Consider a network where resistor R1 is in series with a parallel combination of R2, R3, and R4 across a fixed DC source. If only R2 increases in resistance (R1, R3, R4, and the source remain unchanged), how does the voltage across R3 change?

Introduction / Context:
Series–parallel circuits appear in practical voltage dividers and sensor networks. Understanding how a change in one branch resistor affects voltages elsewhere builds strong intuition for troubleshooting and design sensitivity analysis.

Given Data / Assumptions:

• Topology: R1 in series with a parallel network consisting of R2, R3, and R4.
• Only R2 changes (increases). R1, R3, R4, and the supply voltage remain constant.
• We seek the direction of change of the voltage across R3.

Concept / Approach:
The voltage across R3 is the same as the voltage across the entire parallel network (since branches in parallel share the same voltage). Let Rp be the equivalent resistance of the parallel block. The series divider relation for the parallel block is Vp = Vs * (Rp / (R1 + Rp)). If R2 increases, Rp increases (the parallel equivalent increases when one branch resistance increases), and the divider formula shows that Vp increases monotonically with Rp.

Step-by-Step Solution:
Define Rp = (1 / (1/R2 + 1/R3 + 1/R4))When R2 increases, 1/R2 decreases → the sum (1/R2 + 1/R3 + 1/R4) decreases → Rp increases.Voltage across R3 equals Vp = Vs * Rp / (R1 + Rp).Differentiate or reason monotonicity: dVp/dRp = Vs * R1 / (R1 + Rp)^2 > 0.Therefore, Vp (and hence VR3) increases.

Verification / Alternative check:
Pick numbers: R1 = 1 kΩ, R2 = 1 kΩ → Rp_initial; then double R2 to 2 kΩ → Rp grows; computing the divider shows the parallel-node voltage rises, confirming the conclusion.

Why Other Options Are Wrong:
Decrease/remain the same: contradict the divider dependence on Rp; as Rp rises, the parallel node takes a larger share of the source voltage.

Cannot tell/oscillate: the effect is deterministic for fixed DC resistive networks.

Common Pitfalls:
Confusing voltage sharing in the parallel block (common voltage) with current sharing (different in each branch). Also, remember that raising one branch resistance increases the overall parallel equivalent (though with diminishing sensitivity).

Final Answer:
increase