Kirchhoff's Loop Rule

Imagine carrying one coulomb of charge around a complete circuit and returning to the starting point. Its final electric potential must equal its starting potential, so all voltage rises and drops must balance.

A cell normally provides a voltage rise, while a resistor produces a voltage drop in the direction of current.

Kirchhoff's loop rule states that the algebraic sum of potential changes around every closed loop is zero. It extends energy conservation to circuits containing several sources and resistors.

Why a closed-loop sum is zero

Potential difference is energy transferred per unit charge. After a charge travels around a closed path and returns to its starting point, its net change in electric potential energy must be zero.

Equivalently, the energy per coulomb supplied by sources equals the energy per coulomb transferred in loads and internal resistances.

Choosing the loop direction

You may travel clockwise or anticlockwise. The physical answer does not depend on this choice because reversing the traversal reverses every term in the equation and leaves the same solution.

Potential change across a resistor

Conventional current passes through a resistor from higher to lower potential. Moving through the resistor with the current therefore gives a drop; moving against current gives a rise.

Potential change across a source

Moving through an ideal source from its negative terminal to its positive terminal raises potential by ε. Moving from positive to negative lowers potential by ε.

Single-source loop

A 12 V ideal source drives current through 2 Ω and 4 Ω in series. Traverse from the negative to positive source terminal and then through both resistors with the current.

The resistor drops are 4 V and 8 V. Together they equal the 12 V rise supplied by the source.

Opposing sources

Suppose 12 V and 6 V sources oppose each other in one loop with total resistance 3 Ω. Choose the direction favored by the larger source.

The smaller source is crossed from positive to negative in this traversal, so its emf appears with a minus sign.

Including internal resistance

Internal resistance behaves like an ordinary series resistor for sign purposes. A cell of emf ε and internal resistance r connected to external resistance R obeys:

Several loops and unknown currents

In a network, first assign a current to each branch. Use junction equations to relate branch currents, then write loop equations for enough independent closed paths to solve the remaining unknowns.

A component shared by two loops carries the algebraic combination of the chosen loop currents. Its voltage term must use the actual branch-current expression.

Equation-solving checklist

• Draw and label every emf, resistance and current arrow.
• Choose a loop traversal direction.
• Walk around the loop once and write each voltage change in order.
• Use −IR with current and +IR against current.
• Use +ε from negative to positive and −ε from positive to negative.
• Set the algebraic sum equal to zero.
• Combine with junction equations when the circuit has branches.
• Check that the final voltage rises equal the final voltage drops.

Common conceptual checks

A closed-loop equation must have voltage units in every term. If all resistor terms have the same sign in a one-current loop, the emf terms must supply the balancing rise. Reversing the complete loop direction should produce the same equation multiplied by −1.

Exam-ready response

• State the rule and its energy-conservation basis.
• Show the four traversal sign cases.
• Write the loop equation before substituting or simplifying.
• Preserve units in the final answer.
• Interpret negative current direction rather than discarding the result.