Series and parallel resistance

What is "equivalent resistance"?

We derive everything from just two laws:

Ohm's law: $V = IR$ (WHAT: voltage across a resistor = current × its resistance).
Kirchhoff's laws:
- KVL (loop): voltages around a loop sum to zero → in a single line, applied $V$ splits among resistors.
- KCL (node): current in = current out → at a junction, currents split/recombine.

Series — derivation from scratch

Setup: $R_1, R_2, \dots, R_n$ in a single line, total voltage $V$ across the chain, current $I$ .

HOW (derive): The voltage across each is (Ohm): $V_k = I R_k$ . By KVL the applied voltage equals the sum of the drops: $V = V_1 + V_2 + \dots + V_n = I R_1 + I R_2 + \dots + I R_n = I\sum_k R_k$ Since $R_{eq} = V/I$ : $\boxed{R_{series} = R_1 + R_2 + \dots + R_n}$

Parallel — derivation from scratch

Setup: $R_1,\dots,R_n$ all connected between node A and node B. Common voltage $V$ , total current $I$ entering A.

HOW (derive): Current through each (Ohm): $I_k = V/R_k$ . By KCL the total current splits among the branches: $I = I_1 + I_2 + \dots + I_n = \frac{V}{R_1} + \frac{V}{R_2} + \dots + \frac{V}{R_n} = V\sum_k \frac{1}{R_k}$ Since $\dfrac{1}{R_{eq}} = \dfrac{I}{V}$ : $\boxed{\frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}}$

Worked examples

Common mistakes (Steel-man + fix)

Recall Feynman: explain to a 12-year-old

Imagine water pipes. Series = one long pipe with several narrow chokes in a row; the water must squeeze through every choke, so the chokes' "difficulty" piles up — total is harder. Parallel = the same water reaches a place where it can split into several pipes side-by-side; more pipes means water flows easier overall, so total difficulty drops below even the easiest single pipe. Same water-per-second through a series chain everywhere; same water-pressure across each side-by-side pipe.

Flashcards

In series, which quantity is identical for every resistor?

The current

I

(single path, KCL).

In parallel, which quantity is identical for every resistor?

The voltage

V

(same two nodes).

Series equivalent resistance formula?

R_{eq}=R_1+R_2+\dots+R_n

Parallel equivalent resistance formula?

\frac{1}{R_{eq}}=\frac{1}{R_1}+\dots+\frac{1}{R_n}

Two-resistor parallel shortcut?

R_{eq}=\dfrac{R_1R_2}{R_1+R_2}

(product over sum).

Is series

R_{eq}

bigger or smaller than each resistor?

Bigger than any single one.

Is parallel

R_{eq}

bigger or smaller than each resistor?

Smaller than the smallest one.

n

equal resistors

R

in parallel give?

R/n

In parallel, which resistor carries more current?

The smaller resistance (

I_k=V/R_k

Which two laws derive these formulas?

Ohm's law + Kirchhoff's (KVL for series, KCL for parallel).

How do voltages relate in a series chain?

They add up to the applied voltage (KVL).

How do currents relate at a parallel junction?

They add up to the total current (KCL).

Connections

Ohm's Law — the $V=IR$ used at every step.
Kirchhoff's Laws — KVL (series) and KCL (parallel) are the real source of these rules.
Resistivity and Resistance — where a single $R$ comes from ( $R=\rho L/A$ ).
EMF and Internal Resistance — internal resistance sits in series with the battery.
Wheatstone Bridge — networks that are neither purely series nor parallel.
Power in Circuits — $P=I^2R=V^2/R$ distributed across the combination.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, resistors basically charge ke flow me rukawat hai. Do hi cases samajhne hai. Series matlab ek hi single raasta — current ko har resistor se ek ke baad ek guzarna padta hai, isliye sabhi me current same rehta hai. Kyunki rukawat add hoti jaati hai, total $R_{eq}=R_1+R_2+\dots$ , aur ye hamesha sabse bade resistor se bhi bada hota hai. Voltage drops add hokar battery ka voltage banate hai (yeh KVL hai).

Parallel me dono resistor ke dono sire same do nodes pe jude hote hai, isliye har resistor pe same voltage lagta hai. Yaha currents add hote hai: $I=I_1+I_2$ , aur formula banta hai $\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}$ . Sabse bada confusion yahi hota hai — last me reciprocal flip karna mat bhulna. Aur ek baat: parallel me $R_{eq}$ hamesha sabse chhote resistor se bhi chhota aata hai, kyunki zyada raaste matlab current ko nikalna asaan.

Trick yaad rakho: Series = Same current, Sum karo. Parallel = Potential same, 1/R add karo. Smaller resistance zyada current kheechta hai (kyunki same voltage par $I=V/R$ ). Mixed circuit aaye to ghabrao mat — pehle parallel block ko ek single resistance me collapse karo, phir series me add karte jao. Har step pe check: series me voltages jodke total banna chahiye, parallel me branch currents jodke total banna chahiye.

Yeh topic important isliye hai kyunki koi bhi bada circuit, chahe kitna bhi complex ho, in do rules se step-by-step ek single $R_{eq}$ me simplify ho jaata hai — phir battery se total current Ohm's law se ek line me nikal jaata hai.

What is "equivalent resistance"?

Series — derivation from scratch

Parallel — derivation from scratch

Worked examples

Common mistakes (Steel-man + fix)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections