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Current Electricity

Ch. 9Std 12

Easy Overview

Electric current is simply charges in motion. When you flip a switch and a bulb lights up, charges are flowing through the wire, through the bulb, and back through the circuit. But what determines how much current flows? What happens when you put multiple resistors together? How do we measure current and voltage in complex circuits? Current electricity answers all these questions. Electric current (I) is defined as the rate of flow of charge: I = ΔQ/Δt. The SI unit is the ampere (A), where 1 A = 1 C/s. By convention, current direction is the direction of flow of positive charge — even though in a metal wire, it is actually electrons that move. For current to flow, there must be a potential difference (voltage) across the conductor. Ohm's law states that V = IR, where R is resistance. The unit of resistance is the ohm (Ω). Resistance depends on the material, length, cross-sectional area, and temperature. For a uniform conductor: R = ρL/A, where ρ is resistivity. Copper has low resistivity (1.7 × 10⁻⁸ Ω·m), while rubber has extremely high resistivity (~10¹³ Ω·m). For most metals, resistance increases with temperature: R_T = R₀(1 + αΔT). Circuits with multiple resistors can be analysed using Kirchhoff's laws. Kirchhoff's current law (KCL): sum of currents entering a junction equals sum leaving. Kirchhoff's voltage law (KVL): sum of voltages around any closed loop is zero. These laws, combined with Ohm's law, can solve any resistive circuit. The Wheatstone bridge uses a balanced circuit to measure unknown resistance. A potentiometer measures EMF without drawing current. Understanding these principles is essential for electrical engineering, electronics, and power systems.

Electric Current and Drift Velocity

Electric current I = dQ/dt. In a conductor, free electrons drift slowly under an electric field. Drift velocity v_d ≈ 0.1 mm/s — extremely slow, yet the light turns on instantly because the signal propagates at near light speed. I = nAe v_d, where n is free electron density. Current density J = I/A = ne v_d = σE, where σ = neμ is conductivity.

Ohm's Law and Electrical Resistance

Ohm's law: V = IR (at constant temperature). R = ρL/A. Resistivity ρ is a material property. Conductivity σ = 1/ρ. Ohmic materials (metals at constant temp) give a straight-line V-I characteristic. Non-ohmic materials (diodes, thermistors) have curved V-I characteristics. Ohm's law is empirical, not fundamental.

Resistivity and Conductivity

Resistivity ρ is intrinsic, independent of shape. SI unit: Ω·m. Conductivity σ = 1/ρ. Classification: conductors (ρ ≈ 10⁻⁸ Ω·m), semiconductors (ρ ≈ 10⁻⁵ to 10⁵ Ω·m), insulators (ρ ≈ 10¹² to 10¹⁸ Ω·m). For metals: ρ_T = ρ₀(1 + αΔT). For semiconductors: ρ decreases with temperature (negative α).

Temperature Dependence of Resistance

For metals: R_T = R₀(1 + αΔT). α ≈ 0.0039/°C for copper. For alloys like constantan, α is very small — ideal for standard resistors. For semiconductors: R decreases exponentially with temperature: R = R₀e^(E_g/(2kT)). Thermistors exploit this for temperature sensing.

Combination of Resistors — Series and Parallel

Series: same current, R_eq = R₁ + R₂ + R₃ + ... Equivalent R > any individual. Parallel: same voltage, 1/R_eq = 1/R₁ + 1/R₂ + ... Equivalent R < smallest individual. For two resistors in parallel: R_eq = (R₁R₂)/(R₁ + R₂).

Kirchhoff's Current Law (KCL)

KCL: ΣI_in = ΣI_out at a junction. Equivalently, ΣI = 0 at a junction (taking sign convention). Based on conservation of charge — charge cannot accumulate at a junction. Used to relate currents in different branches.

Kirchhoff's Voltage Law (KVL)

KVL: ΣV = 0 around any closed loop. Based on conservation of energy — returning to starting point gives zero net potential change. Sign convention: battery from − to + = +V, resistor in direction of current = −IR. Used with KCL to solve multiloop circuits.

Wheatstone Bridge

Four resistors P, Q, R, S in a diamond shape. Galvanometer between two opposite nodes. Battery across the other two. Balanced when P/Q = R/S (I_g = 0). Used to measure unknown resistance S = (Q/P)R. Very sensitive — used in strain gauge measurements.

Meter Bridge (Slide Wire Bridge)

A practical Wheatstone bridge using a 1 m uniform wire. Unknown X in one gap, known R in the other. Jockey moved along wire until galvanometer shows zero deflection. If balance at L cm: X = RL/(100 − L). Can measure from fractions of an ohm to thousands of ohms.

Joule Heating and Electrical Power

Power dissipated: P = VI = I²R = V²/R. Heat generated: H = I²Rt (Joule's law). Applications: electric heaters, incandescent bulbs, fuses. Rated power at rated voltage. If voltage changes, P ∝ V² (fixed R).

Electromotive Force (EMF) and Internal Resistance

EMF (ε) is terminal voltage when no current flows. When current flows: V = ε − Ir, where r is internal resistance. Current I = ε/(R + r). Maximum power transfer when load R = r. Potentiometer measures true EMF unaffected by internal resistance.

Potentiometer

A potentiometer measures potential difference without drawing current. A long uniform wire with constant potential gradient. At balance: ε_x = KL (L is balance length). Uses: comparing EMFs, measuring internal resistance, calibrating voltmeters and ammeters.

Electrical Energy Consumption

Energy = P × t. Commercial unit: kilowatt-hour (kWh). 1 kWh = 3.6 × 10⁶ J. A 100 W bulb used for 10 hours = 1 kWh. Electricity bills charge per kWh. Power factor in AC: P = VI cosφ.

Superconductivity

Zero resistance below critical temperature T_c. Discovered by Onnes in 1911 for mercury at 4.2 K. BCS theory: Cooper pairs. High-temperature superconductors (YBCO, T_c = 92 K) cooled by liquid nitrogen. Applications: MRI magnets, particle accelerators, maglev trains, SQUIDs.

Key Points

  • •Current I = dQ/dt. Drift velocity v_d = I/(nAe).
  • •Ohm's law: V = IR. Resistivity ρ = RA/L.
  • •R_T = R₀(1 + αΔT) for metals.
  • •Series: R_eq = ΣRᵢ. Parallel: 1/R_eq = Σ1/Rᵢ.
  • •KCL: ΣI_in = ΣI_out. KVL: ΣV = 0.
  • •Wheatstone bridge balanced: P/Q = R/S.
  • •Meter bridge: X = RL/(100 − L).
  • •Joule heating: P = I²R = V²/R = VI.
  • •EMF: ε = V + Ir. Maximum power when R = r.
  • •Potentiometer: null measurement of EMF.
  • •1 kWh = 3.6 × 10⁶ J.
  • •Superconductivity: R = 0 below T_c.

Practice Questions

  • State Ohm's law. Derive expression for resistivity in terms of relaxation time. Explain temperature dependence for metals and semiconductors.
  • State and explain Kirchhoff's laws. Apply to find currents in a circuit with 2 Ω, 3 Ω, 5 Ω resistors and 4 V, 6 V cells.
  • Describe Wheatstone bridge and derive balance condition. In a meter bridge, balance at 40 cm with known resistor 10 Ω. Find unknown resistance.
  • Derive power dissipated in a resistor. A 100 W, 220 V bulb connected to 110 V. Calculate power consumed.
  • Explain internal resistance. A cell of EMF 2 V, internal resistance 0.5 Ω, connected to 4.5 Ω resistor. Find current and terminal voltage.
  • What is a potentiometer? Explain how it compares EMFs of two cells. Why is it preferred over a voltmeter?
  • Series and parallel combinations. Three resistors 2 Ω, 3 Ω, 6 Ω in parallel. Find equivalent R. If connected to 12 V battery, find total current.
  • Explain Joule's law. A heater draws 5 A from 220 V. Find power and heat in 30 minutes. Cost for 2 hours at ₹8/kWh?