Kirchoff's Node Law

Two different examples are given below. Both examples involve the circuit shown at right that consists of a battery and five capacitors. The charges on capacitors B, C and D are known. Positive and negative signs have also been drawn next to the plates of capacitors B, C and D to indicate the signs of the charges on each plate. Notice that the capacitor plates connected to the positive terminal of the battery bear a positive charge. The goal of the two examples is to determine the charges on capacitors E and F.

Example #1

The circuit at right is identical to the one above except that parts of the circuit have been colored (red and green). Because every capacitor has a gap between its plates, the colored part of the circuit does not touch the part that is uncolored. The top plate of capacitor F is colored differently (red) to remind you that you are trying to determine the charge of capacitor F.

Before the battery was connected to the circuit, all of the capacitors were uncharged. To transfer charge to an object, you must touch it. Since the colored part of the circuit does not touch anything, it remains neutral, even after the battery is connected.

The total charge of the colored part of the circuit is:

Q_{bottom plate of B} + Q_{left plate of E} + Q_{top plate of F} = 0

Substituting in values give:

+2C + -3C + Q_{top plate of F} = 0

Solving reveals:

Q_{top plate of F} = +1C

This means that the charge of capacitor F is 1C and its top plate is positively charged.

Example #2

The circuit at right is identical to the circuit in the previous example, except that a different part of the circuit has been highlighted. As before, most of the circuit elements are colored green. The red capacitor plate is colored differently to remind you that you are trying to determine the charge of that capacitor.

• The colored part of the circuit is not touching anything, so its net charge must be zero.
• You learned in example #1 that the charge of capacitor F is 1C and its bottom plate is negatively charged.
• Kirchoff’s Node Law gives:

Q_{bottom plate of C} + Q_{bottom plate of E} + Q_{bottom plate of F} = 0

• Substituting in values gives:

5C + -1C + Q_{bottom plate of F} = 0

• Solving gives Q_{bottom plate of F} = -4C. Hence, capacitor F bears a charge of 4C and its bottom plate is negative. Notice, this also means the top plate of capacitor F, which is directly connected to the positive terminal of the battery, is indeed positive.

An alternative method of finding the charge on capacitor F is to consider the highlighted portion of the circuit shown in the diagram at right. As always, the top plate of F is colored differently (red) to remind you that it is the charge of F that we are trying to determine. Notice that this time, the colored portion of the circuit contains the entire battery. Remember, however, that the battery is always neutral. As before, the colored part of the circuit must be neutral because it is not touching anything.

Q_{top plate of C} + Q_{top plate of B} + Q_battery + Q_{right plate of D} + Q_{top plate of F} = 0

-5C + -2C + 0C +3C + Q_{top plate of F} = 0

• Solving gives Q_{top plate of F} = +4C. This means that the charge on capacitor C is 4C and its top plate is positively charged. This is the same answer as before.
• The first method for finding the charge of capacitor F was a bit simpler, but both methods are valid.