Current Electricity
'Current is the flow of charge — but the real story is about ENERGY transfer, not electron speed.'
1. Chapter Overview
This chapter moves from ELECTROSTATICS (charges at rest) to CURRENT ELECTRICITY (charges in motion). Topics include: electric CURRENT (I = dq/dt), OHM'S LAW (V = IR), RESISTIVITY and its temperature dependence, the DRIFT VELOCITY of electrons, KIRCHHOFF'S LAWS (junction and loop rules) for analysing complex circuits, the WHEATSTONE BRIDGE for precise resistance measurement, the POTENTIOMETER for comparing emfs, and ELECTRIC POWER.
2. Electric Current and Drift Velocity
Current
- I = dq/dt. Direction: from + to − (conventional current). Unit: Ampere (A).
- Current density: J = I/A = σE. J = n e v_d.
Drift Velocity
- v_d = (eE/m)τ. 'The small net velocity of electrons under an applied field.'
- I = n e A v_d — where n is number density of free electrons.
3. Ohm's Law and Resistivity
Ohm's Law
- V = IR (at constant temperature). 'The current through a conductor is directly proportional to the potential difference across it.'
- Ohmic materials: V-I graph is a STRAIGHT LINE through origin (metals). Non-ohmic: Diodes, semiconductors, electrolytes.
Resistivity and Conductivity
- Resistivity ρ = RA/L. Unit: Ω·m. Intrinsic property of material.
- Conductivity σ = 1/ρ. Unit: S/m.
- Temperature dependence: ρ = ρ₀[1 + α(T − T₀)]. α = temperature coefficient of resistance.
| Material | Resistivity ρ (Ω·m) | Temperature Coefficient α |
|---|---|---|
| Copper | 1.7×10⁻⁸ | +0.0039/°C |
| Silicon | 2.3×10³ (intrinsic) | −0.075/°C (NEGATIVE) |
| Glass | 10¹⁰ to 10¹⁴ | — |
4. Combination of Resistors
| Combination | Formula | Characteristic |
|---|---|---|
| SERIES | R_eq = R₁ + R₂ + ... | Same current, potential divides |
| PARALLEL | 1/R_eq = 1/R₁ + 1/R₂ + ... | Same potential, current divides |
5. Kirchhoff's Laws
Junction Law (KCL)
- Σ I_in = Σ I_out. 'Charge is CONSERVED at a junction.'
- 'What goes into a junction MUST come out.'
Loop Law (KVL)
- Σ V = 0 around any closed loop. 'Energy is CONSERVED in a closed loop.'
- 'The sum of potential rises equals the sum of potential drops.'
Worked Example 1
Problem: Find the current through the 2 Ω resistor in a circuit with E₁=6V (r₁=1Ω), E₂=4V (r₂=1Ω), and R=2Ω connected in a loop. Solution: Apply KVL: E₁ − Ir₁ − IR − Ir₂ − E₂ = 0 (assuming E₁ > E₂ direction). 6 − I(1) − I(2) − I(1) − 4 = 0 ⇒ 2 − 4I = 0 ⇒ I = 0.5 A.
6. Wheatstone Bridge
- A circuit for measuring UNKNOWN resistance accurately.
- Balanced condition: R₁/R₂ = R₃/R₄ (galvanometer shows ZERO current).
- 'At balance, the ratio of resistances in one branch equals the ratio in the other.'
- Applications: Metre bridge (practical form of Wheatstone bridge).
7. Potentiometer
- A device for comparing emfs WITHOUT drawing current.
- Principle: V = kl (potential drop proportional to length).
- Comparing emfs: E₁/E₂ = l₁/l₂.
- Advantage: It draws ZERO current from the source being measured — gives TRUE emf.
8. Comparison Table: Voltmeter vs Potentiometer
| Aspect | Voltmeter | Potentiometer |
|---|---|---|
| What it measures | Potential difference | Emf or potential difference |
| Current drawn | Small but non-zero | ZERO (null-point method) |
| Accuracy | Limited by internal resistance | VERY HIGH |
| True emf? | No (draws some current) | YES (no current drawn) |
9. Electric Power and Energy
- Power: P = VI = I²R = V²/R. Unit: Watt (W).
- Joule heating: H = I²Rt. 'The heat produced by current in a resistor.'
- Kilowatt-hour (kWh) : Commercial unit of energy. 1 kWh = 3.6×10⁶ J.
10. Common Mistakes
- Conventional current vs electron flow: Conventional current flows from + to −. Electrons flow from − to +. For circuit analysis, use CONVENTIONAL current.
- Kirchhoff's sign convention: 'Potential RISE from − to + (battery), Potential DROP from + to − (battery).' Be consistent in your loop direction.
- Resistors in parallel: 1/R_eq is the SUM of RECIPROCALS. Students often add resistances directly (that's series).
- Voltmeter connection: Always in PARALLEL. Ammeter in SERIES. Connecting them wrong can damage the instrument.
11. CBSE Exam Focus
- Ohm's law and V-I characteristics — ohmic vs non-ohmic materials
- Drift velocity — derivation of I = n e A v_d
- Resistivity — temperature dependence, α
- Kirchhoff's laws — solving complex circuits (junction and loop equations)
- Wheatstone bridge — balanced condition, metre bridge problems
- Potentiometer — comparing emfs, finding internal resistance of a cell
12. Self-Test
Q1: A copper wire of length 2 m and area 0.5 mm² has resistivity 1.7×10⁻⁸ Ω·m. Find its resistance. A1: R = ρL/A = (1.7×10⁻⁸)(2)/(0.5×10⁻⁶) = (3.4×10⁻⁸)/(5×10⁻⁷) = 0.068 Ω.
Q2: Find the current through a 10 Ω resistor when connected to a 12 V battery with internal resistance 2 Ω. A2: I = E/(R+r) = 12/(10+2) = 12/12 = 1 A.
Q3: In a Wheatstone bridge, R₁=2Ω, R₂=4Ω, R₃=3Ω. Find R₄ for balance. A3: R₁/R₂ = R₃/R₄ ⇒ 2/4 = 3/R₄ ⇒ R₄ = 6 Ω.
Q4: A potentiometer wire has length 10 m and resistance 20 Ω. A 2V driver cell is connected. Find the potential gradient. A4: Total current I = 2/20 = 0.1 A. Potential gradient k = IR/L = (0.1×20)/10 = 0.2 V/m.
Q5: Compare the emfs of two cells using a potentiometer. Balance lengths are 80 cm and 60 cm. A5: E₁/E₂ = l₁/l₂ = 80/60 = 4/3. E₁:E₂ = 4:3.
13. Conclusion
Current electricity is the ENGINEERING side of electrostatics:
- OHM'S LAW: 'The simplest relation between V and I — linear, predictable, fundamental.'
- KIRCHHOFF: 'The two rules that let you analyse ANY circuit — no matter how complex.'
- WHEATSTONE BRIDGE: 'Precision resistance measurement — balanced when the galvanometer reads zero.'
- POTENTIOMETER: 'The instrument that measures TRUE emf — no current, no error.'
'Current electricity transforms abstract charge into usable energy — the flow of electrons powers our world.'
