In a circuit, a $ 10 ext{ volt}$ battery and three resistors $ R_1 = 2\ \Omega $, $ R_2 = 3\ \Omega $, and $

Question

In a circuit, a $ 10	ext{ volt}$ battery and three resistors $ R_1 = 2\ \Omega $, $ R_2 = 3\ \Omega $, and $ R_3 = 6\ \Omega $ are connected in parallel to each other. Which of the following is the correct value of effective resistance $ R_e $ and current $ I $ flowing through the circuit?

Examoogle · Accepted Answer

The correct answer is Option D: $R_e = 1\ \Omega, I = 10\ 	ext{A}$. In a parallel circuit, the voltage across each resistor is identical to the source voltage, and the total effective resistance is calculated using the reciprocal sum formula. With a voltage of $10	ext{ V}$ and parallel resistors of $2\,\Omega$, $3\,\Omega$, and $6\,\Omega$, the calculation leads to an effective resistance of $1\,\Omega$ and a total current of $10	ext{ A}$ based on Ohm's Law.

Step-by-Step Solution & Concepts:
In a parallel circuit, the voltage across each resistor is identical to the source voltage, and the total effective resistance is calculated using the reciprocal sum formula. With a voltage of $10	ext{ V}$ and parallel resistors of $2\,\Omega$, $3\,\Omega$, and $6\,\Omega$, the calculation leads to an effective resistance of $1\,\Omega$ and a total current of $10	ext{ A}$ based on Ohm's Law.

Working Principle: For resistors in parallel, the reciprocal of the effective resistance $R_e$ is the sum of the reciprocals of the individual resistances: $\frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$. Given $R_1=2\,\Omega$, $R_2=3\,\Omega$, and $R_3=6\,\Omega$, we find $\frac{1}{R_e} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3+2+1}{6} = \frac{6}{6} = 1\,\Omega^{-1}$, so $R_e = 1\,\Omega$. The total current $I$ drawn from the battery is determined by $I = \frac{V}{R_e} = \frac{10	ext{ V}}{1\,\Omega} = 10	ext{ A}$.

Key Formulas: $\frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ — Formula for equivalent resistance in parallel; $I = \frac{V}{R_e}$ — Ohm's Law to calculate total current

Key Points: In parallel connections, potential difference across each resistor is constant and equal to the supply voltage.; The equivalent resistance in a parallel circuit is always less than the value of the smallest individual resistor.; Total current is the sum of currents flowing through each branch ($I = I_1 + I_2 + I_3$).

Advantages: Each component operates independently.; Failure of one component does not stop the current flow in other branches.

Disadvantages: Requires more wiring than a series circuit.; Cannot provide different voltage levels to components.

Applications: Household electrical wiring.; Power distribution in vehicles.

Additional Info: The calculation yields $1/R_e = 0.5 + 0.333 + 0.166 = 1.0$, hence $R_e = 1\,\Omega$.. Current is $I = 10	ext{ V} / 1\,\Omega = 10	ext{ A}$.

Answer: D is correct — The calculation $\frac{1}{R_e} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6}$ yields $R_e = 1\,\Omega$, and applying Ohm's law with $V=10	ext{V}$ gives $I=10	ext{A}$.

Exam Tip: Whenever you see resistors in parallel, always check if the equivalent resistance is smaller than the smallest resistor in the set—if not, you have miscalculated!

Examoogle · Answer

Option A: $R_e = 2\ \Omega, I = 1\ 	ext{A}$

Examoogle · Answer

Option B: $R_e = 1\ \Omega, I = 1\ 	ext{A}$

Examoogle · Answer

Option C: $R_e = 2\ \Omega, I = 10\ 	ext{A}$

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