Electricity

Electricity is the study of electric charge, current, potential difference, and the behaviour of components in electrical circuits. This section covers the fundamental principles of DC circuits, resistance, and capacitance that form the basis of all electronic systems.

Topics Covered

Current and Resistance

• Electric current — rate of flow of charge: $I = \frac{\Delta Q}{\Delta t}$; conventional current vs. electron flow
• Potential difference — energy per unit charge: $V = \frac{W}{Q}$; the volt
• Resistance — $R = \frac{V}{I}$; Ohm’s law (for ohmic conductors); $I$-$V$ characteristic curves
• Resistivity — $R = \frac{\rho L}{A}$; how material, length, and cross-sectional area affect resistance
• Temperature dependence — positive temperature coefficient in metals; negative temperature coefficient in thermistors (NTC)

DC Circuits

• Series circuits — $I$ is constant, $V$ splits, $R_{\text{total}} = R_1 + R_2 + \cdots$
• Parallel circuits — $V$ is constant, $I$ splits, $\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$
• Kirchhoff’s laws — current law (conservation of charge at junctions) and voltage law (conservation of energy around loops)
• Potential dividers — $V_{\text{out}} = V_{\text{in}} \frac{R_2}{R_1 + R_2}$; sensors and variable resistors
• Internal resistance — $\varepsilon = V + Ir$; terminal p.d. vs. EMF; power delivery and maximum power transfer
• Electrical power and energy — $P = IV = I^2R = \frac{V^2}{R}$; $W = Pt$

Capacitance

• Definition — $C = \frac{Q}{V}$; the farad; parallel plate capacitor $C = \frac{\varepsilon_0 A}{d}$
• Energy stored — $E = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$
• Charging and discharging — exponential curves; $Q = Q_0 e^{-t/RC}$; time constant $\tau = RC$
• Capacitors in series and parallel — analogous to springs: series adds reciprocals, parallel adds directly

Internal Resistance and EMF

The EMF ($\varepsilon$) of a cell is the energy provided per coulomb of charge when no current flows. In practice, all cells have internal resistance $r$, so the terminal potential difference is less than the EMF when current flows:

$\varepsilon = V + Ir$

Where $V$ is the terminal p.d., $I$ is the current, and $r$ is the internal resistance. To measure $r$ experimentally, vary the external resistance and plot $V$ against $I$ — the gradient is $-r$ and the $y$-intercept is $\varepsilon$.

Resistivity and Superconductivity

Resistivity $\rho$ is a material property independent of dimensions: $R = \frac{\rho L}{A}$. For metals, $\rho$ increases with temperature (more lattice vibrations scatter electrons). Superconductors have $\rho = 0$ below a critical temperature $T_c$ — applications include loss-free power transmission and strong electromagnets in MRI machines.

Study Tips

1. Draw circuit diagrams — use standard symbols, label all currents and voltages, and indicate direction of conventional current.
2. Apply Kirchhoff’s laws systematically — choose a loop, go around consistently, and set up equations. Solve simultaneously.
3. Understand $I$-$V$ characteristics — sketch and explain the curves for a resistor (linear), filament lamp (curved, resistance increases with temperature), and diode (threshold voltage).
4. Practise internal resistance problems — they combine circuit analysis with the concept of EMF and are frequently examined.
5. For capacitor discharge — always identify the time constant $\tau = RC$ first. After one time constant, the charge drops to $37\%$ of its initial value.

How to Use These Notes

Follow the sidebar order. Each page provides definitions, derivations, worked examples with circuit diagrams, and exam-style problems. Start with current and resistance, then DC circuits, then capacitance.