Current Electricity

Electric current is the rate of flow of charge through a conductor: $I = Q/t$ (for steady current), measured in amperes (A). By convention, current direction is taken as the direction of flow of positive charge, even though in metals it's actually electrons (negative charge) that physically move, in the opposite direction.

Free electrons in a conductor move randomly at high speeds due to thermal motion, but this random motion averages to zero net displacement. When an electric field is applied, electrons acquire a small, net average velocity superimposed on this random motion, called the drift velocity ($v_d$) — it is this slow drift, not the electrons' actual (much faster) random thermal speed, that constitutes the electric current.

The relaxation time (τ) is the average time between successive collisions of an electron with the lattice ions. Drift velocity can be expressed as $v_d = \dfrac{eE\tau}{m}$, showing it depends directly on the applied field and relaxation time. Mobility ($\mu$) is defined as drift velocity per unit electric field: $\mu = v_d/E = e\tau/m$ — a measure of how easily charge carriers move through a specific material under a given field.

At constant temperature, the current through a conductor is directly proportional to the potential difference across it: $V = IR$, where $R$ (resistance) is a constant of proportionality for that conductor under those conditions.

Resistivity (ρ) is a material-specific property (independent of the conductor's actual shape or size) describing its intrinsic resistance to current flow: $R = \rho\dfrac{L}{A}$, where $L$ = length, $A$ = cross-sectional area.

Conductivity ($\sigma$) is simply the reciprocal of resistivity: $\sigma = 1/\rho$ — materials with high conductivity (like copper, silver) are good conductors; materials with very low conductivity (high resistivity) are good insulators.

For most metals, resistance increases with rising temperature, since increased thermal vibration of the metal's lattice ions causes more frequent collisions with the flowing electrons, impeding their drift: $R_T = R_0(1+\alpha\Delta T)$, where $\alpha$ is the temperature coefficient of resistance.

For semiconductors, increasing temperature provides more thermal energy to free additional charge carriers (electrons and holes) from the material's covalent bonds — this increase in the number of available charge carriers dominates over any increased collision effects, causing semiconductor resistance to decrease with rising temperature, unlike the behaviour of metals.

Resistors in series carry the same current, while the total voltage divides among them. Equivalent resistance is the sum: $R_{eq} = R_1+R_2+\cdots$ — always larger than the largest individual resistance.

Resistors in parallel have the same voltage across each, while the total current divides among them. Equivalent resistance: $\dfrac{1}{R_{eq}} = \dfrac{1}{R_1}+\dfrac{1}{R_2}+\cdots$ — always smaller than the smallest individual resistance.

The EMF of a cell/battery is the total energy provided per unit charge by the source, including the energy needed to overcome its own internal resistance ($r$) — it represents the maximum possible potential difference the cell could provide, measured when no current is drawn (open circuit).

For $n$ identical cells (each of EMF $\varepsilon$ and internal resistance $r$) connected in series: total EMF $= n\varepsilon$, total internal resistance $= nr$ — useful for obtaining a higher voltage.

For $n$ identical cells connected in parallel: total EMF remains $\varepsilon$ (unchanged), but total internal resistance $= r/n$ (reduced) — useful for supplying a larger current while keeping the same voltage, since the reduced internal resistance allows more current to be drawn without as much internal voltage drop.

The sum of currents entering any junction in a circuit equals the sum of currents leaving that junction — a direct consequence of conservation of charge (charge cannot accumulate indefinitely at a junction under steady-state conditions).

The sum of all potential differences (EMFs and IR drops) around any closed loop in a circuit is exactly zero — a direct consequence of conservation of energy (a charge returning to its starting point after going around any closed loop must have zero net change in potential energy).

A Wheatstone bridge is a four-resistor arrangement, with a galvanometer connected across the bridge's midpoint, used to precisely measure an unknown resistance. The bridge is balanced when the galvanometer shows zero deflection (no current flows through it), which occurs when:

where $P, Q, R, S$ are the four resistances forming the bridge's four arms.

A metre bridge is a practical application of the Wheatstone bridge principle, using a 1-metre-long uniform resistance wire in place of two of the bridge's arms — a sliding "jockey" contact finds the balance point where the galvanometer shows zero deflection, and the two wire lengths on either side of this point act as two of the bridge's resistance ratio arms (since resistance of a uniform wire is directly proportional to its length).

The rate at which electrical energy is converted to other forms (heat, light, mechanical work) in a circuit element:

Electrical energy consumed is $E = Pt$, commonly measured in kilowatt-hours (kWh) for household billing purposes — 1 kWh (also called 1 "unit" of electricity) is the energy consumed by a 1000 W appliance running for 1 hour.

A potentiometer is a long, uniform resistance wire used to measure an unknown EMF or potential difference by comparison, again using a null (balance point) method — this makes it more accurate than a simple voltmeter, since at the balance point, no current is drawn from the source being measured at all, avoiding any loading effect or voltage drop due to internal resistance.