Parallel circuits — comparison with smallest branch: In any parallel network of resistors, is the equivalent resistance always greater than the lowest-value branch resistor, or is it smaller? Choose the correct statement.

Introduction / Context:A key hallmark of parallel connections is that the equivalent resistance is always less than the smallest individual branch resistance. This rule is frequently used to sanity-check calculations and quickly bound answers during design and exams.

Given Data / Assumptions:

• Resistive network with two or more branches in parallel.
• All components are ideal resistors.
• No reactive effects or frequency dependence considered.

Concept / Approach:The parallel identity is 1/R_eq = 1/R_1 + 1/R_2 + ... + 1/R_n. Since at least one positive conductance term equals 1/R_min and others are nonnegative, the total conductance exceeds 1/R_min, implying R_eq < R_min. Hence, the claim that R_eq is greater than the lowest-value branch is incorrect in all ideal cases.

Step-by-Step Solution:

Verification / Alternative check:Example: R_min = 100 Ω in parallel with anything else. Even with a very large resistor in parallel, the equivalent becomes slightly less than 100 Ω. This corroborates the general rule for parallel networks.

Why Other Options Are Wrong:

• Correct: Incorrect choice because the statement itself is false.
• True only for two branches: It is false even for two branches.
• True only when resistors are equal: If equal, R_eq = R/n which is smaller than each branch.
• Depends on source voltage: R_eq is independent of source magnitude.

Common Pitfalls:Comparing sums of resistances rather than conductances; applying series intuition to parallel networks; overlooking that R_eq is dominated by the lowest branch value downward.

Final Answer:Incorrect