DC Circuits

Current, Voltage, and Resistance

• Current ($I$): flow of charge, $I = \Delta q / \Delta t$. Unit: ampere (A).
• Voltage ($V$ or $\Delta V$): electric potential difference. Unit: volt (V).
• Resistance ($R$): opposition to current flow. Unit: ohm ($\Omega$).

Ohm's Law

$V = IR$

Resistors in Series

$R_{\text{eq}} = R_1 + R_2 + R_3 + \cdots$

• Same current flows through each.
• Voltage divides: $V_{\text{total}} = V_1 + V_2 + \cdots$

Resistors in Parallel

$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots$

• Same voltage across each.
• Current divides: $I_{\text{total}} = I_1 + I_2 + \cdots$

Kirchhoff's Rules

1. Junction Rule (conservation of charge): Current in = Current out at any junction.
2. Loop Rule (conservation of energy): The sum of voltage changes around any closed loop = 0.

Power in Circuits

$P = IV = I^2R = \frac{V^2}{R}$

Unit: watt (W).

EMF and Internal Resistance

$V_{\text{terminal}} = \mathcal{E} - Ir$ where $\mathcal{E}$ is the EMF and $r$ is the internal resistance of the battery.

Problem: Three resistors ($2\ \Omega$, $3\ \Omega$, $6\ \Omega$) are connected in parallel to a $12\ \text{V}$ battery. Find the total current and power.

Solution:

Equivalent resistance: $\frac{1}{R_{\text{eq}}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3+2+1}{6} = 1$ $R_{\text{eq}} = 1\ \Omega$

Total current: $I = V/R_{\text{eq}} = 12/1 = 12\ \text{A}$

Total power: $P = IV = 12 \times 12 = 144\ \text{W}$

• Ohm's law ($V = IR$) is the fundamental circuit equation.
• Series: same current, voltages add, resistances add.
• Parallel: same voltage, currents add, reciprocal resistances add.
• Kirchhoff's rules ensure conservation of charge and energy.
• Power: $P = IV = I^2R = V^2/R$.