# Multiloop Circuits

Multiloop Circuits

assume
the batteries are ideal. There are two junctions in this circuit, at b and d, and there are three branches connecting these junctions. The branches are the left branch (bad), the right branch (bad), and the central branch (bd). What are the currents in the three branches? We arbitrarily label the currents, using a different subscript for each branch. Current iI has the same value eve  anywhere in branch bad, i2 has the same value everywhere in branch bed, and i3 is the current through branch bd. The directions  of the currents are assumed arbitrarily.Consider junction d for a moment: Charge comes into that junction via incoming currents iI and i3, and it leaves via outgoing current i2• Because there is no variation in the charge at the junction, the total incoming current must equal the total outgoing current:

i + i = I2

You can easily check that applying this condition to junction b leads to exactly the same equation. Equation 28-15 thus suggests a general principle:

JUNCTION RULE: The sum of the currents entering any junction must be equal to the sum of the currents leaving that junction.

This rule is often called Kirchhoff s junction rule (or Kirchhoff s current law). It is simply a statement of the conservation of charge for a steady flow of charge-there is neither a build-up nor a depletion of charge at a junction. Thus.iour basic tools for solving complex circuits are the loop rule (based on the conservation of energy) and the junction rule (based on the conservation of charge). Equation 28-15 is a single equation involving three unknowns. To solve the circuit completely (that is. to find all three currents), we need two more equations involving those same unknowns. We obtain them by applying the loop rule twice. In the circuit of  we have three loops from which to choose: the left-hand loop (had), the right-hand loop (bib), and the big loop (bad).’Which two loops we choose does not matter-let’s choose the left-hand loop and the right-hand loop. If we traverse the left-hand’ loop in a counterclockwise direction from point b, the loop rule gives us. 