A single resistor across a battery is solved by Ohm's law alone. Real circuits have many components wired in tangles, and to handle them the subject rests on two laws so simple they can feel like restatements of the obvious, which is exactly their strength. They are Kirchhoff's laws, and each is a conservation principle already met, now sharpened into a tool. Together with Ohm's law they are complete: no resistive circuit exists that these three cannot solve.
The current law, charge is conserved at a junction
A junction is any point where wires meet and current can divide or combine. Kirchhoff's current law states that the total current flowing into a junction equals the total current flowing out. Nothing accumulates there, because charge is conserved and a junction is not a reservoir. If 3 A arrive along one wire and split into two, the two must carry 3 A between them, in whatever division the rest of the circuit dictates. Written as a balance, the currents in equal the currents out at every junction, and that is the law that governs how current shares itself among parallel paths.
The voltage law, energy is conserved around a loop
A loop is any closed path through the circuit that returns to where it began. Kirchhoff's voltage law states that around any loop the voltage rises and the voltage drops cancel exactly, summing to zero. This is energy conservation seen per unit charge: a charge carried once around a loop returns to its starting potential, so whatever energy it gained at sources it must have given up at components, with nothing left over. The battery lifts each coulomb by its EMF, and the resistors around the loop drop it back down by their share of that voltage, and the shares add up to the lift. This law is what lets a source voltage be split among the components in series with it.
Series, resistances add
Components are in series when they lie on a single path so the same current flows through all of them, one after another. Because the same current runs through each, and the voltage law says their voltage drops add up to the source voltage, their resistances simply add:
Three resistors of 100, 220 and 330 ohms in series behave as one 650 ohm resistor. The current is the same everywhere along the chain, and the source voltage divides among them in proportion to their resistances, since the bigger resistor drops more voltage for the same current. This last fact is the voltage divider, one of the most used patterns in electronics: two resistors in series split a voltage in the ratio of their values, so that the voltage across the second is the source voltage times .
Example. A 9 V battery is across two series resistors, 1000 ohms and 2000 ohms. What current flows, and what is the voltage across the 2000 ohm resistor?
The total resistance is ohms, so the current is mA, the same through both. The voltage across the 2000 ohm resistor is V, which is also V by the divider, agreeing.
Now you. Two resistors, 470 ohms and 330 ohms, are in series across 8 V. What is the current?
Answer
Total resistance ohms, so A, or 10 mA.
Parallel, conductances add
Components are in parallel when they are connected across the same two points, so the same voltage sits across all of them and the current divides among them. Here it is the conductances, the reciprocals of resistance, that add, which for resistance itself gives the reciprocal rule:
The result is always smaller than the smallest resistor in the group, because adding another path can only make it easier for current to flow, never harder. Two equal resistors in parallel give half the resistance; a 100 ohm and a 100 ohm in parallel act as 50 ohms. The reason conductance is the natural quantity here is exactly this: another lane in parallel adds its ease of flow to the total, so the eases add while the resistances do the awkward reciprocal dance. The current law then splits the total current among the branches in proportion to their conductances, the current divider, so the lower-resistance branch takes the larger share.
Example. A 100 ohm and a 100 ohm resistor are in parallel across 5 V. What is the combined resistance, and the total current from the source?
Two equal resistors in parallel halve the resistance, giving 50 ohms. The total current is A, which splits into 0.05 A through each branch, since equal resistances share the current equally.
Now you. A 200 ohm and a 300 ohm resistor are in parallel. What is their combined resistance?
Answer
, so ohms, smaller than either.
Reducing a network
The power of these rules is that any ladder of resistors, however complicated, can be collapsed step by step. Find a group that is clearly series or clearly parallel, replace it with its single equivalent, and the circuit gets simpler; repeat, and a page of resistors becomes one number, from which the source current follows by Ohm's law. Working back outward then recovers the voltage across and current through every original component. This reduce-and-expand method handles the great majority of circuits a builder meets. A minority are wired so that no two components are cleanly in series or parallel, a bridge being the classic case, and those need the more systematic machinery of the next lesson. But the two laws are the whole foundation, and they are worth stating once more in their plainest form: charge does not pile up at a junction, and energy does not appear or vanish around a loop.