Kirchhoff's Second Law: The Voltage Rule
The Core Idea: Energy in a Loop
Imagine you are on a roller coaster. You start at the highest point with a lot of potential energy. As you go down the hills and through the loops, that energy is converted into kinetic energy (speed) and some is lost to friction and air resistance. When the ride is over and you return to the starting platform, all the energy you started with has been accounted for—it was either used for motion or lost as heat. This is the essence of the law of conservation of energy: energy cannot be created or destroyed, only transformed.
Kirchhoff's Second Law applies this exact same idea to electricity. In an electrical circuit, batteries act like the initial lift hill, providing electrical potential energy, which we measure in volts. Components like resistors, bulbs, and motors act like the friction and loops on the roller coaster; they use up this electrical energy, converting it into other forms like light, heat, or motion. The "loop" is the closed path the electricity follows.
For any closed loop in a circuit, the algebraic sum of all voltages is zero: $$\sum V = 0$$ This means: (Sum of EMFs) = (Sum of Potential Drops), or more commonly written as: $$\sum \epsilon = \sum IR$$ where $\epsilon$ is the electromotive force (EMF) of a battery and $IR$ is the voltage drop across a resistor.
Voltage Drops and Rises: Assigning the Signs
The trickiest part of applying KVL is getting the signs right. It's like keeping track of whether you are going uphill (gaining potential) or downhill (losing potential) on our roller coaster. To do this, we use a consistent method:
- Choose a direction to travel around the loop (clockwise or counterclockwise). The choice is arbitrary; if you guess the current direction wrong, your final answer will simply be a negative number, indicating the current flows the other way.
- Assign a sign to voltage changes:
- When you move from the negative to the positive terminal of a battery, it's an increase in voltage (a +EMF).
- When you move from the positive to the negative terminal, it's a decrease in voltage (a -EMF).
- When you move across a resistor in the same direction as the assumed current, you experience a drop in voltage (a -IR).
- When you move across a resistor in the opposite direction to the assumed current, you experience a rise in voltage (a +IR).
Once all the signs are assigned, you simply add them all up and set the sum equal to zero.
A Section with the Theme of Practical Application or Concrete Example
Let's apply KVL to a simple circuit with two resistors and a single battery to find the current flowing. This is a direct, practical application of the law.
Consider a $9\text{V}$ battery connected to two resistors in series: $R_1 = 2\Omega$ and $R_2 = 1\Omega$.
| Step-by-Step KVL Analysis | |
|---|---|
| Step 1: Assume Current Direction | We assume a clockwise current, $I$. |
| Step 2: Travel the Loop | Starting from the battery's negative terminal, we move clockwise. |
| Step 3: Encounter Battery | We go from negative to positive: this is a voltage rise of $+9\text{V}$. |
| Step 4: Encounter Resistor R1 | We traverse $R_1$ in the direction of current $I$: this is a voltage drop of $-I \times R_1 = -I \times 2$. |
| Step 5: Encounter Resistor R2 | We traverse $R_2$ in the direction of current $I$: this is a voltage drop of $-I \times R_2 = -I \times 1$. |
| Step 6: Apply KVL Formula | Sum all voltages and set to zero: $$+9V - (2\Omega)I - (1\Omega)I = 0$$ |
| Step 7: Solve for Current (I) | $$9 - 2I - I = 0$$ $$9 - 3I = 0$$ $$3I = 9$$ $$I = 3\text{A}$$ The positive result confirms our assumption of a clockwise current is correct. |
Analyzing Complex Circuits with Multiple Loops
KVL truly shines when circuits become more complex, with multiple batteries and loops. For these, we use a system of equations derived from KVL and Kirchhoff's First Law[1] (the Current Law).
Example: A two-loop circuit with two batteries. Battery A ($12\text{V}$) and Resistor A ($4\Omega$) are in the left loop. Battery B ($5\text{V}$) and Resistor B ($1\Omega$) are in the right loop. A central Resistor C ($2\Omega$) is shared by both loops.
We define three different currents: $I_1$ for the left loop, $I_2$ for the right loop, and $I_3$ for the shared branch. Applying KCL at the junction gives us $I_1 = I_2 + I_3$.
Now, we apply KVL to each independent loop:
- Left Loop (clockwise): $+12 - 4I_1 - 2I_3 = 0$
- Right Loop (clockwise): $-5 - 1I_2 + 2I_3 = 0$ (Notice the $-5\text{V}$ because we are moving from positive to negative terminal of the battery, and the $+2I_3$ because we are moving against the assumed direction of $I_3$ across Resistor C).
We now have three equations with three unknowns, which can be solved simultaneously to find all the currents in the circuit.
Common Mistakes and Important Questions
Q: I keep getting the signs wrong for the batteries and resistors. Is there a simpler way?
A: A common and effective mental model is the "Hiker Analogy." Think of voltage as elevation. A battery is a lift that raises you up by its voltage value. A resistor is a downhill slope that lowers you by an amount $IR$. If you take a closed walk (a loop) and end up back where you started, the total uphill must equal the total downhill. If you find yourself "climbing" a resistor, that means you are going against the current, which is like an uphill, so you add $+IR$.
Q: Does Kirchhoff's Second Law apply to circuits with capacitors and inductors?
A: Yes, but with an important extension. For circuits with changing currents (AC circuits), the law still holds, but the voltage across a capacitor ($V_C = Q/C$) and an inductor ($V_L = L(dI/dt)$) must be included in the sum. The core principle—that the sum of voltages around any closed loop is zero—remains a cornerstone of circuit analysis.
Q: What happens if I choose the wrong direction for the current when I start my KVL analysis?
A: Don't worry! This is a common concern for beginners. If you assume the wrong direction for a current, the mathematics will correct it for you. When you solve your KVL equations, the value for that current will simply come out as a negative number. A negative current simply means the actual flow of charge is in the opposite direction to your initial guess. The magnitude (size) of the current will still be correct.
Footnote
[1] KCL (Kirchhoff's Current Law): Also known as Kirchhoff's First Law, it states that the total current entering a junction (or node) in a circuit must equal the total current leaving the same junction. This is based on the conservation of electric charge.
[2] EMF (Electromotive Force): This is not actually a force. It is the voltage generated by a source of electrical energy, like a battery or generator. It represents the energy provided per unit charge to drive a current around a circuit, measured in volts (V).
[3] Potential Drop (or IR Drop): The decrease in electrical potential energy that occurs when charge moves through a component that resists its flow, such as a resistor. It is calculated by multiplying the current (I) through the component by its resistance (R), hence $V = IR$.