Kirchhoff’s laws

Kirchhoff’s laws are fundamental rules for analyzing electrical circuits. They are based on the conservation of charge and energy and allow us to calculate currents and voltages in complex circuits.

Kirchhoff’s First Law (Current Law)

Kirchhoff’s first law states that the total current entering a junction (or node) in a circuit is equal to the total current leaving the junction. This is a consequence of the conservation of electric charge—charge cannot be created or destroyed.

Mathematically:

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

Kirchhoff’s Second Law (Voltage Law)

Kirchhoff’s second law states that the sum of the electromotive forces (emfs) around any closed loop in a circuit is equal to the sum of the potential drops (products of current and resistance) in that loop. This is a consequence of the conservation of energy.

Mathematically:

$$\sum \text{emf} = \sum IR$$

Combined Resistance in Series

When resistors are connected in series, the total resistance is the sum of the individual resistances. This can be derived using Kirchhoff’s laws.

Suppose resistors $R_1$, $R_2$, ..., $R_n$ are connected in series to a battery of emf $E$.

• The same current $I$ flows through each resistor.
• The total voltage across the combination is $E$.

By Kirchhoff’s second law:

$$E = IR_1 + IR_2 + \cdots + IR_n = I(R_1 + R_2 + \cdots + R_n)$$

So, the combined resistance $R_\text{total}$ is:

$$R_\text{total} = R_1 + R_2 + \cdots + R_n$$

Combined Resistance in Parallel

When resistors are connected in parallel, the total resistance is less than any individual resistance. This can also be derived using Kirchhoff’s laws.

Suppose resistors $R_1$, $R_2$, ..., $R_n$ are connected in parallel across a voltage $V$.

• The voltage across each resistor is $V$.
• The total current $I$ is the sum of the currents through each resistor.

By Kirchhoff’s first law:

$$I = I_1 + I_2 + \cdots + I_n$$

But $I_k = \frac{V}{R_k}$ for each resistor, so:

$$I = \frac{V}{R_1} + \frac{V}{R_2} + \cdots + \frac{V}{R_n}$$

The total resistance $R_\text{total}$ is defined by $I = \frac{V}{R_\text{total}}$, so:

$$\frac{V}{R_\text{total}} = \frac{V}{R_1} + \frac{V}{R_2} + \cdots + \frac{V}{R_n}$$

Dividing both sides by $V$:

$$\frac{1}{R_\text{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$$

Using Kirchhoff’s Laws to Solve Circuit Problems

To analyze a circuit using Kirchhoff’s laws:

1. Assign current directions (arbitrary if unknown).
2. Apply Kirchhoff’s first law at junctions to relate currents.
3. Apply Kirchhoff’s second law around loops to relate voltages and currents.
4. Solve the resulting simultaneous equations for unknown currents and voltages.