Kirchhoff's Current and Voltage Laws

The two laws

Kirchhoff's Current Law (KCL)is just conservation of charge: at any junction, the current flowing in must equal the current flowing out. Charge can't pile up at a point.

Kirchhoff's Voltage Law (KVL) is conservation of energy: going all the way around any closed loop in a circuit, the total voltage gained (from sources) equals the total voltage dropped (across resistors). You end up back where you started, at the same potential.

Worked example: a two-loop circuit

A circuit has two loops sharing a middle branch. The left loop has an 8V source in series with a 1Ω resistor, the right loop has an 8V source in series with a 2Ω resistor, and the shared middle branch has a 2Ω resistor. Current $I_1$ flows from the left source into the shared node, $I_2$ from the right source into the same node, and $I_3$ flows out through the middle resistor. Find all three currents.

Solution: KCL at the shared node gives $I_3 = I_1+I_2$. Applying KVL around each loop:

Substituting $I_3 = I_1+I_2$ into both:

From (ii): $I_1 = 4 - 2I_2$. Substituting into (i):

Then $I_1 = 4-2(1) = 2\text{ A}$, and $I_3 = I_1+I_2 = 3\text{ A}$.

Check: left loop: $2(1)+3(2)=2+6=8$ ✓. Right loop: $1(2)+3(2)=2+6=8$ ✓. Both match the source voltages.