Physics · Ch 2 — Current Electricity

Kirchhoff's Second Rule (Voltage Rule or Loop Rule)

2.5.2

Kirchhoff's Second Rule (Voltage Rule or Loop Rule)

Kirchhoff's second rule, the voltage rule (or loop rule), states that in any closed circuit, the algebraic sum of the products of current and resistance for each part of the circuit is equal to the total emf included in that circuit. This rule follows directly from the law of conservation of energy applied to a closed loop: the energy supplied by every emf source around the loop must exactly equal the total energy delivered to every resistor in that same loop.

A careful sign convention is essential when applying this rule (Figure 2.24): the product of current and resistance, IR, is taken as POSITIVE when the loop is traversed in the SAME direction as the assumed current through that resistor, and NEGATIVE when traversed opposite to the assumed current. Likewise, an emf $\varepsilon$ is taken as POSITIVE when the loop is traversed from the cell's negative terminal to its positive terminal, and NEGATIVE when traversed the other way, from positive to negative. It is important to note that the voltage rule may only be validly applied once every current in the circuit has settled into a steady state (i.e. the currents in every branch have become constant in time, not still changing). Applying both of Kirchhoff's rules together to a network with several loops and junctions -- writing one current-rule equation …

Figure 2.24Kirchhoff's voltage rule sign convention

What this figure shows. Four small labelled panels (a)-(d) illustrating the sign convention for the loop rule. In (a), a resistor R is traversed in the SAME direction as its current I (going from a to b), giving $V=+IR$ . In (b), the same resistor is traversed OPPOSITE to its current (going from b to a), giving $V=-IR$ . In (c), a cell of emf $\varepsilon$ is traversed from its negative terminal to its positive terminal (going from a to b), giving $V=+\varepsilon$ . In (d), the same cell is traversed from positive to negative ( …

Misc ~note-steady-stateWhen the voltage rule may be applied

Worked out. A short but important caveat: Kirchhoff's voltage rule may only be applied once every current in the circuit has settled into a steady-state condition, i.e. the current in each branch has become constant in time; it cannot be validly applied to a circuit whose currents are still changing. …

Misc Example 2.21Setting up the loop equations for a general five-resistor network

Worked out. A complex network of conductors, containing resistors $R_1$ through $R_5$ carrying currents $I_1$ through $I_5$ and a single emf source $\varepsilon$ , can be split into two closed loops, EACE and ABCA; the loop equations are required. Applying Kirchhoff's voltage rule to loop EACE gives $I_1R_1+I_2R_2+I_3R_3=\varepsilon$ . Applying it to loop ABCA (which contains no emf source of its own) gives $I_4R_4+I_5R_5-I_2R_2=0$ . This example is purely about setting up the correct loop equations symbolically -- it demonstrates the general method of picking closed loops and summing $IR$ drops around each, which is then solved as simultaneous equations for a specifi …

Misc Example 2.22Solving a two-loop, two-battery network for the current in a shared resistor

Worked out. A network with points A, B, C, D, E, F contains a $2\ \Omega$ resistor and a 9 V battery in one branch, a $3\ \Omega$ resistor in another, a $1\ \Omega$ resistor shared between two loops, and a 6 V battery, with $I_1$ the current from the 9 V battery splitting into $I_2$ and $(I_1-I_2)$ at junction E; the current in the $1\ \Omega$ resistor is required. Applying Kirchhoff's voltage rule to loop EFCBE gives $1\cdot I_2+3I_1+2I_1=9$ , i.e. $5I_1+I_2=9$ . Applying it to loop EADFE gives $3(I_1-I_2)-1\cdot I_2=6$ , i.e. $3I_1-4I_2=6$ . Solving these two simultaneous equations gives $I_1=1.83$ A and $I_2=-0.13$ A; since $I_2$ comes out negative, the actual current in the $1\ \Omega$ resistor flows from F to E, opposite …

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