Comprehensive theory, key formulas, diagrams, and memory aids for Electricity.
Electric charge ($Q$) is a fundamental property of matter. It comes in two types: positive (protons) and negative (electrons). The SI unit is the coulomb (C). The charge of one electron is $e = 1.6 \times 10^{-19}$ C.
Electric current ($I$) is the rate of flow of charge: $$I = \frac{\Delta Q}{\Delta t}$$
The SI unit is the ampere (A). Current is carried by free electrons in metals (conventional current is defined as flowing from + to −, opposite to electron flow).
Number of charge carriers: If there are $n$ conduction electrons per unit volume (number density), each carrying charge $q$, moving at drift velocity $v_d$ through a conductor of cross-sectional area $A$: $$I = nAqv_d$$
At typical current densities, electrons drift at only ~mm s⁻¹ — much slower than the electrical signal speed (~c), which is the speed at which the electric field propagates.
Potential difference (voltage, $V$) between two points is the work done per unit charge in moving charge between those points: $$V = \frac{W}{Q}$$
Unit: volt (V) = J C⁻¹.
Electromotive force (e.m.f., $\varepsilon$) is the energy given per unit charge by the source (e.g., battery, generator) to charge carriers. It is NOT a force, despite its name: $$\varepsilon = \frac{W}{Q}$$
The distinction: e.m.f. is the energy supplied per coulomb by the source; p.d. is the energy transferred per coulomb by the component.
Ohm's Law: For a metallic conductor at constant temperature, the current through it is proportional to the potential difference across it: $$V = IR$$
Resistance ($R$) is the opposition to current flow: $$R = \frac{V}{I}$$
Unit: ohm (Ω) = V A⁻¹. Ohm's Law is not a universal law — it is obeyed only by ohmic conductors (metals at constant temperature). Non-ohmic components (filament lamps, diodes, thermistors, LDRs) have resistance that varies with conditions.
Resistivity ($\rho$): An intrinsic material property linking resistance to geometry: $$R = \frac{\rho L}{A}$$
Where $L$ = length (m), $A$ = cross-sectional area (m²), $\rho$ = resistivity (Ω m). Copper: $\rho \approx 1.7 \times 10^{-8}$ Ω m. Good insulators: $\rho \sim 10^{12}$ Ω m.
For metals, resistivity increases with temperature (more lattice vibrations impede electron flow). For semiconductors, resistivity decreases with temperature (more charge carriers become available).
Series: All charge flows through each resistor. Total resistance: $$R_{total} = R_1 + R_2 + R_3 + \ldots$$
Current is the same through each; voltages add to give the total EMF.
Parallel: Current divides between branches. Total resistance: $$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$$
Voltage is the same across each branch; currents add.
Adding more resistors in series increases total resistance. Adding more resistors in parallel decreases total resistance (more pathways for current).
graph LR
A[Battery] --> B[R1]
B --> C[R2]
C --> D[R3]
D --> E[Back to Battery]
style A fill:#1e293b,color:#fff
Power dissipated in a component: $$P = IV = I^2R = \frac{V^2}{R}$$
Energy dissipated: $$W = Pt = IVt = I^2Rt = \frac{V^2t}{R}$$
The energy is transferred as thermal energy (heat) in resistors. In electrolysis or motors, some energy is transferred usefully as chemical or mechanical energy.
A real battery has internal resistance ($r$) — the resistance of the materials inside the cell itself. When current flows, some voltage is dropped across this internal resistance (the "lost voltage"):
$$\varepsilon = V_{terminal} + Ir = IR + Ir = I(R + r)$$
The terminal voltage (voltage across the external circuit): $$V_{terminal} = \varepsilon - Ir$$
When no current flows (open circuit): $V_{terminal} = \varepsilon$ (the terminal voltage equals the e.m.f.)
As more current is drawn (lower external $R$), terminal voltage decreases more. A "flat" battery has high internal resistance.
Graphical determination: Plot $V_{terminal}$ against $I$. Gradient = $-r$ (internal resistance), y-intercept = $\varepsilon$ (e.m.f.).
Kirchhoff's First Law (KCL): The sum of currents entering any junction equals the sum of currents leaving. (Conservation of charge) $$\sum I_{in} = \sum I_{out}$$
Kirchhoff's Second Law (KVL): The sum of e.m.f.s around any closed loop equals the sum of potential differences (p.d.s) around the same loop. (Conservation of energy) $$\sum \varepsilon = \sum IR$$
These laws allow analysis of any circuit, however complex.
A potential divider uses two (or more) resistors in series to produce a fraction of the supply voltage: $$V_{out} = V_{in} \times \frac{R_2}{R_1 + R_2}$$
Practical sensing circuits use resistors whose resistance changes with the environment: - Thermistor (NTC): resistance decreases as temperature increases. - Light-Dependent Resistor (LDR): resistance decreases as light intensity increases.
Combining a thermistor or LDR with a fixed resistor in a potential divider creates a sensor circuit whose output voltage varies with temperature or light — used in alarm systems, automatic lighting, and temperature controllers.