Chapter 3: Current Electricity

ELECTRIC CURRENT

Basic Concept

  • Definition: Rate of flow of electric charge through a cross-section
  • Mathematical Expression: I = dQ/dt
  • Unit: Ampere (A) = Coulomb/second (C/s)
  • Conventional Current Direction: From positive to negative terminal (opposite to electron flow)

Current Density

  • Definition: Current per unit area perpendicular to flow
  • Vector Form: J = nqvd (n = number density, q = charge, vd = drift velocity)
  • Relation to Current: I = J·A (where A is area vector)
  • Unit: A/m²

OHM'S LAW

Statement

Current through a conductor is proportional to potential difference across it (at constant temperature)

\[ V = IR \]
\[ R = \rho \frac{L}{A} \]

where ρ is resistivity, L is length, and A is cross-sectional area

Resistance and Resistivity

  • Resistance (R): Opposition to current flow, unit: Ohm (Ω)
  • Resistivity (ρ): Material property, unit: Ohm-meter (Ω·m)
  • Conductivity (σ): 1/ρ, unit: Siemens/meter (S/m)

Drift Velocity

  • Definition: Average velocity of charge carriers in response to electric field
  • vd = (eE)/(m·τ) = (eE)/m × τ = μE
  • μ = eτ/m (mobility)
  • Typical values: 10⁻⁴ m/s (very slow compared to thermal velocities)

Temperature Dependence

  • For metals: R = R₀[1 + α(T - T₀)]
  • α = temperature coefficient of resistance
  • For semiconductors: Resistance decreases with temperature

LIMITATIONS OF OHM'S LAW

  • Not valid for non-linear elements (diodes, transistors)
  • Not applicable at high fields or frequencies
  • Not valid for semiconductors and gaseous conductors
  • Fails for superconductors (R = 0 below critical temperature)

ELECTRICAL ENERGY AND POWER

Power

\[ P = VI = I^2R = \frac{V^2}{R} \]
  • Electrical power
  • Unit: Watt (W) = Joule/second (J/s)

Joule Heating

  • Heat produced: H = I²Rt
  • Energy dissipated as heat in resistor

CELLS AND BATTERIES

EMF (Electromotive Force)

  • Definition: Work done per unit charge to maintain potential difference
  • Unit: Volt (V)
  • EMF (ε) > Terminal voltage (V) due to internal resistance

Internal Resistance

\[ V = \varepsilon - Ir \]
  • V = terminal voltage
  • r = internal resistance of cell
  • Power delivered to external circuit: P = VI = I(ε - Ir)
  • Maximum power transfer when external resistance equals internal resistance

Cells in Series

  • εeq = ε₁ + ε₂ + ... + εₙ
  • req = r₁ + r₂ + ... + rₙ

Cells in Parallel

  • For identical cells: εeq = ε, req = r/n
  • For different cells: More complex calculation required

KIRCHHOFF'S LAWS

Kirchhoff's Current Law (KCL)

  • Junction Rule: Sum of currents entering a junction equals sum of currents leaving
  • ΣI = 0 (algebraic sum of currents at a junction is zero)
  • Based on conservation of charge

Kirchhoff's Voltage Law (KVL)

  • Loop Rule: Sum of potential differences around any closed loop is zero
  • ΣV = 0 (algebraic sum of potential differences in a closed loop is zero)
  • Based on conservation of energy

ELECTRICAL NETWORKS

Resistors in Series

\[ R_{eq} = R_1 + R_2 + ... + R_n \]
  • Same current through each resistor
  • Voltage divides proportionally to resistance

Resistors in Parallel

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} \]
  • Same voltage across each resistor
  • Current divides inversely proportionally to resistance

Wheatstone Bridge

  • Used for precise resistance measurement
  • Balanced when R₁/R₂ = R₃/R₄
  • No current flows through galvanometer when balanced

Meter Bridge

  • Special case of Wheatstone bridge
  • R₁/R₂ = l₁/l₂ (where l₁, l₂ are lengths of wire)

POTENTIOMETER

Working Principle

  • Measures EMF by balancing against known potential drop
  • No current drawn from source being measured
  • More accurate than voltmeter (which draws current)

Applications

  • Comparing EMFs of cells
  • Measuring internal resistance
  • Measuring small potential differences

KEY FORMULAS

  • Current: I = dQ/dt
  • Current Density: J = nqvd
  • Ohm's Law: V = IR
  • Resistance: R = ρL/A
  • Temperature dependence: R = R₀[1 + α(T - T₀)]
  • Power: P = VI = I²R = V²/R
  • Terminal voltage: V = ε - Ir
  • Resistors in series: Req = R₁ + R₂ + ... + Rₙ
  • Resistors in parallel: 1/Req = 1/R₁ + 1/R₂ + ... + 1/Rₙ
  • Wheatstone bridge balance: R₁/R₂ = R₃/R₄