In a ____________ circuit, the total resistance is smaller than the smallest resistance in the circu

Question

In a ____________ circuit, the total resistance is smaller than the smallest resistance in the circuit.

Examoogle · Accepted Answer

Parallel

Explanation: In a parallel circuit, the equivalent resistance ($R_{eq}$) is always less than the value of the smallest individual resistor in the network. This occurs because providing multiple parallel paths decreases the overall opposition to the flow of current.

Working Principle: When resistors are connected in parallel, the total conductance ($G_{eq}$) is the sum of individual conductances, where $G = 1/R$. Since $G_{eq} = G_1 + G_2 + ... + G_n$, the total conductance is always greater than any individual conductance. Consequently, since $R_{eq} = 1/G_{eq}$, the total resistance must be smaller than any of the individual resistances present in the circuit.

Key Formulas: $\frac{1}{R_{eq}} = \sum_{i=1}^{n} \frac{1}{R_i}$ — Total equivalent resistance for $n$ resistors in parallel; $R_{eq} = \frac{R_1 	imes R_2}{R_1 + R_2}$ — Equivalent resistance for two resistors in parallel

Key Points: Adding more resistors in parallel always decreases the total resistance of the circuit.; The voltage remains constant across all branches in a parallel circuit.; Parallel circuits are the standard configuration for household electrical wiring to ensure independent device operation.

Advantages: Independent operation of electrical components.; Total circuit resistance decreases as more branches are added.; Constant voltage supply across all parallel components.

Disadvantages: Current increases significantly as more parallel branches are added, potentially leading to overheating.; Requires more wiring compared to simple series connections.

Applications: Household lighting and appliance circuits.; DC power distribution systems.; Multi-load electronic bias circuits.

Additional Info: For $n$ equal resistors of value $R$ connected in parallel, the total resistance is $R_{eq} = R/n$.. Option A (Series) results in a total resistance greater than the largest individual resistance, making it the opposite of the given condition.

Answer: B is correct — In a parallel circuit, the reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances, mathematically forcing the total resistance to be lower than the smallest component.

Exam Tip: Always remember that in parallel, conductance adds up; since resistance is the inverse of conductance, adding branches always lowers the barrier to current flow.

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