Kirchhoff's Law

Kirchhoff’s Law, combined with Ohm’s Law, form the fundamentals for circuit analysis. It uses the conservation of energy, as well as the conservation of Charge.

Definition §

Kirchhoff’s law has two main rules:

$$I_{in}=I_{out}\text{in a junction}$$

$$V_{battery}/\mathcal{E}=\sum IR\text{in a loop}$$

• Note that a junction is defined to be a point where wires intersect.
• A loop is any closed path starting from the positive terminal of a battery and ending at the negative terminal

[!info ] Convention… again To maintain consistency in Kirchhoff’s law, we need to define the sign of current flowing between some points. Generally, current is defined as the direction a positive charge would flow through a circuit, i.e. from +terminal to -terminal of a battery.

Conservation of Charge §

The first rule states that no build up of charge can occur in a junction of a circuit, i.e. the sum of a charge in a junction is always equal to zero:

$$\sum i=\sum i_{in}+\sum i_{out}=0$$

Generally, the current going into a circuit is defined to be positive, while current leaving a circuit is defined to be negative.

Conservation of Energy §

The second rule states the total change in Electrical Potential Energy in a circuit must be zero. We can use the definition of voltage to extend this:

$$U_f-U_i=0\to \Delta V=0$$

In regards to the circuit, we can define two ‘paths’ for $V_f$ and $V_i$ - one going through the battery, to give us the EMF, $\mathcal{E}$, and one going through the normal circuit path.

$$\mathcal{E}=\sum V_{loop}$$

Again, conventionally current flows from positive to negative, and electric potential difference (voltage) is positive as we move along the conventional current.

Another convention is that for a loop (circuit) through a resistor that follows the direction of conventional current, we define the change in potential, $\Delta V=-IR$ .

Essentially, if our direction of loop is the same as the current, the voltage drops, but if they are opposite, the voltage rises.

Similarly if we go against the direction of conventional current, the change is now $+IR$. EMF is defined the opposite way, with $\mathcal{E}=+V$ from negative terminal to positive terminal.