NCERT Plus Two (Class 12) Physics • Chapter 3

Current Electricity

Electric current, drift velocity & mobility, Ohm's law (J = σE), resistivity & temperature dependence, EMF & internal resistance, cell combinations, Kirchhoff's rules & Wheatstone bridge.

📄 Official NCERT Textbook PDF

Topic 3.2 – 3.4: Electric Current, Drift Velocity & Mobility Derivation

Flip a wall switch in your Kochi bedroom and the ceiling fan spins the VERY SAME INSTANT — yet each individual electron inside the copper wire only crawls at a snail's pace of 1 mm/s! Why? Because pushing one electron sends an electromagnetic domino ripple through the entire wire at near light speed!
CURRENT (SCALAR!)I = dqdt (Ampere, C/s)  |  Current Density J = IA (Vector!)
DRIFT VELOCITYvd = e E τm  |  Current link: I = n e A vd
MOBILITY (μ)μ = |vd|E = e τm (Unit: m² V⁻¹ s⁻¹, always positive!)
🧠 Memory Trick: Remember "I = n e A v_d" as "I = n-e-A-v-d (Needy!)". Electrons drift at tortoise speed (~1 mm/s) opposite to E, while thermal jitter is rabbit-fast (~10⁵ m/s)!
💡 Simple Kid-Check: Without a battery, free electrons bounce around metal atoms wildly like bees inside a closed jar (huge random speed, ZERO net progress!). Connect a battery, and a gentle steady breeze (E) makes the whole swarm slowly drift toward the positive end!
  • 1
    Scalar Current vs Vector Current Density: Even though current has direction along a wire, I is a SCALAR (wire currents add algebraically, NOT by vector parallelogram rule!). Current density J = I/A n̂ IS a vector!
  • 2
    Drift Velocity Derivation: Electric field accelerates each electron with a = −eE/m. Between collisions (average relaxation time τ), random thermal velocities average to zero, leaving drift velocity vd = −(eEm)τ.
  • 3
    Relating I and vd: In time Δt, electrons inside cylinder volume A vd Δt cross the section: Δq = (n A vd Δt) e ⇒ I = n e A vd (and J = n e vd).
🧒 Super-Easy Check: A copper wire has n ≈ 8.5 × 10²⁸ m⁻³, area 1 mm², carrying 1.36 A. Then vd = I/(neA) ≈ 10⁻⁴ m/s = 0.1 mm/s — slower than an ant, yet the bulb lights instantly because the electric field wave travels at ~c!
Watch three classic 1-mark traps: (1) Is electric current a vector or scalar? SCALAR! Because currents at a junction add like numbers (3 A + 4 A = 7 A regardless of wire angle!), whereas current density J is a vector! (2) If a wire has varying cross-section (thick to narrow), steady current I stays constant, so drift speed vd ∝ 1/A is FASTER in the narrow section! (3) Mobility μ = |vd|/E is defined as POSITIVE for both electrons and holes.

Topic 3.5 – 3.6: Ohm's Law (J = σE) & Ohmic vs Non-Ohmic Graphs

Push water harder through a garden hose and the water flow rises smoothly in step — double the pressure, double the flow! George Simon Ohm discovered the exact same tidy rule for metals at steady temperature: double the voltage across a nichrome wire and the current doubles cleanly!
OHM'S LAWV = I R (Constant T!)  |  Microscopic form: J = σ E
RESISTANCE & ρR = ρ lA (Unit: Ω)  |  ρ = mn e² τ = 1σ
NON-OHMIC CASESCurved I–V (Bulb), asymmetric (p-n Diode), negative slope (GaAs)!
🧠 Memory Trick: Check axes on I–V vs V–I graphs! On V on y-axis vs I on x-axis, Slope = R; on I on y-axis vs V on x-axis, Slope = 1/R!
💡 Simple Kid-Check: Ohm's Law is NOT a universal law of nature like Newton's gravity — it's an everyday habit obeyed nicely by cool metals (Ohmic), while diodes, LEDs, and glowing bulb filaments happily break it (Non-ohmic)!
  • 1
    Microscopic Derivation (J = σE): Substitute vd = eEτ/m into J = n e vd: J = (n e² τm) E = σ E! Thus conductivity σ = ne²τ/m and resistivity ρ = m / (ne²τ)!
  • 2
    Macroscopic Link: Using E = V/l and J = I/A inside V/l = ρ (I/A) gives V = (ρ lA) I = R I!
  • 3
    3 Ways Devices Break Ohm's Law (NCERT 3.6):
    (a) Non-proportionality: Filament bulb heats up at high V, raising R so I bends downward.
    (b) Sign depends on polarity: p-n junction diode lets huge current pass forward and tiny microamps in reverse.
    (c) Multi-valued V for same I: Gallium Arsenide (GaAs) shows a negative-resistance region where rising V drops I!
Top Kerala HSE graph trap: Examiners draw two straight lines T₁ and T₂ on a graph — ALWAYS read the axes first! If I is on the y-axis and V is on the x-axis, the STEEPER line has larger 1/R ⇒ LOWER resistance R (cooler temperature)! If V is on y-axis, steeper means HIGHER R!

Topic 3.7 – 3.9: Resistivity, Temperature Dependence & Electric Power

Compare the bright orange heating coil inside an iron box (Nichrome) with the thick cool copper wires inside your wall! Why does the coil glow white-hot while the wall wire stays cool? Because resistance, temperature coefficient α, and heating power I²R control where energy drops!
WIRE STRETCH TRICKStretch wire to n times length (constant volume) ⇒ Rnew = n² R! (ρ stays SAME!)
TEMP EFFECT (α)ρT = ρ₀[1 + α(T − T₀)] • Metals: α > 0 • Semiconductors: α < 0 • Manganin: α ≈ 0
ENERGY & POWERP = VI = I²R = V²R  |  1 kWh = 3.6 × 10⁶ J
🧠 Memory Trick: Why do we use Manganin / Constantan for lab standard resistors? High resistivity + almost ZERO α! And remember: stretching a wire 2× makes R 4×!
💡 Simple Kid-Check: Heat up a copper wire and its metal ions vibrate harder like restless dancers in a crowded hallway, making electrons collide more often (shorter relaxation time τ ⇒ higher resistance!). Heat up silicon instead, and millions of fresh charge carriers pop free, so resistance DROPS!
  • 1
    Why ρ Depends Only on Material & T: Since ρ = m / (n e² τ), cutting or stretching a wire NEVER changes ρ! When stretched to length l' = n l, area shrinks to A' = A/n (same volume!), so R' = ρ(nl)/(A/n) = n² R!
  • 2
    Three ρ vs T Signatures (Must-Draw Graphs!):
    • Metals (Cu): ρ rises nearly linearly at room temp (curves up gently from tiny residual value near 0 K).
    • Alloys (Nichrome, Manganin): Very large baseline ρ and almost flat line (α ≈ 0).
    • Semiconductors (Si, Ge): ρ drops exponentially with rising T (α < 0)!
  • 3
    Why High-Voltage Power Lines Save Energy: Power delivered to town is P = VI, while cable heat loss is Ploss = I² Rc = P² Rc / V²! Stepping up voltage 10× slashes cable heat loss 100×!
🧒 Classic Bulb Exam Puzzle: A 60 W, 220 V bulb and a 100 W, 220 V bulb are wired in SERIES across 220 V. Which glows brighter? Since R = V²/P, the 60 W bulb has HIGHER resistance! In series both carry identical current I, so Pactual = I²R is bigger for the 60 W bulb — it glows brighter! (In parallel, 100 W wins!)
Watch three giant DHSE traps: (1) Doubling wire length by stretching multiplies R by 4 (because area halves simultaneously!), NOT 2! (2) Does resistivity ρ change when you double wire radius or length? NO! ρ is a material property (changes ONLY with material or temperature!). (3) In semiconductors (Si, Ge), heating DECREASES resistivity (negative α)!

Topic 3.10: Cells, EMF, Internal Resistance & Terminal PD

Have you noticed how a car's headlights dim slightly the instant you turn the ignition key to start the engine? A battery isn't an ideal magic pump — its own acid/electrolyte pushes back with an Internal Resistance r, stealing part of the voltage whenever heavy current flows!
EMF (ε)Open-circuit PD (I = 0); work per unit charge by non-electrostatic force (Unit: Volt!)
DISCHARGING CELLI = εR + r  |  Terminal PD: V = ε − I r (V < ε!)
CHARGING & GRAPHCharging: V = ε + I r (V > ε) • V vs I slope = −r, intercept = ε
🧠 Memory Trick: Think of EMF ε as your salary, Ir as internal tax inside the cell, and V = ε − Ir as take-home pay delivered to load R! When charging, you pay extra: V = ε + Ir!
💡 Simple Kid-Check: Despite its name "Electromotive Force", EMF is NOT a force in Newtons — it's an energy-per-charge push measured in Volts! When zero current leaves the cell, no internal tax Ir is paid, so voltmeter reads pure ε!
  • 1
    Deriving Terminal Potential Difference: External resistor R gets V = IR. Inside cell, chemical electrolyte opposes current with internal resistance r. Total loop drop: ε = IR + Ir = I(R + r) ⇒ I = εR + r and V = ε − Ir!
  • 2
    Formula for Internal Resistance r: Rearranging V = ε − (V/R)r gives r = R (εV − 1). Larger electrode area cuts r; wider plate gap or colder electrolyte raises r!
  • 3
    Maximum Power Transfer Theorem: Power delivered to external load P = I²R = ε² R(R + r)² peaks when external load matches internal resistance: R = r (Pmax = ε² / (4r)).
🧒 Super-Easy Example: A car battery has ε = 12 V and r = 0.4 Ω. If starter draws I = 10 A, terminal voltage drops to V = 12 − (10)(0.4) = 8 V! What's the max possible short-circuit current? Imax = ε/r = 12/0.4 = 30 A!
Two classic Kerala HSE exam traps: (1) Can terminal potential difference V EVER be greater than EMF ε? YES — while CHARGING the battery from an external charger (current enters positive terminal, so V = ε + Ir > ε)! (2) On the V vs I graph of a cell, y-intercept (at I = 0) gives ε, while magnitude of negative slope gives internal resistance r!

Topic 3.11: Series & Parallel Combination of Cells

Ever popped batteries inside a TV remote only to realise putting ONE cell backwards makes the remote barely flicker — or die completely? Cells wired head-to-tail add their pushes (Series), while cells wired side-by-side share the load to deliver heavy current (Parallel)!
SERIES CELLSSame way: εeq = ε₁ + ε₂ • Opposing: εeq = |ε₁ − ε₂| • ALWAYS req = r₁ + r₂!
PARALLEL CELLSεeqreq = ε₁r₁ + ε₂r₂  |  1req = 1r₁ + 1r₂
n IDENTICAL CELLSSeries: (nε, nr) (best if R ≫ r) • Parallel: (ε, r/n) (best if R ≪ r)
🧠 Memory Trick: Flipping one cell backwards in series subtracts its EMF (ε₁ − ε₂), yet its internal resistance STILL ADDS (r₁ + r₂)! And in Parallel, think Conductance-weighted average: εeq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂)!
💡 Simple Kid-Check: Notice the clever cross-product in Parallel cells: εeq = ε₁r₂ + ε₂r₁r₁ + r₂! The cell with SMALLER internal resistance dominates the output voltage — just like the quieter, stronger helper carrying more weight!
  • 1
    Series Derivation: Connect negative of cell 1 to positive of cell 2: VAC = VAB + VBC = (ε₁ − Ir₁) + (ε₂ − Ir₂) = (ε₁ + ε₂) − I(r₁ + r₂). If cell 2 is reversed (+ meets +), εeq = ε₁ − ε₂, yet req = r₁ + r₂!
  • 2
    Parallel Derivation (2 Cells): Same terminal voltage V across both branches: I₁ = (ε₁ − V)/r₁ and I₂ = (ε₂ − V)/r₂. Adding I = I₁ + I₂ gives:
    εeq = ε₁r₂ + ε₂r₁r₁ + r₂  &  req = r₁ r₂r₁ + r₂!
  • 3
    When to Pick Series vs Parallel? For n identical cells: Series gives I = nε/(R + nr) (wins when R ≫ r); Parallel gives I = ε/(R + r/n) = nε/(nR + r) (wins when R ≪ r)!
🧒 Super-Easy Remote-Control Trap: A torch holds 4 identical 1.5 V cells in series (total 6 V). You accidentally insert 1 cell backwards! How much EMF drops? Not just 1.5 V — the flipped cell stops adding +1.5 AND subtracts −1.5, leaving 3(1.5) − 1.5 = 3.0 V (lost 2 × 1.5 = 3 V!), while total internal resistance stays 4r!
Top Kerala HSE 2-mark trap: (1) If n identical cells are in series and m cells are accidentally reversed, net EMF drops by 2mε to (n − 2m)ε (because each flipped cell costs 2ε!), while internal resistance stays nr! (2) If one cell in parallel has reversed polarity, replace +ε₂ with −ε₂ in the numerator: (ε₁r₂ − ε₂r₁)/(r₁ + r₂)!

Topic 3.12 – 3.13: Kirchhoff's Rules & Wheatstone Bridge

When a complex power grid has three batteries and five criss-crossing loops, plain series-parallel shortcuts break down! Gustav Kirchhoff gave engineers two super-simple traffic & altitude accounting rules — and using them, the diamond-shaped Wheatstone Bridge measures unknown resistances to pin-drop accuracy!
1. JUNCTION RULE (KCL)ΣIin = ΣIout (ΣI = 0) → Based on Conservation of Charge!
2. LOOP RULE (KVL)ΣΔV = 0 around any closed loop → Based on Conservation of Energy!
WHEATSTONE BRIDGENull deflection (Ig = 0, VB = VD) when R₁R₂ = R₃R₄!
🧠 Memory Trick: J-C = Junction Rule conserves Charge! L-E = Loop Rule conserves Energy! And when Wheatstone balances (R₁R₄ = R₂R₃), erase the middle galvanometer wire!
💡 Simple Kid-Check: Imagine water pipes meeting at a junction: every drop flowing IN must flow OUT (zero leaks = Junction Rule!). Now hike any closed trail up and down hills back to your starting hut: total altitude climbs minus drops MUST equal zero (Loop Rule!).
  • 1
    Kirchhoff's First Rule (Junction / KCL): At any circuit junction, sum of currents entering equals sum of currents leaving (ΣI = 0), because charge never piles up at a wire node.
  • 2
    Kirchhoff's Second Rule (Loop / KVL): Algebraic sum of potential changes around any closed loop is zero (ΣΔV = 0). Walk rule: crossing resistor along assumed current ⇒ −IR; crossing battery − → + ⇒ +ε!
  • 3
    Wheatstone Bridge Balance Proof: Four resistors form diamond ABDA and BCDB with galvanometer across BD. At null point (Ig = 0):
    • Junction B & D: I₁ = I₃ and I₂ = I₄.
    • Loops ABDA & BCDB give I₁R₁ = I₂R₂ and I₁R₃ = I₂R₄. Dividing both gives R₁ / R₂ = R₃ / R₄!
🧒 Super-Easy Wheatstone Check: A bridge has arms R₁ = 4 Ω, R₂ = 8 Ω, R₃ = 6 Ω, and R₄ = 12 Ω with a 50 Ω galvanometer in the middle. Check cross-products: 4 × 12 = 48 and 8 × 6 = 48 → BALANCED! Galvanometer current Ig = 0 (ignore the middle 50 Ω completely when finding Req!).
Top Kerala HSE 1-mark & 3-mark traps: (1) Always state the conservation laws clearly: First Law = Conservation of Charge; Second Law = Conservation of Energy! (2) In a Wheatstone bridge, swapping the battery and galvanometer STILL leaves the balance condition R₁/R₂ = R₃/R₄ valid! (3) Don't forget: Wheatstone bridge is most sensitive when all four resistances are of the same order of magnitude.

Solved Previous Year Questions (Kerala DHSE & NCERT)

Q1 (Kerala HSE March • 3 Marks): Define drift velocity. Derive the relation I = n e A v_d and hence obtain the microscopic form of Ohm's law J = σ E.

Drift velocity (vd): Average velocity gained by free electrons opposite to applied electric field E, vd = eEτ/m.
In time Δt, charge crossing area A is Δq = n (A vd Δt) e ⇒ I = n e A vd.
Dividing by A: J = n e vd = (ne²τ/m) E = σ E.

Q2 (Kerala HSE Say • 2 Marks): State any two limitations of Ohm's law. Sketch the I–V characteristic of a p-n junction diode or GaAs.

1. Non-linear V–I relation at high currents/temperatures (e.g. filament bulb).
2. Depends on sign of V (asymmetric in p-n diode) or gives non-unique V for same I (negative resistance region in GaAs).

Q3 (Kerala HSE March • 2 Marks): Why are alloys like Manganin or Constantan used for making standard laboratory resistors?

1. They possess high resistivity (allowing compact coils of reasonable length).
2. They have a negligibly small temperature coefficient of resistivity (α ≈ 0), so their resistance stays virtually constant despite room temperature swings.

Q4 (Kerala HSE Model • 2 Marks): A wire of resistance 8 Ω is stretched to triple its original length at constant temperature. Calculate its new resistance.

Volume remains constant: A l = A' l' with l' = 3l ⇒ A' = A/3.
R' = ρ l'/A' = ρ (3l)/(A/3) = 9 (ρl/A) = 9 × 8 = 72 Ω.

Q5 (Kerala HSE March • 2 Marks): The V vs I graph for a cell shows y-intercept 6 V and hits the I-axis at 3 A. Find its EMF and internal resistance.

Using V = ε − Ir:
• At I = 0 (y-intercept): ε = 6 V.
• At V = 0 (short circuit I = 3 A): 0 = 6 − 3r ⇒ r = 2 Ω.

Q6 (Kerala HSE Say • 3 Marks): Two cells of EMFs ε₁, ε₂ and internal resistances r₁, r₂ are connected in parallel. Derive expressions for equivalent EMF and internal resistance.

Both cells share terminal PD V: I₁ = (ε₁ − V)/r₁, I₂ = (ε₂ − V)/r₂.
Total current I = I₁ + I₂ = (ε₁/r₁ + ε₂/r₂) − V(1/r₁ + 1/r₂).
Comparing with I = εeq/req − V/req:
req = r₁r₂/(r₁ + r₂) and εeq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂).

Q7 (Kerala HSE March • 3 Marks): State Kirchhoff's laws. Apply them to obtain the balance condition of a Wheatstone bridge.

KCL: ΣI = 0 at junction (charge conservation). KVL: ΣΔV = 0 in closed loop (energy conservation).
In Wheatstone bridge with Ig = 0: I₁ = I₃ and I₂ = I₄.
Loop ABDA: −I₁P + I₂R = 0 ⇒ I₁P = I₂R.
Loop BCDB: −I₁Q + I₂S = 0 ⇒ I₁Q = I₂S.
Dividing gives P / Q = R / S.

Q8 (NCERT / DHSE Numerical • 2 Marks): Two bulbs rated 40 W, 220 V and 100 W, 220 V are connected in series across 220 V. Which bulb consumes more power and glows brighter?

Bulb resistances: R = V²/P ⇒ R₄₀ = 220²/40 = 1210 Ω, R₁₀₀ = 220²/100 = 484 Ω.
In series both carry equal current I, so Pactual = I² R ⇒ the 40 W bulb (higher resistance) dissipates more heat and glows brighter!