Is it correct to say: Confusing terminal voltage V with EMF \varepsilon when current flows?

V equals \varepsilon only for an ideal cell (r = 0) or open circuit (I = 0). Otherwise V = \varepsilon - Ir.

Is it correct to say: Writing R = \rho A/L instead of \rho L/A?

Resistance grows with length, falls with area. Remember: long thin wire → high R.

Is it correct to say: Applying Ohm's law V = IR to non-ohmic devices (diode, thermistor, filament)?

V = IR only for ohmic materials at fixed temperature. For diodes and semiconductors, the V–I graph is non-linear.

Is it correct to say: Missing sign convention in KVL — treating all resistor drops as positive?

Pick a loop direction. If you traverse a resistor along the current, drop is −IR; against current, +IR. EMF: -→+ inside cell is +\varepsilon.

Is it correct to say: Assuming drift velocity is high because current flows fast?

v_d is typically 10⁻⁴ m/s (0.1 mm/s). The *signal* travels at nearly c because the field sets up almost instantly.

Is it correct to say: Using resistivity temperature formula \rho_0(1+\alpha\Delta T) with T in Kelvin vs Celsius inconsistently?

\Delta T is a temperature *difference*, so K and °C give the same number. But T_0 must match \rho_0's reference.

Current Electricity

Drift velocity, Ohm's law, Kirchhoff's laws, Wheatstone bridge, cells and internal resistance — NCERT Class 12 Physics Ch 3

📐 13 formulas✏️ 3 examples🎯 6 practice⚖️ 7 marks🏫 CBSE📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Wheatstone bridge derivation using Kirchhoff's laws is a repeat 3-mark favourite. Practise labelling arms P, Q, R, S consistently.
→Drift velocity derivation of Ohm's law (v_d = eE\tau/m) frequently appears as a 3-mark question — memorise the four-step chain.
→Never forget internal resistance r: terminal voltage V = \varepsilon - Ir, not \varepsilon.
→Units matter: current density J is A/m², resistivity \rho is \Omega\cdot m, conductivity \sigma is S/m. Marks lost for missing units.
→Kirchhoff's laws are always in the exam — practise sign conventions before entering the hall.

📊 Diagram

Cell with internal resistance driving current through an external resistor

📐 Formulas(13)

Electric Current

I = \frac{d q}{d t}

Symbol	Meaning
$I$	Current (A = C/s)
$q$	Charge crossing a section (C)
$t$	Time (s)

Drift Velocity and Current★ Board fav

I = n e A v_{d}

Symbol	Meaning
$n$	Free-electron density (m⁻³)
$e$	Electron charge = 1.6×10⁻¹⁹ C
$A$	Cross-sectional area (m²)
$v_{d}$	Drift velocity (m/s)

Drift Velocity in Terms of Field★ Board fav

v_{d} = \frac{e E τ}{m}

Ohm's Law (microscopic form)

J = σ E, σ = \frac{n e ^{2} τ}{m}

Ohm's Law (macroscopic form)★ Board fav

V = I R

Resistance of a Uniform Conductor

R = \frac{ρ L}{A}

Symbol	Meaning
$ρ$	Resistivity (\Omega\cdot m)
$L$	Length (m)
$A$	Cross-section area (m²)

Temperature Dependence of Resistivity

ρ_{T} = ρ_{0} [1 + α (T - T_{0})]

Kirchhoff's Junction Rule (KCL)

\sum I_{in} = \sum I_{o u t}

Kirchhoff's Loop Rule (KVL)★ Board fav

\sum_{l oo p} ε - \sum_{l oo p} I R = 0

EMF and Terminal Voltage

V = ε - I r

Symbol	Meaning
$ε$	EMF of cell (V)
$r$	Internal resistance (\Omega)
$V$	Terminal voltage (V)

Wheatstone Bridge Balance★ Board fav

\frac{P}{Q} = \frac{R}{S}

Cells in Series and Parallel

ε_{e q}^{ser i es} = ε_{1} + ε_{2}, \frac{ε _{e q}}{r _{e q}} = \frac{ε _{1}}{r _{1}} + \frac{ε _{2}}{r _{2}} (p a r a l l e l)

Electric Power

P = V I = I^{2} R = \frac{V ^{2}}{R}

✏️ Solved Examples

1Solved Exampleeasy4 steps

A copper wire of cross-section 1 mm² carries a current of 1.6 A. If n = 8.5×10²⁸ electrons/m³, find the drift velocity of electrons.

Convert to SI units

A = 1 mm^{2} = 1 0^{- 6} m^{2}, I = 1.6 A

2Solved Exampleboard4 steps

A cell of EMF 2.0 V and internal resistance 0.5 \Omega is connected to an external resistor of 4.5 \Omega. Find (a) current in the circuit, (b) terminal voltage, (c) power dissipated in the external resistor.

Total resistance in circuit

R_{t o t a l} = R + r = 4.5 + 0.5 = 5.0 Ω

3Solved ExampleHOTS4 steps

In a Wheatstone bridge P = 10 \Omega, Q = 15 \Omega, R = 20 \Omega. Find the value of S for balance. If a 2 V battery of negligible internal resistance is connected, find the current drawn from it.

Apply balance condition

\frac{P}{Q} = \frac{R}{S} \Rightarrow S = \frac{QR}{P} = \frac{15 \times 20}{10} = 30 Ω

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Confusing terminal voltage V with EMF \varepsilon when current flows
✓V equals \varepsilon only for an ideal cell (r = 0) or open circuit (I = 0). Otherwise V = \varepsilon - Ir.
2
Writing R = \rho A/L instead of \rho L/A
✓Resistance grows with length, falls with area. Remember: long thin wire → high R.
3
Applying Ohm's law V = IR to non-ohmic devices (diode, thermistor, filament)
✓V = IR only for ohmic materials at fixed temperature. For diodes and semiconductors, the V–I graph is non-linear.
4
Missing sign convention in KVL — treating all resistor drops as positive
✓Pick a loop direction. If you traverse a resistor along the current, drop is −IR; against current, +IR. EMF: -→+ inside cell is +\varepsilon.
5
Assuming drift velocity is high because current flows fast
✓v_d is typically 10⁻⁴ m/s (0.1 mm/s). The *signal* travels at nearly c because the field sets up almost instantly.
6
Using resistivity temperature formula \rho_0(1+\alpha\Delta T) with T in Kelvin vs Celsius inconsistently
✓\Delta T is a temperature *difference*, so K and °C give the same number. But T_0 must match \rho_0's reference.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
A wire of resistance 4 \Omega is stretched to twice its original length. Find its new resistance (volume constant).
Q2
Two resistors 6 \Omega and 12 \Omega are in parallel. This combination is in series with a 4 \Omega resistor. Total resistance?
Q3
A cell of EMF 6 V has internal resistance 1 \Omega. When short-circuited, what current flows?
Q4
The resistance of a metal wire is 20 \Omega at 20°C and 30 \Omega at 120°C. Find its temperature coefficient of resistance.
Q5
In a Wheatstone bridge, if arm resistances are 100, 200, 300 \Omega and the fourth arm has an unknown X, find X for balance (arrangement: 100 opposite 300).
Q6
A 100 W bulb and a 60 W bulb both rated 220 V are connected in series across 220 V mains. Which bulb glows brighter?

📝 Notes

Current Electricity — key concepts

Study of electric charge in motion: how it flows through conductors, what resists it, and how circuits with batteries and resistors behave.

Two pictures of Ohm's law

Microscopic: J = \sigma E — current density is proportional to the electric field inside the conductor. Follows from the drift-velocity model (v_d = eE\tau/m).
Macroscopic: V = IR — the total voltage across a wire is proportional to the current through it. Same physics, different scale.

Both hold only at constant temperature and only for ohmic materials.

Cell, EMF and internal resistance

EMF (\varepsilon): the energy per unit charge supplied by the source when no current flows.
Internal resistance (r): unavoidable resistance inside the cell.
When a current I is drawn, the terminal voltage drops to V = \varepsilon − Ir. When charging, V = \varepsilon + Ir.

Kirchhoff's laws — always applicable

Junction rule (KCL): conservation of charge — \sum I_{in} = \sum I_{out}.
Loop rule (KVL): conservation of energy — \sum \varepsilon = \sum IR around any closed loop.

For multi-loop networks, write one KVL equation per independent loop and one KCL per independent junction.

Wheatstone bridge in one line

Balance ⇔ P/Q = R/S ⇔ no current through galvanometer. This is the basis of the metre-bridge, used to measure an unknown resistance to good precision.

Quick sanity checks

Doubling wire length at constant volume → resistance rises 4× (since A halves too).
Series resistors ⇒ R_{eq} > any single R. Parallel ⇒ R_{eq} < smallest R.
If terminal voltage V < EMF \varepsilon, current is flowing out of the cell (discharge).
1 \Omega × 1 A = 1 V. 1 V × 1 A = 1 W. Cross-check units before finalising numerical answers.