Kirchhoff's rules

Why Kirchhoff's rules are needed

Many circuits cannot be simplified by simple series and parallel reductions. Bridges, multi-loop networks, and circuits with multiple batteries require a more general method. Kirchhoff's rules provide that method.

The rules are based on fundamental conservation laws: conservation of charge and conservation of energy.

Junction rule

Kirchhoff's junction rule states that the total current entering a junction equals the total current leaving it:

$sum I_{in}=sum I_{out}$

Equivalently,

$sum I=0$

if currents entering and leaving are assigned signs consistently.

This follows from charge conservation. Charge cannot pile up indefinitely at an ideal junction in a steady DC circuit. If more charge entered than left, the junction's charge would grow, changing electric fields until balance was restored.

Loop rule

Kirchhoff's loop rule states that the sum of potential changes around any closed loop is zero:

$sum Delta V=0$

This follows from energy conservation. A charge returning to its starting point in a circuit must return to the same electric potential, so the total potential rise and drop around a loop must cancel.

Sign conventions for resistors

When moving through a resistor in the direction of current, potential decreases:

$Delta V=-IR$

When moving through a resistor opposite the current, potential increases:

$Delta V=+IR$

This convention reflects that conventional current flows from higher potential to lower potential through a resistor.

Sign conventions for batteries

For an ideal battery with emf $mathcal{E}$, moving from the negative terminal to the positive terminal gives a potential rise:

$Delta V=+mathcal{E}$

Moving from positive to negative gives a potential drop:

$Delta V=-mathcal{E}$

Solving circuits

A systematic method is to assign current directions in each branch. If you guess wrong, the resulting current will be negative, indicating the true direction is opposite your assumption.

Next, write junction equations and loop equations until you have enough independent equations for the unknown currents. Then solve algebraically.

It is important to use independent equations. Writing more loops than necessary can produce redundant equations.

Example structure

For a loop with one battery and two resistors in series, the loop rule gives

$mathcal{E}-IR_1-IR_2=0$

so

I=rac{mathcal{E}}{R_1+R_2}

This agrees with series resistance. Kirchhoff's rules generalize familiar results.

Internal resistance

Real batteries have internal resistance. A simple model treats a real battery as ideal emf $mathcal{E}$ in series with internal resistance $r$. If current $I$ leaves the positive terminal, the terminal voltage is

$V_{terminal}=mathcal{E}-Ir$

The battery voltage under load is less than its emf.

Meters in circuits

An ideal ammeter has zero resistance and is placed in series to measure current. An ideal voltmeter has infinite resistance and is placed in parallel to measure potential difference. Real meters approximate these ideals but can disturb circuits slightly.

The big idea

Kirchhoff's junction rule expresses charge conservation, and Kirchhoff's loop rule expresses energy conservation. Together they provide a general method for analyzing DC circuits that cannot be reduced by simple series-parallel combinations. Careful sign conventions and systematic equation writing are the key skills.