4.3ElectricityCore

Kirchhoff’s Laws and Circuit Rules

However complicated a circuit diagram looks, it only ever obeys two underlying rules: charge cannot pile up anywhere, and energy must balance around any complete loop. Everything else — every series and parallel resistor formula you’ll use — follows directly from those two ideas.

40 min Video by ZPhysics Module 4.3 Electrical circuits

A Level Physics: Kirchoff’s second law and series and parallel circuitsWatch the full walkthrough before the notes below.

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What you'll be able to do

State Kirchhoff’s first law and link it to conservation of charge
State Kirchhoff’s second law and link it to conservation of energy
Derive and use the combined resistance formula for resistors in series
Derive and use the combined resistance formula for resistors in parallel

Kirchhoff’s first law

Kirchhoff’s first law states that the total current flowing into a junction equals the total current flowing out of it — a direct statement of conservation of charge, since charge cannot accumulate anywhere in a circuit.

1Total current in

= 2.5 + 1.5 = 4.0

2Current out must equal current in:

1.0 + I = 4.0

Answer

I = 3.0

Kirchhoff’s second law

Kirchhoff’s second law states that, around any complete closed loop, the sum of the e.m.f.s equals the sum of the potential difference drops — a statement of conservation of energy, since every joule a source supplies must be accounted for by the time charge returns to its starting point.

Series and parallel resistors

In series, the same current flows through every component (there is only one path), while the source’s e.m.f. is shared out as individual p.d. drops — giving the series resistance rule. In parallel, every branch shares the same p.d. (they connect the same two points), while the total current splits between branches according to their resistance — giving the parallel resistance rule.

R_{series} = R_{1} + R_{2} + \dots \frac{1}{R _{parallel}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots

Combined resistance: series (add directly) and parallel (add reciprocals).

1Total resistance:

R = 5.0 + 8.0 + 12.0 = 25.0 Ω

2Current:

I = \frac{V}{R} = \frac{20}{25.0}

Answer

R = 25.0 Ω

I = 0.80

Tip — Series resistance is always bigger than the largest individual resistor; parallel resistance is always smaller than the smallest individual resistor — a quick sanity check on any answer.

Equation recap

R_{series} = R_{1} + R_{2} + \dots

Series combined resistance.

\frac{1}{R _{parallel}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots

Parallel combined resistance.

Common mistakes to avoid

Adding parallel resistances directly, as if they were in series.

Parallel resistances combine via reciprocals — find 1/R_total first, then take the reciprocal of that sum.

Assuming current is the same everywhere in any circuit.

Current is the same only along an unbranched (series) path — in parallel branches, current divides while p.d. stays the same across each branch.

Applying Kirchhoff’s second law to only part of a loop.

The law must be applied around a COMPLETE closed loop, including every e.m.f. and every p.d. drop encountered along the way.

Key takeaways

Kirchhoff’s first law (conservation of charge): current into a junction = current out.
Kirchhoff’s second law (conservation of energy): total e.m.f. around a loop = total p.d. drop.
Series: same current, resistances add. Parallel: same p.d., reciprocal resistances add (total always smaller than the smallest branch).

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Kirchhoff’s Laws and Circuit Rules

40 min Video by ZPhysics Module 4.3 Electrical circuits