Learning Objectives
- Define electric current, drift velocity, and current density
- Understand Ohm's law and electrical resistance
- Study resistivity and its dependence on temperature
- Analyse series and parallel combinations of resistors
- Apply Kirchhoff's laws and understand Wheatstone bridge and potentiometer
Key Concepts
Electric Current
Current: I = Q/t = dQ/dt (rate of flow of charge). SI unit: ampere (A).
Conventional current flows from higher potential (+) to lower potential (-), opposite to electron flow.
Current density: J = I/A = neν_d (A/m²), where n = number density, e = charge, ν_d = drift velocity.
Drift Velocity
Average velocity acquired by electrons due to the applied electric field.
ν_d = eEτ/m, where τ is the relaxation time (average time between collisions).
I = neAν_d. Drift velocity is very small (≈ 10⁻⁴ m/s), but electrical signal travels at speed of light.
Mobility: μ = ν_d/E = eτ/m (m²/V·s)
Ohm's Law
V = IR (potential difference = current × resistance).
Ohm's law is valid for metallic conductors at constant temperature. Non-ohmic devices: diode, LED, thermistor.
Resistance: R = ρL/A, where ρ = resistivity, L = length, A = cross-sectional area.
SI unit: ohm (Ω). Conductance: G = 1/R (siemens, S).
Resistivity and Temperature Dependence
ρ = m/(ne²τ) (from microscopic model).
Temperature dependence: ρ(T) = ρ₀[1 + α(T - T₀)], where α is the temperature coefficient of resistivity.
For metals: α > 0 (resistivity increases with temperature).
For semiconductors: α < 0 (resistivity decreases with temperature).
For alloys (constantan, manganin): α ≈ 0 (nearly constant).
Combinations of Resistors
Series: R_eq = R₁ + R₂ + R₃ + ... (same current, voltage divides).
Parallel: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + ... (same voltage, current divides).
EMF and Internal Resistance
EMF (ε): Work done per unit charge by the source in maintaining current.
Terminal voltage: V = ε - Ir (during discharge), V = ε + Ir (during charging).
For cells in series: ε_eq = ε₁ + ε₂, r_eq = r₁ + r₂.
For cells in parallel: ε_eq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂), 1/r_eq = 1/r₁ + 1/r₂.
Kirchhoff's Laws
Junction Rule (KCL): Sum of currents at a junction = 0 (ΣI = 0). Based on charge conservation.
Loop Rule (KVL): Sum of potential differences in a closed loop = 0 (ΣV = 0). Based on energy conservation.
Wheatstone Bridge
Balanced condition: P/Q = R/S (no current through galvanometer).
Potentiometer
A device for measuring EMF without drawing current. Principle: V ∝ L (potential drop is proportional to length).
Comparing EMFs: ε₁/ε₂ = L₁/L₂
Internal resistance: r = R(L₁/L₂ - 1)
Advantages over voltmeter: draws no current, measures true EMF.
Electrical Energy and Power
Power: P = VI = I²R = V²/R. SI unit: watt (W).
Electrical energy: E = Pt. 1 kWh = 3.6 × 10⁶ J.
Summary
Electric current is the rate of charge flow, driven by potential difference. Ohm's law (V = IR) applies to ohmic conductors. Resistance depends on material (resistivity), length, and area. Kirchhoff's laws enable circuit analysis. The Wheatstone bridge measures unknown resistance. The potentiometer measures EMF accurately by null method.
Important Terms
- Drift Velocity: Average velocity of electrons in a conductor under electric field
- Resistivity: Material property, ρ = RA/L (Ω·m)
- EMF: Energy per unit charge provided by a source
- Internal Resistance: Resistance offered by the electrolyte and electrodes of a cell
- Wheatstone Bridge: Circuit for measuring unknown resistance
- Potentiometer: Device for measuring EMF by null method
Quick Revision
- I = neAν_d; ν_d = eEτ/m; V = IR
- R = ρL/A; ρ(T) = ρ₀[1 + α(T - T₀)]
- Series: R = R₁ + R₂; Parallel: 1/R = 1/R₁ + 1/R₂
- V = ε - Ir; P = VI = I²R = V²/R
- KCL: ΣI = 0; KVL: ΣV = 0
- Wheatstone: P/Q = R/S; Potentiometer: ε₁/ε₂ = L₁/L₂