Key Points

Electric current is the rate of flow of electric charge through a conductor. For a steady current, it is given by $I = \frac{q}{t}$, and more generally as $I(t) = \frac{dQ}{dt}$. The SI unit of current is the Ampere (A).

Ohm's law states that the potential difference (V) across a conductor is directly proportional to the current (I) flowing through it, provided the temperature and other physical conditions remain unchanged. The formula is $V = IR$, where R is the resistance.

The resistance (R) of a conductor depends on its length (l), cross-sectional area (A), and the material's resistivity ($\rho$). The relationship is given by the formula $R = \rho \frac{l}{A}$. The SI unit of resistance is Ohm ($\Omega$) and resistivity is Ohm-meter ($\Omega \text{m}$).

Current density (j) is the current per unit area, $j = \frac{I}{A}$. In vector form, Ohm's law is expressed as $\mathbf{j} = \sigma \mathbf{E}$, where $\sigma$ is the conductivity of the material and $\mathbf{E}$ is the electric field. Conductivity is the reciprocal of resistivity, $\sigma = \frac{1}{\rho}$.

Drift velocity ($v_d$) is the average velocity attained by charge carriers in a material due to an electric field. The current is related to drift velocity by $I = n e A v_d$, where n is the number density of free electrons, e is the charge of an electron, and A is the cross-sectional area.

Resistivity arises from collisions of electrons with the ions of the conductor. It is given by $\rho = \frac{m}{ne^2\tau}$, where m is the mass of an electron, n is the electron number density, e is the electron charge, and $\tau$ is the average relaxation time between collisions.

Mobility ($\mu$) is a measure of how quickly a charge carrier moves through a material in an electric field. It is defined as the magnitude of the drift velocity per unit electric field, $\mu = \frac{|v_d|}{E} = \frac{e\tau}{m}$. Its SI unit is $\text{m}^2\text{V}^{-1}\text{s}^{-1}$.

For metallic conductors, resistivity increases with temperature, approximately as $\rho_T = \rho_0[1 + \alpha(T - T_0)]$, where $\alpha$ is the temperature coefficient of resistivity. For semiconductors, resistivity decreases with an increase in temperature.

The power (P) dissipated in a resistor is the rate at which electrical energy is converted into heat. It is calculated using the formulas $P = VI = I^2R = \frac{V^2}{R}$. The SI unit of power is the Watt (W).

The electromotive force (emf, $\varepsilon$) of a cell is the potential difference between its terminals in an open circuit. When current (I) is drawn, the terminal voltage (V) is $V = \varepsilon - Ir$, where r is the internal resistance of the cell.

For n cells connected in series, the equivalent emf is the sum of individual emfs, $\varepsilon_{eq} = \varepsilon_1 + \varepsilon_2 + ... + \varepsilon_n$. The equivalent internal resistance is the sum of individual internal resistances, $r_{eq} = r_1 + r_2 + ... + r_n$.

For cells connected in parallel, the equivalent emf ($\varepsilon_{eq}$) and internal resistance ($r_{eq}$) are given by $\frac{\varepsilon_{eq}}{r_{eq}} = \frac{\varepsilon_1}{r_1} + \frac{\varepsilon_2}{r_2} + ...$ and $\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} + ...$.

At any junction in an electrical circuit, the sum of the currents entering the junction is equal to the sum of the currents leaving it. This rule is based on the law of conservation of charge.

The algebraic sum of the changes in potential around any closed loop in a circuit is zero. This rule is based on the law of conservation of energy.

A Wheatstone bridge is a circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit. The bridge is balanced when no current flows through the galvanometer, and the condition is $\frac{R_1}{R_2} = \frac{R_3}{R_4}$.