SCIENCE · INVESTIGATE AND EXPLAIN

Reduce a network by nodes rather than its drawing

Series elements share a current because no junction offers an alternative path between them.

Learning outcome: Redraw the same node connections in a different layout before calculating. Compare equal and unequal branches, retain the junction residual and one complete loop voltage account, and predict the open-branch limit without performing any physical switching.

Starting recap

Integrate current or resistance, preserve node topology and derive transient state with a signed energy account.

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Try

For branches 4 and 6 ohm, the parallel equivalent is 2.4 ohm and total is 4.4 ohm. At 12 V, common current is 30/11 A. Branch voltage is 72/11 V; currents are 18/11 and 12/11 A, which add to the common current.

Open the experiment

Predict

What proves two branches are parallel?

Your prediction and reason

Experiment

Redraw the same node connections in a different layout before calculating. Compare equal and unequal branches, retain the junction residual and one complete loop voltage account, and predict the open-branch limit without performing any physical switching.

Preserve topology while reducing a network

Worked teaching model · synthetic values · 12 V ideal source, 2 ohm common series resistor, then selected parallel branches returning to source. Nodes rather than drawing proximity define connectivity. No wire, meter or source resistance beyond stated components.

First branch resistance: 4 ohm Second branch resistance: 6 ohm

Reset keeps your prediction, captures and practice history.

Read the node connections before calculating

Two distinct branches connect J to B. Their drawing offsets do not change their shared endpoint nodes. Source positive terminal is S; conventional source-delivered current proceeds S to J. Component orientation and drawing lengths are schematic; source polarities are specified in the caption and node labels.

First node	Second node	Element
S	J	2 Ω
J	B	4 Ω
J	B	6 Ω
B	S	12 V source

Common schematic symbols and conventions

The symbol key is not a wiring recommendation. An open switch removes a conducting path; an ideal closed switch supplies a zero-drop connection. A current arrow is a chosen conventional-charge reference, not electron drift. A junction dot signifies connection; a crossing without a dot need not connect.

Potential around source-series-first-branch loop

Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.

Potential around source-series-first-branch loop: numerical coordinates

ordered node index, not physical distance	potential (V)
0	0
1	12
2	6.54545
3	0

Calculated values under the declared model
Quantity	Value	Unit
Parallel equivalent resistance	2.4	ohm
Total resistance	4.4	ohm
Common series current	2.72727	A
Parallel branch voltage	6.54545	V
First branch current	1.63636	A
Second branch current	1.09091	A
Junction residual	0	A
Series resistor power	14.876	W
First branch power	10.7107	W
Second branch power	7.1405	W
Total source power	32.7273	W

The common current divides only at the actual junction, while both parallel branches share the same node-to-node voltage.

Series and parallel are connectivity relations, not horizontal or vertical placement in a drawing. Removing one branch increases equivalent resistance here. An ideal bypass across a resistor fixes its voltage to zero; an ideal zero-resistance path directly across an ideal source is a singular model, not a finite predicted current or a safe experiment.

Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.

The common current divides only at the actual junction, while both parallel branches share the same node-to-node voltage.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Identify actual connection constraints

Series elements share a current because no junction offers an alternative path between them. Parallel branches share both endpoint nodes and therefore the same voltage. Moving a symbol on paper without changing its node connections changes no circuit; crossing lines without a junction dot do not automatically create a connection.

Derive equivalent resistance

For series elements, the same I produces total drop I(R1 + R2), so resistances add. For parallel elements, branch currents V/R1 and V/R2 add, giving 1/Req = 1/R1 + 1/R2. A positive parallel equivalent is smaller than either branch resistance. In this packet the common 2 ohm resistor remains in series with the parallel equivalent, not in parallel with one chosen branch.

Use conservation at nodes and loops

The junction relation Icommon = I1 + I2 expresses charge conservation without sustained node accumulation. Around a closed quasistatic loop, signed potential rises and drops sum to zero, expressing energy per charge conservation. A potential-versus-ordered-node graph is not a spatial field map: the horizontal index orders elements rather than measuring physical distance.

Change the topology as a virtual counterfactual

Opening one branch removes that conducting path and increases this network’s equivalent resistance. Shorting across one resistor with an ideal bypass makes its voltage zero, but must still respect the rest of the source model. A closed path is not a guarantee of nonzero current without a driving source; an open branch can have nonzero voltage while carrying no steady current. The symbol key distinguishes source, resistor, bulb, switch, capacitor, inductor and meters; a diagonal arrow across a component denotes an adjustable parameter.

Worked original case

Investigation

Changed-case practice

Try a new practice case

With both branches 4 ohm and the common resistor 2 ohm, what is total current at 12 V in amperes?

Numerical result in the requested units

This public, unsaved check is not a secure assessment. Explain the source integral, symmetry or charge boundary in the retained course evidence workspace.

Sources and evidence boundary

Original teaching checked 31 October 2026 against the current official framework, Unit 11 and OpenStax series and parallel networks. All diagrams and numerical records are virtual, supplied or generated. No physical wiring, battery connection, short circuit, meter operation or electrical handling is requested. Public feedback and retained self-reports do not certify independent reasoning, actual laboratory participation or institutional credit.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which source, topology, initial condition or instrument assumption would invalidate this inference?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Close a source, internal-loss and load-power ledger Next: Solve coupled junctions and interpret a signed bridge current

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