Effect of adding more branches — as additional resistors are connected in parallel to an existing network, how does the total (equivalent) resistance of the circuit change?
Correct Answer: decrease
Introduction / Context:Understanding how equivalent resistance changes with circuit topology is essential for sizing power supplies and estimating currents. This item examines the well-known behavior of parallel combinations when more branches are added.
Given Data / Assumptions:
- One or more resistors are already in parallel.
- An additional resistor is connected in parallel across the same two nodes.
- Ideal lumped components and wiring.
Concept / Approach:The reciprocal relation for parallel resistances is 1 / R_eq = 1 / R1 + 1 / R2 + ... Adding another positive term (1 / R_new) to the right-hand side increases the total conductance, which means R_eq (the reciprocal of conductance) must decrease.
Step-by-Step Solution:
1) Start with 1 / R_eq(old) = 1 / R1 + 1 / R2 + ...2) Add a new branch: 1 / R_eq(new) = 1 / R_eq(old) + 1 / R_new.3) Since 1 / R_new is positive, the right side increases.4) Therefore R_eq(new) = 1 / (larger positive number) is strictly smaller than R_eq(old).Verification / Alternative check:For two equal resistors R in parallel, R_eq = R / 2; adding a third equal branch makes R_eq = R / 3 — a clear decrease with each added branch.
Why Other Options Are Wrong:
- Increase: Opposite to the conductance sum rule.
- Increase by the value of the resistor being connected: That describes series, not parallel.
- Remain unchanged: Only true if the added branch has infinite resistance (i.e., not actually connected).
Common Pitfalls:Confusing series (resistances add) with parallel (conductances add); forgetting that any finite added branch lowers R_eq.
Final Answer:Total resistance decreases when more resistors are added in parallel.