The Kirchhoff Laws
Charge does not accumulate anywhere. Energy does not accumulate anywhere. These are the only two laws, and they govern every circuit that has ever existed.
The Junction Rule
At any node—a junction of wires, a crossroads of potentials—the sum of currents entering equals the sum of currents leaving. $I_{\text{in}} = I_{\text{out}}$. This is conservation of charge, nothing more, nothing less. Charge has no storage facility at a node. It cannot park there. It cannot take a nap on the copper. It flows through or it flows around, but it does not linger.
Consider a node with three wires. Two currents arrive: $I_1$ and $I_2$. A third departs: $I_3$. The node demands $I_3 = I_1 + I_2$. There is no debate. The universe has already decided. The node simply announces the verdict.
This feels trivial because it is trivial at its root, but its consequences are anything but. Loop analysis, nodal analysis, the entire edifice of circuit theory rests on this single observation about a point in space. You can analyze a circuit with a thousand nodes using only this rule and the loop rule, and you will get the right answer every time.
The Loop Rule
Walk around any closed loop in a circuit, summing potential differences, and you return to where you started with zero net change. $\sum V = 0$. This is conservation of energy, expressed in the language of electric fields and work.
When you traverse a battery from negative to positive, you gain potential: $+V$. When you traverse a resistor in the direction of current, you lose potential: $-IR$. These are the signs of the game. Respect them and the math sings. Disrespect them and the math curses you.
A simple loop with a battery and a resistor gives $V - IR = 0$, so $I = V/R$. The familiar relationship emerges naturally, without mysticism, without authority. It follows from two observations about a conserved world.
The Combined Power
Kirchhoff's laws together form a system of equations. For a circuit with $N$ nodes and $B$ branches, you need $(N-1)$ junction equations and $B - (N-1)$ loop equations. The loop count is the number of independent meshes—the windows in your circuit diagram, the holes in a donut. Euler's formula connects them. Circuit theory is topology.
This is not merely a calculation method. It is a way of seeing. Every circuit, from the simplest flashlight to the most complex processor, obeys these laws. No exceptions. The laws were not discovered by Kirchhoff—he simply wrote down what nature was already doing.
The genius of the laws is their universality. They apply to DC and AC. To resistors, capacitors, and inductors (with the appropriate impedance substitutions). To ideal sources and real components. They are the grammar of electric circuits, and every circuit tells its story in this language.
What They Mean
Conservation of charge means no miracles. Charge cannot appear or disappear at a node. It flows. Conservation of energy means no free lunch. Every volt drop costs energy. Every volt gain provides it.
These are not rules imposed on circuits. These are descriptions of a world that does not allow violations. Kirchhoff simply noticed. He wrote it down. And we have been solving equations ever since.