Identical resistors in parallel — equivalent resistance rule: For like-value resistors connected in parallel, is the equivalent resistance equal to the individual resistor value multiplied by the number of resistors, or does it follow a different relationship?
Correct Answer: Incorrect
Introduction / Context:Memorizing the right shortcut for identical components saves time. In series, resistances add. In parallel, conductances add, giving a very different shortcut for identical resistors that many students initially mix up.
Given Data / Assumptions:
- n identical resistors, each of value R, connected in parallel.
- Ideal components; parasitics neglected.
- We compare with an incorrect series-like rule.
Concept / Approach:For identical parallel resistors: 1/R_eq = n * (1/R) so R_eq = R / n. The false statement claims R_eq = R * n, which is the series rule. Therefore, in parallel, the more identical branches you add, the smaller the equivalent resistance becomes, not larger.
Step-by-Step Solution:
1) Write the parallel identity: 1/R_eq = sum(1/R_i). 2) Substitute R_i = R for all i, giving 1/R_eq = n/R. 3) Invert to obtain R_eq = R / n. 4) Compare to the claim R * n and note the contradiction.Verification / Alternative check:Example with R = 120 Ω and n = 3: R_eq = 120/3 = 40 Ω, clearly not 360 Ω. Bench tests confirm decreasing R_eq with added parallel branches.
Why Other Options Are Wrong:
- Correct: Not correct; that is the series behavior.
- True only for series connections: This option is closer to the truth about series, but the given statement concerns parallel and remains incorrect.
- True when resistors are wirewound: Construction type does not invert the rule.
- True for temperature-compensated parts: Compensation does not convert parallel math to series math.
Common Pitfalls:Applying series addition to parallel networks; failing to switch to conductance thinking for parallel combinations.
Final Answer:Incorrect