in parallel.. Option D is an incorrect algebraic manipulation for equivalent resistance.

Answer: B is correct — By Kirchhoff's Current Law, the total current $I$ entering a parallel circuit node is equal to the sum of currents flowing through each parallel resistor.

Exam Tip: Always remember that in parallel circuits, voltage is common, and in series circuits, current is common.

Examoogle · Answer

Option A: $R_{	ext{eq}} = R_1 + \frac{R_2 R_3}{R_2 + R_3}$

Examoogle · Answer

Option C: $\frac{1}{I} = \frac{1}{I_1} + \frac{1}{I_2} + \frac{1}{I_3}$

Examoogle · Answer

Option D: $R_{	ext{eq}} = R_3 + \frac{R_2 R_1}{R_2 + R_1}$

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