Electric Charge and Current

All matter is built from atoms containing charged particles. Protons carry positive charge, electrons carry negative charge of equal magnitude, and neutrons are electrically neutral. When charges move freely through a conductor, we have an electric current.

**Electric charge ($Q$) is measured in coulombs (C)**. One coulomb represents the combined charge of approximately $6.24 \times 10^{18}$ electrons.

**Electric current ($I$)** is the rate of flow of charge — the quantity of charge passing a cross-section of conductor per unit time:

$$I = \frac{Q}{t}$$

where $I$ is in amperes (A), $Q$ is in coulombs (C), and $t$ is in seconds (s).

Example: A charge of 36 C flows through a lamp in 12 s. The current is:

$$I = \frac{36}{12} = 3\ \text{A}$$

By convention, current flows from the positive terminal to the negative terminal of a source. Electrons actually travel in the opposite direction (from negative to positive) — this distinction between conventional current and electron flow is important in circuit analysis.

Voltage and Resistance

**Voltage ($V$), or potential difference, is the energy transferred per unit charge between two points in a circuit — the "electrical pressure" that drives current. Unit: volts (V)**.

**Resistance ($R$) is the opposition a material presents to the flow of electric current. It arises from collisions between moving electrons and the ions of the conductor's lattice. Unit: ohms (Ω)**. Metals such as copper are good conductors (low $R$); rubber and glass are insulators (very high $R$).

Ohm's Law states that the voltage across a conductor is directly proportional to the current through it, provided temperature remains constant:

$$V = IR$$

Rearranged forms:

$$I = \frac{V}{R} \qquad R = \frac{V}{I}$$

Worked example: A resistor of $R = 15\ \Omega$ is connected to a $6\ \text{V}$ battery.

$$I = \frac{V}{R} = \frac{6}{15} = 0.4\ \text{A}$$

Note: Ohm's Law holds only for ohmic conductors — materials whose resistance stays constant with temperature. Components such as filament bulbs and diodes are non-ohmic because their resistance changes with temperature or current direction.

Series and Parallel Circuits

Series Circuits

In a series circuit, components are connected end-to-end in a single loop — the same current passes through every element.

• Same current everywhere: $I_1 = I_2 = \ldots = I_{total}$
• Voltages add: $V_{total} = V_1 + V_2 + \ldots$
• Resistances add directly:

$$R_{total} = R_1 + R_2 + R_3 + \ldots$$

Example: $R_1 = 4\ \Omega$ and $R_2 = 6\ \Omega$ in series give $R_{total} = 10\ \Omega$.

Limitation: If one component fails (open circuit), the entire circuit breaks — a drawback of series wiring.

Parallel Circuits

In a parallel circuit, components share the same two nodes — the same voltage appears across every branch.

• Same voltage across each branch: $V_1 = V_2 = \ldots = V_{total}$
• Currents add: $I_{total} = I_1 + I_2 + \ldots$
• Reciprocal rule governs total resistance:

$$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$$

Example: $R_1 = 6\ \Omega$ and $R_2 = 12\ \Omega$ in parallel:

$$\frac{1}{R_{total}} = \frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} \implies R_{total} = 4\ \Omega$$

Connecting resistors in parallel always reduces the total resistance below the smallest individual value — adding more paths makes it easier for current to flow overall. Household wiring uses parallel circuits so each appliance receives full mains voltage and can be switched independently.