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Concept module

Equivalent Resistance

Reduce one highlighted resistor group into an equivalent block, then collapse the whole mixed circuit honestly and watch how the total current and grouped behavior change together.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 12.0 sLiveReduction graphs stay resistance-based while the time rail inspects live charge flow through the grouped circuit honestly.

0.00 s12.0 s

Equivalent Resistance

Keep one outer resistor in series with a highlighted two-resistor group, then reduce that group honestly before collapsing the whole circuit to one equivalent load.

Graphs

Switch graph views without breaking the live stage and time link.

Reduction sweep vs R3

Sweep only R3 while keeping the battery, R1, R2, and the group mode fixed. The grouped-pair curve is the first reduction step. The total-equivalent curve is always that grouped curve shifted upward by the fixed outer resistor because R1 is still added in series afterward.

R3 (ohm): 2 to 18equivalent resistance (ohm): 0 to 32

Grouped pair equivalentTotal equivalent

Hover or scrub to link the graph back to the stage.R3 (ohm) / equivalent resistance (ohm)

Controls

Adjust the physical parameters and watch the motion respond.

Battery voltage

V

12 V

Changes the source push for the whole reduced circuit.

Outer resistor R1

R_{1}

4 ohm

This resistor stays in series with the whole grouped block.

Grouped resistor R2

R_{2}

6 ohm

Changes the top or first resistor inside the highlighted group.

Grouped resistor R3

R_{3}

6 ohm

Changes the bottom or second resistor in the highlighted group and drives the response-graph sweep.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Hide

Use a parallel grouped pairOff keeps the highlighted pair in series. On makes the highlighted pair parallel before R1 is added.

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

Graph readingPrompt 1 of 2

The total-equivalent curve stays a fixed amount above the grouped-pair curve because R1 is added only after the highlighted pair has already been reduced.

Try this

Hold R1 at 4 ohm and compare the vertical gap between the two curves as you preview different R3 values.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

V

Battery voltage

12 V

Raises or lowers the current through the whole reduced circuit without changing the reduction order itself.

Graph: Current sweep vs R3Graph: Voltage share vs R3Overlay: Current flowOverlay: Voltage labelsOverlay: Reduction guide

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Stay with one grouped circuit and look for the reduction step that the live stage, the simplification card, and the response graphs all agree on.

Graph readingPrompt 1 of 2

Graph: Reduction sweep vs R3

The total-equivalent curve stays a fixed amount above the grouped-pair curve because R1 is added only after the highlighted pair has already been reduced.

Try this

Hold R1 at 4 ohm and compare the vertical gap between the two curves as you preview different R3 values.

Why it matters

The graph makes the two-step reduction visible instead of purely symbolic.

Control: Outer resistor R1Control: Grouped resistor R3Graph: Reduction sweep vs R3Overlay: Reduction guideEquation

R_{1}

Equation

R_{3}

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Current flow

Shows live current arrows and charge counters through the grouped circuit.

What to notice

R1 always carries the total current, but the grouped pair only splits current when the highlighted pair is parallel.

Why it matters

The reduction step should still match the live charge flow you can see on the stage.

Control: Battery voltageControl: Outer resistor R1Control: Grouped resistor R2Control: Grouped resistor R3Control: Use a parallel grouped pairGraph: Current sweep vs R3Equation

V

Equation

R_{1}

Equation

R_{2}

Equation

R_{3}

Challenge mode

Use the same grouped circuit for compact reduction targets. The checklist reads the live reduction card, equivalent sweep, and current response from one honest circuit.

0/1 solved

TargetCore

5 of 8 checks

Collapse the parallel group

Starting from the balanced series group, switch the highlighted pair into the parallel case until the reduction card reads about 3 ohm for the grouped pair and about 7 ohm for the full circuit.

Graph-linkedGuided start2 hints

Suggested start

Use the reduction guide and the equivalent sweep together while you change only the grouped-pair mode.

Matched

Open the Reduction sweep vs R3 graph.

Reduction sweep vs R3

Matched

Keep the Reduction guide visible.

Matched

Keep resistance1 between 3.8 ohm and 4.2 ohm.

4 ohm

Matched

Keep resistance2 between 5.8 ohm and 6.2 ohm.

6 ohm

Matched

Keep resistance3 between 5.8 ohm and 6.2 ohm.

6 ohm

Pending

Keep group equivalent resistance between 2.9 ohm and 3.1 ohm.

12 ohm

Pending

Keep equivalent resistance between 6.9 ohm and 7.1 ohm.

16 ohm

Pending

Keep total current between 1.67 A and 1.76 A.

0.75 A

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, a 12 V source drives R1 = 4 ohm in series with a grouped pair of R2 = 6 ohm and R3 = 6 ohm. The grouped pair reduces to 12 ohm, so the total equivalent resistance is 16 ohm and the total current is 0.75 A. The highlighted group is in series, so the same current crosses both grouped resistors before you add that reduced block to R1. The grouped block has 9 V across it. R2 carries 0.75 A and R3 carries 0.75 A. The grouped resistors are dissipating nearly the same power right now.

Equation detailsDeeper interpretation, notes, and worked variable context.

Series grouped pair

If the highlighted pair sits in one current path, reduce it by direct addition first.

R_{group} = R_{2} + R_{3}

R_{2}

Grouped resistor 2 6 ohm

R_{3}

Grouped resistor 3 6 ohm

Parallel grouped pair

If the highlighted pair spans the same two group nodes, reduce it with the parallel rule first.

R_{group} = \frac{R _{2} R _{3}}{R _{2} + R _{3}}

R_{2}

Grouped resistor 2 6 ohm

R_{3}

Grouped resistor 3 6 ohm

Total equivalent resistance

After the highlighted group is reduced, the outer resistor still adds in series with the whole block.

R_{eq} = R_{1} + R_{group}

R_{1}

Outer resistor 4 ohm

R_{2}

Grouped resistor 2 6 ohm

R_{3}

Grouped resistor 3 6 ohm

Whole-circuit Ohm's law

The battery current is set by the battery voltage and the final equivalent resistance.

I_{total} = \frac{V}{R _{eq}}

V

Battery voltage 12 V

R_{1}

Outer resistor 4 ohm

Grouped-block voltage

The reduced block still takes a definite part of the source voltage in the full circuit.

V_{group} = I_{total} R_{group}

V

Battery voltage 12 V

R_{1}

Outer resistor 4 ohm

R3 current in a parallel group

When the highlighted pair is parallel, the R3 branch current follows the shared group voltage.

I_{3} = \frac{V _{group}}{R _{3}}

V

Battery voltage 12 V

R_{3}

Grouped resistor 3 6 ohm

Same current in a series group

When the highlighted pair is series, both grouped resistors carry the same current as the full circuit.

I_{2} = I_{3} = I_{total}

R_{2}

Grouped resistor 2 6 ohm

R_{3}

Grouped resistor 3 6 ohm

Charge passed

At one fixed current, the amount of charge through a resistor grows linearly with time.

Q = I t

R_{3}

Grouped resistor 3 6 ohm

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 1 compact task ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 6 of 60 / 6 complete

Electricity

Earlier steps still set up Equivalent Resistance.

1. Electric Fields2. Electric Potential3. Basic Circuits4. Power and Energy in Circuits+2 more steps

Previous step: Series and Parallel Circuits.

Start from track beginning

Short explanation

What the system is doing

Equivalent resistance lets you replace a bounded group of resistors with one simpler block that draws the same total current from the same source. The reduction is only honest when the grouped resistors really share one series path or the same two nodes.

This module stays intentionally small. One outer resistor always sits in series with a highlighted two-resistor group, and that group can switch between series and parallel. That is enough to teach reduction order, grouped voltage and current behavior, and the effect on the total circuit without turning the page into a symbolic circuit solver.

Key ideas

01Reduce the highlighted group first, then combine that one-number result with the outer resistor that stays in series with the whole block.

02When the grouped pair is in series, the same current crosses both grouped resistors before you add their resistances. When the grouped pair is in parallel, both grouped branches share the same group voltage before you combine them.

03A smaller total equivalent resistance lets the same battery drive a larger total current, so the reduction step is not bookkeeping only. It changes the live circuit behavior everywhere.

Live reduction checks

Solve the exact state on screen.

Use the exact circuit on screen. The same resistor values, group mode, inspected time, overlays, and response graphs drive the reduction steps below.

Live valuesFollowing current parameters

For the current circuit, what equivalent resistance does the highlighted pair reduce to, what total equivalent resistance does the source feel, and what total current follows?

V

Battery voltage

12 V

R_{1}

Outer resistor

4 ohm

R_{2}

Grouped resistor 2

6 ohm

R_{3}

Grouped resistor 3

6 ohm

1. Start from the live grouped pair

The highlighted pair is a series grouped pair, so

R_{group} = R_{2} + R_{3}

2. Reduce the grouped pair numerically

With

R_{2} = 6

and

R_{3} = 6

R_{group} = 6 + 6

, so

R_{group} = 12

3. Add the outer series resistor

Because

R_{1}

stays in series with the whole block,

R_{eq} = 4 + 12

, so

R_{eq} = 16

4. Use the total equivalent in Ohm's law

I_{total} = \frac{V}{R _{eq}} = \frac{1}{2} 16 = 0.75

Grouped and total equivalent resistance

R_{group} = 12, R_{eq} = 16, I_{total} = 0.75

The grouped pair reduces by direct addition first, so the total equivalent stays larger before the source current is found.

Reduction-order checkpoint

The highlighted pair starts as two 6 ohm resistors in series with R1 = 4 ohm and the same 12 V battery. If you switch only the highlighted pair to parallel, why does the total current jump so much?

Prediction prompt

Predict whether the big change comes from R1 changing, from the grouped pair changing, or from both before you switch the group mode.

Check your reasoning

The grouped pair changes from 12 ohm in series to 3 ohm in parallel, while R1 stays the same 4 ohm resistor. That drops the total equivalent from 16 ohm to 7 ohm, so the same battery drives a much larger total current.

Equivalent resistance is causal here, not decorative. The live source current follows the one total load that remains after the grouped pair is reduced honestly.

Common misconception

Equivalent resistance is just a shortcut number, so it does not really tell you anything about the current or voltage in the original circuit.

The equivalent resistance is defined by matching the total current drawn from the same source. If the source sees the same voltage and the same total current, the simplified circuit is telling you something physically real about the original one.

What you cannot do is ignore the reduction order. You must first identify a group that truly behaves like one series block or one parallel block before replacing it.

Quick test

Reasoning

Question 1 of 5

Answer from the live grouped circuit, not from detached formulas. Each question asks what the stage, reduction card, or linked graphs must mean.

Which part of the circuit should you reduce first in this module?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows one battery on the left, one outer resistor labeled R1 on the top path, and a highlighted two-resistor group to the right. The highlighted group can appear either as two series resistors on the same path or as two parallel branches between the same two group nodes.

Moving charge markers follow the actual current paths, and optional overlays add current arrows, voltage labels, node markers, charge counters, and a reduction guide that highlights the pair that should be simplified first. A reduction card on the right shows the live grouped equivalent and the final total equivalent before the readout card lists the current grouped state.

Compare mode adds a dashed ghost circuit for the second setup so only the changed grouped relationship needs to be noticed. The page stays intentionally bounded to one outer resistor and one two-resistor group.

Graph summary

All three graphs sweep only R3 while keeping the battery, R1, R2, and the grouped-pair mode fixed. The reduction sweep shows the grouped equivalent together with the final total equivalent, the current graph shows total current alongside the grouped resistor currents, and the voltage-share graph shows how the outer drop, grouped-block drop, and live R3 drop behave as R3 changes.

The time rail still inspects the same stage honestly while those graphs stay parameter-based. Pausing or scrubbing lets the learner compare the grouped charge counters at one chosen time without changing the graph sweep.

Keep the electricity path moving

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Power consequencesElectricity