Kirchhoff's Circuit Laws
Electric charge can't pile up or vanish, so at any junction the current flowing in must exactly equal the current flowing out. Energy can't appear or disappear either, so if you add up all the voltage gains and drops all the way around any closed loop, they cancel to zero.
The formula
ΣI_in = ΣI_out ; ΣΔV = 0
- I_in — current in (A): total current flowing into a junction
- I_out — current out (A): total current flowing out of that same junction
- ΔV — voltage change (V): a gain or drop in voltage across one part of a loop
Worked example
A wire carries 6 A into a junction, where it splits into two branches. One branch carries 4 A. How much current flows through the other branch?
- I_in = 6 A
- I_out1 = 4 A
- I_out2 = I_in − I_out1
- I_out2 = 6 A − 4 A
I_out2 = 2 A
Test yourself
A junction now has 9 A flowing in, splitting into two branches. One branch carries 5 A. What flows through the other branch?
- Correct answer: 4 A
- 14 A
- 5 A
Right! Currents in must equal currents out: 9 A − 5 A = 4 A.
Going all the way around a closed loop in a circuit, adding up every voltage rise and drop, what must the total equal?
- The total current
- Correct answer: Zero
- The battery's voltage doubled
Exactly — energy is conserved, so all the voltage gains and drops around a loop sum to zero.
Where you see this
Your home's wiring is a tree of junctions: the mains feed splits at the breaker panel, splits again at every room, and every amp that flows in at one end flows back out the other — nothing piles up. A river forking around an island shows the junction rule in nature: the two branches carry exactly the stream the single channel brought to the fork.
Common mistakes
The classic error is that current gets used up as it passes through devices — charge is conserved, so whatever flows into a junction flows out: 9 A in with 5 A down one branch means exactly 4 A down the other, not less. The loop rule gets misapplied its own way: with several components in one loop, the battery's voltage is not dropped across your favorite component — ΣΔV = 0 balances the whole trip, every rise against every drop. Keep the two conservations straight: junctions conserve charge, loops conserve energy.
How it connects
Ohm's law prices a single resistor; these two laws do the bookkeeping for whole networks — and both are conservation laws from mechanics in circuit clothing (ΣI_in = ΣI_out is charge conservation, ΣΔV = 0 is energy conservation). The RC circuit, next, is the first circuit whose behavior changes as it runs, and solving it takes both rules at once.