The equivalent resistance of the combination of resistors, when $ 1 ext{ ohm}$ and $ 2 ext{ ohm}$ are connec

Question

The equivalent resistance of the combination of resistors, when $ 1	ext{ ohm}$ and $ 2	ext{ ohm}$ are connected in parallel, is ________.

Examoogle · Accepted Answer

The correct answer is Option A: $(2/3)	ext{ ohm}$. When two resistors $R_1 = 1\ \Omega$ and $R_2 = 2\ \Omega$ are connected in parallel, their equivalent resistance $R_{eq}$ is given by the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. Substituting the given values, $\frac{1}{R_{eq}} = \frac{1}{1} + \frac{1}{2} = \frac{3}{2}\ \Omega^{-1}$, which gives $R_{eq} = \frac{2}{3}\ \Omega$.

Step-by-Step Solution & Concepts:
When two resistors $R_1 = 1\ \Omega$ and $R_2 = 2\ \Omega$ are connected in parallel, their equivalent resistance $R_{eq}$ is given by the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. Substituting the given values, $\frac{1}{R_{eq}} = \frac{1}{1} + \frac{1}{2} = \frac{3}{2}\ \Omega^{-1}$, which gives $R_{eq} = \frac{2}{3}\ \Omega$.

Working Principle: In a parallel electrical connection, the potential difference across each resistor is the same, while the total current divides among the parallel branches. The reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the individual resistances, which always results in an equivalent resistance smaller than the smallest individual resistor in the combination.

Key Formulas: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$ — Equivalent resistance in parallel; $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$ — Shortcut formula for two parallel resistors

Key Points: In a parallel circuit, the voltage across each resistor remains the same.; The total equivalent resistance in parallel is always less than the value of the smallest individual resistor.; Total current in a parallel circuit is the sum of currents through individual branches ($I = I_1 + I_2$).

Advantages: If one component fails, other components continue to work normally.; Every appliance gets the same full voltage source.

Disadvantages: Draws higher total current from the source compared to series connection for same resistors.

Applications: Household electrical wiring and appliances; Electronic circuit boards where components require a constant voltage supply

Additional Info: Option A: $(2/3)\ \Omega$ is the mathematically correct value calculated via parallel formula.. Option B ($2\ \Omega$) and Option C ($3\ \Omega$) represent incorrect calculations or confusion with series combination ($1 + 2 = 3$).

Answer: A is correct — The equivalent resistance of 1 ohm and 2 ohm resistors connected in parallel is (2/3) ohm.

Exam Tip: Always remember: Series combination increases total resistance ($R_{eq} = R_1 + R_2$), while parallel combination decreases total resistance below the smallest individual resistor.

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