Kirchhoff's circuit laws are tools for analyzing circuits that cannot be reduced to a single series or parallel path. They matter because real circuits often have multiple loops, junctions, batteries, and resistors connected in complex ways. The laws let you write equations for unknown currents and voltages using conservation principles.
With a clear sign convention, the same method works for simple lab circuits and more advanced electrical networks.
Kirchhoff's current law says charge does not pile up at a junction, so the total current entering a node equals the total current leaving it. Kirchhoff's voltage law says energy is conserved around any closed loop, so the sum of voltage rises and drops is zero. To solve a multi-loop circuit, assign current directions, choose loop directions, write KCL and KVL equations, then solve the simultaneous equations.
If a current comes out negative, the actual current flows opposite to the direction you assumed.
Key Facts
- Kirchhoff's current law: sum I_in = sum I_out at any junction.
- Kirchhoff's voltage law: sum ΔV = 0 around any closed loop.
- Ohm's law connects each resistor's voltage and current: V = IR.
- Across a resistor in the direction of current, ΔV = -IR because electric potential drops.
- Across a battery from negative to positive terminal, ΔV = +ε because electric potential rises.
- A circuit with n junctions has only n - 1 independent current law equations.
Vocabulary
- Junction
- A junction is a point in a circuit where three or more conducting branches meet.
- Loop
- A loop is any closed path in a circuit that starts and ends at the same point.
- Branch current
- A branch current is the current flowing through one specific path between two junctions.
- Voltage rise
- A voltage rise is an increase in electric potential, such as moving through a battery from its negative terminal to its positive terminal.
- Voltage drop
- A voltage drop is a decrease in electric potential, such as moving through a resistor in the direction of the current.
Common Mistakes to Avoid
- Mixing sign conventions within one problem, which makes KVL equations inconsistent. Choose a direction for each loop and apply voltage rise and drop rules the same way every time.
- Assuming the guessed current direction must be correct, which is not required. A negative answer simply means the real current flows opposite the chosen arrow.
- Writing KCL with only some of the branch currents at a junction, which violates charge conservation. Include every current entering and leaving the node.
- Using only one loop equation for a multi-loop circuit, which usually gives too few equations. You need enough independent KCL and KVL equations to match the number of unknown currents.
Practice Questions
- 1 At a junction, 3.0 A and 1.5 A enter from two branches, while 2.2 A leaves through a third branch. What current must leave through the fourth branch?
- 2 A loop contains a 12 V battery and two series resistors of 4.0 ohms and 8.0 ohms. Use KVL to find the current in the loop and the voltage drop across each resistor.
- 3 In a two-loop circuit, you assume a clockwise current in the right loop and solve to get I = -0.40 A. Explain what the negative sign means and whether the circuit laws were used incorrectly.