3.5.1.4ElectricityCore

Kirchhoff’s Laws and Circuit Rules

Two simple conservation principles are all you need to analyse any circuit, however complicated: charge cannot pile up anywhere, and energy must balance around any complete loop. These are $Kirchhoff’s laws$ , and from them the familiar series and parallel resistor rules follow directly.

40 min Video by Science Shorts 3.5.2 DC circuits

Kirchhoff’s Laws — A-level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

State Kirchhoff’s first law and explain it in terms of conservation of charge
State Kirchhoff’s second law and explain it in terms of conservation of energy
Derive and use the total resistance formula for resistors in series
Derive and use the total resistance formula for resistors in parallel
Analyse a circuit combining series and parallel sections

Kirchhoff’s first law

Kirchhoff’s first law states that the total current flowing $into$ any junction equals the total current flowing $out$ of it. This is simply a statement of $conservation of charge$ — charge cannot be created or destroyed, or pile up, at a junction, so whatever current arrives must leave by some route.

1Total current in

= 3.0 + 2.0 = 5.0

2By Kirchhoff’s first law, current out must equal current in.

Answer

I = 5.0

Tip — Kirchhoff’s first law is exactly why the current is the same at every point along a single (unbranched) loop — there’s nowhere else for it to go.

Kirchhoff’s second law

Kirchhoff’s second law states that, around any complete closed loop in a circuit, the sum of the e.m.f.s equals the sum of the potential difference drops. This is $conservation of energy$ : every joule of energy a source gives to charge must be transferred somewhere else (across resistors, lamps, etc.) by the time that charge returns to its starting point.

1By Kirchhoff’s second law, e.m.f. = sum of p.d. drops around the loop.

12 = 5.0 + V_{2}

Answer

V_{2} = 7.0

Resistors in series

In a series circuit, the same current flows through every component (there is only one path, so Kirchhoff’s first law leaves no alternative), while the source’s total e.m.f. is shared out among the components as individual p.d. drops (Kirchhoff’s second law). Adding the p.d.s $V = I R_{1} + I R_{2} + \dots$ and factoring out the common current $I$ gives the series resistance rule.

R_{total} = R_{1} + R_{2} + \dots + R_{n}

Total resistance of resistors in series.

1Total resistance:

R = 4.0 + 6.0 + 10.0 = 20.0 Ω

2Current:

I = \frac{V}{R} = \frac{12}{20.0}

Answer

R = 20.0 Ω

I = 0.60

Tip — In series, resistance always ADDS — the total is bigger than any individual resistor, since the current has to fight through every resistor one after another.

Resistors in parallel

In a parallel circuit, every branch has the same p.d. across it (they all connect the same two points, or "nodes"), while the total current entering the combination splits between the branches (Kirchhoff’s first law) in proportion to how easily each branch conducts. Writing the total current as the sum of each branch’s current, $I = \frac{V}{R _{1}} + \frac{V}{R _{2}} + \dots$ , and factoring out the common p.d. $V$ gives the parallel resistance rule.

\frac{1}{R _{total}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots + \frac{1}{R _{n}}

Total resistance of resistors in parallel.

\frac{1}{R} = \frac{1}{6.0} + \frac{1}{12.0} = \frac{2}{12.0} + \frac{1}{12.0} = \frac{3}{12.0}

R = \frac{12.0}{3} = 4.0 Ω

Answer

R = 4.0 Ω

— always smaller than the smallest individual resistor.

Tip — In parallel, total resistance is always SMALLER than the smallest individual resistor — adding another path always gives current an easier overall route.

Equation recap

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

Series resistors: resistances add.

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

Parallel resistors: reciprocals add.

Common mistakes to avoid

Adding parallel resistances directly, like series resistances.

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

Assuming current is the same everywhere in any circuit, including parallel branches.

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

Forgetting that Kirchhoff’s second law must include every e.m.f. and every p.d. drop around a specific closed loop, not the whole circuit at once.

Apply Kirchhoff’s second law one complete loop at a time — a circuit with multiple loops needs one equation per independent loop.

Key takeaways

Kirchhoff’s first law (conservation of charge): total current into a junction = total current out.
Kirchhoff’s second law (conservation of energy): around any closed loop, total e.m.f. = total p.d. drop.
Series: same current everywhere, resistances add (R = R₁+R₂+…).
Parallel: same p.d. across every branch, reciprocal resistances add (1/R = 1/R₁+1/R₂+…), and total resistance is always less than the smallest branch.

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Kirchhoff’s Laws and Circuit Rules

Two simple conservation principles are all you need to analyse any circuit, however complicated: charge cannot pile up anywhere, and energy must balance around any complete loop. These are

Kirchhoff’s laws

, and from them the familiar series and parallel resistor rules follow directly.

40 min Video by Science Shorts 3.5.2 DC circuits