28.2 Kirchhoff’s Laws and the Basic Circuit

We are now ready to begin analyzing circuits. To analyze a circuit means to find:

The potential difference across each circuit component.
The current in each circuit component.

Because charge is conserved, the total current into the junction of FIGURE 28.4 must equal the total current leaving the junction. That is,

\sum I_{in} = \sum I_{out}

(28.1)

This statement, which you met in Chapter 27, is Kirchhoff’s junction law.

The figure illustrates Kirchhoff’s junction law.

FIGURE 28.4 Full Alternative Text

Because energy is conserved, a charge that moves around a closed path has $Δ U_{elec} = 0$ . We apply this idea to the circuit of FIGURE 28.5 by adding all of the potential differences around the loop formed by the circuit. Doing so gives

Δ V_{loop} = \sum_{}^{} {(Δ V)}_{i} = 0

(28.2)

where ${(Δ V)}_{i}$ is the potential difference of the ith component in the loop. This statement, introduced in Chapter 26, is Kirchhoff’s loop law.

The figure illustrates Kirchhoff’s loop law.

FIGURE 28.5 Full Alternative Text

Kirchhoff’s loop law can be true only if at least one of the ${(Δ V)}_{i}$ is negative. To apply the loop law, we need to establish rules to define which potential differences are positive and which are negative.

TACTICS BOX 28.1 Using Kirchhoff’s loop law

1 Draw a circuit diagram. Label all known and unknown quantities.

2 Assign a direction to the current. Draw and label a current arrow $I$ to show your choice.

If you know the actual current direction, choose that direction.
If you don’t know the actual current direction, make an arbitrary choice. All that will happen if you choose wrong is that your value for $I$ will end up negative.

3 “Travel” around the loop. Start at any point in the circuit, then go all the way around the loop in the direction you assigned to the current in step 2. As you go through each circuit element, $Δ V$ is interpreted to mean $Δ V = V_{downstream} - V_{upstream}$ .

For an ideal battery in the negative-to-positive direction:

$Δ V_{bat} = + ε$

Full Alternative Text
For an ideal battery in the positive-to-negative direction:

$Δ V_{bat} = - ε$

Full Alternative Text
For a resistor: $Δ V_{res} = - Δ V_{R} = - I R$

Full Alternative Text

4 Apply the loop law: $\sum {(Δ V)}_{i} = 0$

Workbook Exercises 4–7

NOTE Ohm’s law gives us only the magnitude $Δ V_{R} = I R$ of the potential difference across a resistor. For using Kirchhoff’s law, $Δ V_{res} = V_{downstream} - V_{upstream} = - Δ V_{R}$ . That is, the potential decreases in the direction of the current.

28.2 Kirchhoff’s Laws and the Basic Circuit

FIGURE 28.4 Kirchhoff’s junction law.

FIGURE 28.5 Kirchhoff’s loop law.

TACTICS BOX 28.1 Using Kirchhoff’s loop law