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Nodal analysis current through 10Ω

Short answer

Node 1 and node 2 (bottom ground). Nodal equations: 8V1 - V2 = 50, -2V1 + 17V2 = 500. Solve to get V1=10.07 V, V2=30.60 V. Current through the 10 Ω resistor = (V2 - V1)/10 = 2.05 A, from node 2 to node 1.

Use the node voltage method (nodal) to find the current flowing through 10Ω resistor in the network shown figure below. [Figure: 2 Ω shunt on the left; 5 Ω in series with a 25 V source; 10 Ω and 2 Ω along the top; 4 Ω shunt; 50 V source on the right.]

Reading the figure

Bottom rail = ground. Node 1 is the top of the 2 Ω and of the branch 5 Ω + 25 V (+ on top, so the branch end is at 25 V when the 5 Ω carries no current). Node 2 is the top of the 4 Ω and of the branch 2 Ω + 50 V (+ on top). The 10 Ω joins node 1 and node 2.

Nodal equations (currents leaving the node)

Node 1: V₁/2 + (V₁ − 25)/5 + (V₁ − V₂)/10 = 0 Multiply by 10: 5 V₁ + 2 V₁ − 50 + V₁ − V₂ = 0 ⇒ 8 V₁ − V₂ = 50 ... (1)

Node 2: V₂/4 + (V₂ − 50)/2 + (V₂ − V₁)/10 = 0 Multiply by 20: 5 V₂ + 10 V₂ − 500 + 2 V₂ − 2 V₁ = 0 ⇒ −2 V₁ + 17 V₂ = 500 ... (2)

Solution

From (1): V₂ = 8 V₁ − 50. Put in (2): −2 V₁ + 136 V₁ − 850 = 500 134 V₁ = 1350, so V₁ = 10.07 V V₂ = 8 × 10.07 − 50 = 30.60 V

Current in the 10 Ω

I = (V₂ − V₁)/10 = (30.60 − 10.07)/10 = 2.05 A

Answer: I₁₀Ω ≈ 2.05 A, flowing from node 2 to node 1

✓ Answer checked by Claude Opus 5.5 (high effort)

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