Physics · Circuits

Equivalent Resistance

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What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Power and Energy in CircuitsUse R_eq to judge current, power, and energy transfer.

Key takeaway

Equivalent resistance is the one resistor that makes the source voltage and total current match the original network.
A valid first reduction comes from the wiring: one current path for series, or the same two nodes for parallel.
After the highlighted pair becomes R_group, the outer resistor R1 adds in series to form the total R_eq.
Lower total equivalent resistance makes the same battery drive a larger total current.
Parallel branches share group voltage but can carry different currents, and their equivalent resistance is less than either branch alone.

Common misconception

Do not combine nearby-looking resistors just because they are drawn close together. Reduce only a true series path or true parallel node pair, and check that a parallel equivalent is smaller than each branch resistance.

Equivalent resistance is defined by matching what the source experiences: the same voltage across the circuit and the same total current through it.

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Equation snapshotSeries grouped pair · Parallel grouped pairShow 3 equations

Choose the group rule from the wiring: series means one shared current path, parallel means the same two nodes. Only after the group becomes R_group should R1 be added to get R_eq.

Series grouped pair
$R_{group} = R_{2} + R_{3}$
Use direct addition only when the highlighted pair lies on one unbranched path and carries the same current.
Parallel grouped pair
$R_{group} = \frac{R _{2} R _{3}}{R _{2} + R _{3}}$
Use the parallel rule only when the highlighted pair connects across the same two group nodes and shares one group voltage.
Total equivalent resistance
$R_{eq} = R_{1} + R_{group}$
After you reduce the highlighted group to one resistor, R1 still adds in series with that reduced block.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current circuit, what does the highlighted pair reduce to, what total equivalent resistance does the battery see, 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. Identify whether the highlighted pair is series or parallel

The highlighted pair is a series grouped pair, so

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

2. Reduce the highlighted pair first

With

R_{2} = 6

and

R_{3} = 6

R_{group} = 6 + 6

, so

R_{group} = 12

3. Combine the reduced group with the outer resistor R1

Because

R_{1}

stays in series with the whole block,

R_{eq} = 4 + 12

, so

R_{eq} = 16

4. Use the final equivalent resistance to find the total current

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.

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Step 6 of 6

Electricity

Equivalent Resistance appears later in this track, so it is cleaner to start from the beginning first.

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