Chapter 3: Current Electricity

General terminology and formulas for Chapter 3: Current Electricity, including Ohm's law, resistance, Kirchhoff's rules, and circuit instrumentation.

Electric Current ($I$)

The rate of flow of electric charge, defined as $I = \frac{dq}{dt}$ (or $I = \frac{q}{t}$).

SI Unit of Electric Current

The ampere ($A$), where $1\,A = 1\,C/s$.

Current Density ($J$)

The current flowing through unit area of cross section of a conductor, given by $J = \frac{I}{A}$.

SI Unit of Current Density

$A/m^2$.

Drift Velocity ($v_d$)

The average velocity acquired by an electron in the presence of an external electric field.

Relaxation Time ($\tau$)

The average time interval between two successive collisions.

Drift Velocity Equation (Magnitude)

$v_d = \frac{eE\tau}{m}$.

Current and Drift Velocity Relation

The total current is given by $I = neAv_d$.

Current Density ($J$) in terms of Drift Velocity

$J = nev_d$.

Mobility ($\mu$)

The ratio of the magnitude of the drift velocity to electric field strength ($\mu = \frac{v_d}{E}$).

SI Unit of Mobility

$CmN^{-1}s^{-1}$ or $m^2V^{-1}S^{-1}$.

Electromotive Force (emf)

The difference in potential between the terminals of a cell when no current is drawn from it.

Potential Difference

The difference in potential between the terminals of a cell or any two points in a circuit when current is drawn from the cell.

Ohm’s Law

At constant temperature, the current flowing through a conductor is directly proportional to potential difference between the ends of the conductor ($I \propto V$).

Resistance ($R$)

The ability of a conductor to oppose electric current, where $V = IR$.

SI Unit of Resistance

The ohm ($\Omega$).

Equation for Resistance ($R$)

$R = \rho \frac{l}{A}$ where $l$ is length, $A$ is area of cross-section, and $\rho$ is resistivity.

Resistivity ($\rho$)

The resistance of a conductor having unit length and unit area of cross section.

SI Unit of Resistivity

Ohm-metre ($\Omega \cdot m$).

Resistivity Equation in terms of Relaxation Time

$\rho = \frac{m}{ne^2\tau}$.

Conductance ($C$)

The reciprocal of resistance ($C = \frac{1}{R}$).

SI Unit of Conductance

$ohm^{-1}$ or mho.

Conductivity ($\sigma$)

The reciprocal of resistivity ($\sigma = \frac{1}{\rho}$).

SI Unit of Conductivity

$\Omega^{-1}m^{-1}$ or $mho\,m^{-1}$.

Conductivity Equation in terms of Relaxation Time

$\sigma = \frac{ne^2\tau}{m}$.

Ohmic Substances

Substances which obey Ohm's Law and have a linear $V\text{-}I$ graph, such as metals.

Non-Ohmic Substances

Substances which do not obey Ohm's law and have nonlinear $V\text{-}I$ graphs (e.g., semiconductors, diodes).

Temperature dependence of Resistance (Equation)

$R_2 = R_1[1 + \alpha(T_2 - T_1)]$.

Temperature Coefficient of Resistance ($\alpha$)

The constant $\alpha$ representing how resistance changes with temperature; its unit is $^0C^{-1}$ or $K^{-1}$.

Sign of $\alpha$ for Metals

Positive.

Sign of $\alpha$ for Semiconductors

Negative.

Carbon Resistor Color Codes: Black to Red

Black = $0$, Brown = $1$, Red = $2$.

Carbon Resistor Color Codes: Orange to Green

Orange = $3$, Yellow = $4$, Green = $5$.

Carbon Resistor Color Codes: Blue to White

Blue = $6$, Violet = $7$, Grey = $8$, White = $9$.

Tolerance Color Codes

Gold = $5\%$, Silver = $10\%$, No colour = $$20\%\,.

Effective Resistance (Series)

$R_s = R_1 + R_2 + R_3 + \dots + R_n$.

Effective Resistance (Parallel)

$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$.

Internal Resistance of a Cell ($r$)

The resistance offered by the electrolyte and electrodes inside the cell.

Terminal Voltage ($V$) Equation

$V = E - Ir$, where $E$ is emf, $I$ is current, and $r$ is internal resistance.

Internal Resistance ($r$) formula

$r = \frac{E - V}{I}$.

Maximum current from a storage battery

$I_{max} = \frac{E}{r}$ (when external resistance $R = 0$).

Current Equation for $n$ cells in Series

$I = \frac{nE}{R + nr}$.

Kirchhoff’s 1st Rule (Junction Rule)

The algebraic sum of the currents meeting at a junction is zero (Total current entering = Total current leaving).

Conservation Law for Kirchhoff’s 1st Rule

Law of conservation of charge.

Kirchhoff’s 2nd Rule (Loop Rule)

In a closed circuit, the algebraic sum of the voltages is zero.

Conservation Law for Kirchhoff’s 2nd Rule

Law of conservation of energy.

Wheatstone’s Bridge Balancing Condition

$\frac{P}{R} = \frac{Q}{S}$ (when galvanometer current $I_g = 0$).

Meter Bridge Principle

It works on the Wheatstone’s bridge principle.

Meter Bridge Unknown Resistance ($X$) Formula

$X = \frac{Rl}{100 - l}$.

Potentiometer Principle

When steady current flows through a wire of uniform area, the potential difference across any part is directly proportional to its length ($V \propto l$).