Calculate the value of v2.

Question

BlackTom AI · Accepted Answer

We start by understanding the circuit configuration: the 90 V source feeds R1 (10 Ω) in series, and after R1 the node splits into two parallel branches consisting of R2 (30 Ω) and R3 (60 Ω) returning to the ground.

Option-by-option analysis:

Option A: 45 Volts. If v2 were 45 V, that would imply the voltage across the parallel combination of R2 and R3 is 45 V. In a series circuit where R1 is in series with the parallel pair, the current through R1 would be I = (Vs - Vparallel)/R1 = (90 V - 45 V)/10 Ω = 4.5 A. But then the current through the parallel network would be Iparallel = Vparallel / Req, where Req = (30 Ω * 60 Ω)/(30 Ω + 60 Ω) = 20 Ω, giving Iparallel = 45 V / 20 Ω = 2.25 A. Conservation of current requires I through R1 equals Iparallel, which would be 4.5 A ≠ 2.25 A. This inconsistency shows 45 V cannot be v2.

Option B: 60 Volts. If v2 = 60 V, the voltage across the parallel network is 60 V. The current through R1 would then be I = (Vs - Vparallel)/R1 = (90 V - 60 V)/10 Ω = 3 A. The equivalent resistance of the parallel branches is Req = 20 Ω, so the current through the parallel network is Iparallel = Vparallel / Req = 60 V / 20 Ω = 3 A. The currents match (3 A through R1 equals 3 A into the parallel branches), so this is consistent. Thus 60 V is a valid v2 value.

Option C: 30 Volts. If v2 were 30 V, then I = (90 - 30)/10 = 6 A through R1. The parallel network would carry Iparallel = 30 V / 20 Ω = 1.5 A. This again violates current conservation at the node after R1 since 6 A would have to split into 1.5 A, which is impossible. Therefore 30 V cannot be v2.

Option D: 55 Volts. If v2 = 55 V, I through R1 would be (90 - 55)/10 = 3.5 A. The parallel branches would carry Iparallel = 55 V / 20 Ω = 2.75 A. Since 3.5 A ≠ 2.75 A, the currents do not balance at the junction, making this value inconsistent with the circuit.

In summary, only the 60 V scenario yields consistent current values throughout the series-parallel network, satisfying Kirchhoff’s laws and Ohm’s law for each element with the given source and resistors.

Calculate the value of v2.

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