NDA Physics · Electricity and Magnetism

Combination of Resistors

Resistors in series add (R = R₁ + R₂ + …); resistors in parallel combine by reciprocals (1/R = 1/R₁ + 1/R₂ + …). Every network reduces by collapsing the innermost series/parallel groups one step at a time.

All Electricity and Magnetism notes NDA Physics strategy

Why this matters

This is the chapter's marquee subtopic — 16 PYQs at 38% HARD, the bank's single biggest HARD pool. Master five shapes and you own these marks: pure series, pure parallel, mixed series-parallel reduction, the 'cut a wire then reconnect' trick, and minimum-vs-maximum resistance. Series always gives the LARGEST equivalent, parallel the SMALLEST — that one fact answers a surprising number of questions outright.

Concept 1 of 5

Resistors in series

Intuition

In series the same current flows through every resistor (one path), and the voltages across them add up. So the resistances simply add — the combination is always BIGGER than the largest single resistor.

Definition

Resistors in series carry the same current; their voltages add. The equivalent resistance is the sum: ** $R_\text{series} = R_1 + R_2 + \cdots$ ** — always larger than the biggest individual resistor.

Series equivalent

R_\text{series} = R_1 + R_2 + \cdots + R_n

Worked example

A 2 Ω, a 3 Ω, and a 5 Ω resistor are connected in series. What is the equivalent resistance?

Series resistances add directly.
$R = 2 + 3 + 5 = 10\,\Omega$ .

Answer:10 Ω.

Practice this concept3 quick reps

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
Two 6 Ω resistors in series give…
2.
In series, which quantity is the same through every resistor?
3.
Is the series equivalent bigger or smaller than the largest resistor?

Series = same current, voltages add

Don't confuse the two combinations. Series: one current path, add the resistances. The combination can never be smaller than any single resistor in it.

Concept 2 of 5

Resistors in parallel

Intuition

In parallel the same voltage sits across every resistor, and the currents through them add. Giving the charge extra paths can only make it easier to flow, so the equivalent resistance is always SMALLER than the smallest single resistor. For n EQUAL resistors R, the parallel value is just R/n.

Definition

Resistors in parallel have the same voltage across them; their currents add. The reciprocal of the equivalent equals the sum of reciprocals: ** $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots$ ** — always smaller than the smallest branch. Two resistors: $R = \dfrac{R_1 R_2}{R_1 + R_2}$ . ** $n$ equal resistors** $R$ : equivalent $= R/n$ .

Parallel equivalent

\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots \qquad (\text{two: } R = \tfrac{R_1R_2}{R_1+R_2})

Worked example

A 6 Ω and a 3 Ω resistor are connected in parallel. What is the equivalent resistance?

Two-resistor shortcut: $R = \dfrac{R_1 R_2}{R_1+R_2}$ .
$R = \dfrac{6\times 3}{6+3} = \dfrac{18}{9} = 2\,\Omega$ .
Note it's smaller than 3 Ω, the smaller branch — as parallel always is.

Answer:2 Ω.

Practice this conceptself-check · 3 quick reps

Try it yourself

Two equal resistors R in parallel are connected across a 12 V battery and draw a total current of 100 mA. Find R.

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
Three 1 Ω resistors in parallel give…
2.
Three equal resistors R in parallel give…
3.
A 4 Ω and 4 Ω in parallel give…

From the bank · past-year question

Example 2Electricity and MagnetismEASY

Three equal resistors are connected in parallel configuration in a closed electrical circuit. Then the total resistance in the circuit becomes

[Q119 · Sep · 2021]

Parallel value is SMALLER than the smallest branch

If you compute a parallel combination and get something bigger than one of the branches, you've made an error. Adding paths reduces resistance. For two equal R the answer is R/2, never 2R.

Drill 3 more on resistors in parallel

Concept 3 of 5

Reducing mixed series-parallel networks

Intuition

Most circuit questions are nested: a parallel cluster sitting inside a series chain, or vice versa. The method is always the same — find the innermost pure-series or pure-parallel group, collapse it to one resistor, redraw, and repeat until a single number is left.

Definition

Reduction algorithm: (1) spot a sub-group that is purely series OR purely parallel; (2) replace it with its equivalent; (3) redraw and repeat. A short-circuit (0 Ω) branch across two points forces those points to the same potential — current takes the zero-resistance path. For self-similar infinite ladders, set the whole network equal to $R_\infty$ and solve the resulting equation (the rest of the ladder is identical to the whole).

Series adds (always larger); parallel combines by reciprocals (always smaller than the smallest branch). Reduce a network innermost-group first.

Worked example

Three resistors 2 Ω, 4 Ω and 4 Ω: the two 4 Ω are in parallel, and that pair is in series with the 2 Ω. Find the equivalent resistance.

Innermost group: 4 Ω ∥ 4 Ω = $4/2 = 2\,\Omega$ (equal pair).
Now in series with the 2 Ω resistor: $2 + 2 = 4\,\Omega$ .

Answer:4 Ω.

Practice this conceptself-check · 3 quick reps

Try it yourself

A 2 Ω, 4 Ω and 8 Ω resistor are all in parallel, and that combination is in series with a 1 Ω resistor. Find the total resistance.

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
First step in reducing any resistor network?
2.
Current through a 0 Ω (short-circuit) branch in parallel with a resistor?
3.
(6 Ω in series with 6 Ω), all in parallel with 12 Ω → equivalent?

From the bank · past-year question

Example 3Electricity and MagnetismMODERATE

Two resistances of 5.0

\Omega

and 7.0

\Omega

are connected in series and the combination is connected in parallel with a resistance of 36.0

\Omega

. The equivalent resistance of the combination of three resistors is

[Q63 · Sep · 2024]

Collapse innermost first — don't add everything blindly

You can't add a series and a parallel resistor in one step. Identify a sub-group that is PURELY one kind, reduce it, redraw, and only then look at the next group. Mixing the two rules in a single step is the most common network error.

Drill 7 more on reducing mixed series-parallel networks

Concept 4 of 5

Cutting a wire and reconnecting it

Intuition

Cut a wire of resistance R into n equal pieces and each piece has resistance R/n (shorter = less resistance). Reconnect those n pieces in parallel and you divide again by n — so the final resistance is R/n². A ring measured across a diameter is just two equal half-rings in parallel.

Definition

Cut a wire of resistance $R$ into $n$ equal pieces ⟹ each piece is $R/n$ . Connect all $n$ pieces in parallel ⟹ equivalent $= \dfrac{R/n}{n} = \dfrac{R}{n^2}$ . A uniform ring of total resistance $R$ , measured across any diameter, behaves as two $R/2$ arcs in parallel = $R/4$ .

Cut into n, reconnect in parallel

R_\text{final} = \dfrac{R}{n^2}

Worked example

A 12 Ω wire is cut into three equal pieces, and the three pieces are connected in parallel. What is the equivalent resistance?

Each piece: $12/3 = 4\,\Omega$ .
Three 4 Ω in parallel (equal): $4/3\,\Omega$ .
Shortcut check: $R/n^2 = 12/9 = 4/3\,\Omega$ . ✓

Answer:4/3 Ω ≈ 1.33 Ω.

Practice this conceptself-check · 3 quick reps

Try it yourself

A uniform circular ring has total resistance 20 Ω. What is the resistance between the two ends of any diameter?

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
A 50 Ω wire cut into 5 equal pieces, reconnected in parallel → ?
2.
A 20 Ω wire cut into 2, reconnected in parallel → ?
3.
Cut R into n pieces and parallel them: final resistance?

From the bank · past-year question

Example 4Electricity and MagnetismMODERATE

An electric wire of resistance 50 ohm is cut into five equal wires. These wires are then connected in parallel. What is the equivalent resistance of this combination?

[Q55 · Apr · 2022]

Cut + parallel = R/n², not R/n

Two effects stack: cutting into n pieces makes each R/n, AND paralleling n of them divides by another n. The combined result is R/n². Stopping at R/n is the dominant wrong answer.

Drill 2 more on cutting a wire and reconnecting it

Concept 5 of 5

Minimum and maximum resistance

Intuition

Given a fixed set of resistors, you get the LARGEST possible resistance by wiring them all in series, and the SMALLEST by wiring them all in parallel. So 'which arrangement gives minimum resistance?' is really 'which is the most-parallel of the cheapest resistors?'

Definition

For any fixed collection of resistors: all in series ⟹ maximum equivalent (the sum); all in parallel ⟹ minimum equivalent (below the smallest branch). To minimise resistance, parallel the smallest-valued resistors; to maximise, put the largest in series.

Worked example

You have two 6 Ω resistors. What are the maximum and minimum resistances you can make with both of them?

Maximum — series: $6 + 6 = 12\,\Omega$ .
Minimum — parallel: $6/2 = 3\,\Omega$ .

Answer:Maximum 12 Ω (series), minimum 3 Ω (parallel).

Practice this conceptself-check · 3 quick reps

Try it yourself

Which gives the smallest resistance between two points: (a) three 3 Ω in parallel, (b) two 3 Ω in parallel, (c) two 1 Ω in series, (d) three 1 Ω in series?

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
All-series or all-parallel for maximum resistance?
2.
All-series or all-parallel for minimum resistance?
3.
Largest resistance from a 2 Ω and 3 Ω?

From the bank · past-year question

Example 5Electricity and MagnetismMODERATE

Which of the following arrangement of resistors offers minimum effective resistance between points X and Y?

[Q109 · Sep · 2023]

Minimum ≠ fewest resistors

Minimum resistance means MOST parallel paths of the SMALLEST resistors — not the smallest count of components. Three 3 Ω in parallel (1 Ω) beats two 1 Ω in series (2 Ω): more parallelism wins even with larger individual values.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Resistors in series
Series equivalent
$R_\text{series} = R_1 + R_2 + \cdots + R_n$
Resistors in parallel
Parallel equivalent
$\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots \qquad (\text{two: } R = \tfrac{R_1R_2}{R_1+R_2})$
Cutting a wire and reconnecting it
Cut into n, reconnect in parallel
$R_\text{final} = \dfrac{R}{n^2}$

Watch out for (5)

Series = same current, voltages add
→ Resistors in series
Parallel value is SMALLER than the smallest branch
→ Resistors in parallel
Collapse innermost first — don't add everything blindly
→ Reducing mixed series-parallel networks
Cut + parallel = R/n², not R/n
→ Cutting a wire and reconnecting it
Minimum ≠ fewest resistors
→ Minimum and maximum resistance

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Electricity and MagnetismMODERATE

Two equal resistors R are connected in parallel, and a battery of 12 V is connected across this combination. A dc current of 100 mA flows through the circuit as shown below. The value of R is

[Q122 · Apr · 2020]

Example 2Electricity and MagnetismHARD

Consider the following circuit: [Five resistors each of resistance R — one in series, then parallel combination of two pairs]. Which one of the following is the value of the resistance between points A and B in the circuit given above?

[Q88 · Sep · 2018]

Example 3Electricity and MagnetismMODERATE

A circular coil of single turn has a resistance of 20

\Omega

. Which one of the following is the correct value for the resistance between the ends of any diameter of the coil ?

[Q86 · Apr · 2017]

Example 4Electricity and MagnetismEASY

If three resistors of 1 Ohm each connect in parallel to each other the resultant resistance is

[Q148 · Apr · 2025]

Example 5Electricity and MagnetismHARD

An infinite combination of resistors, each having resistance

R = 4\,\Omega

, is given. What is the net resistance between the points A and B ? (Each resistance is of equal value,

R = 4

)

[Q51 · Apr · 2024]

Drill every past-year question on this subtopic

16 questions from the bank — paginated, with cart and Word-export support.

Drill the 16 questions