Class 12 : Physics (English) – Chapter 3: Current Electricity

EXPLANATION & SUMMARY

🔵 Introduction

Electricity is the foundation of modern life. While electrostatics explained charges at rest, current electricity studies charges in steady motion. This chapter explores:

🔵 What electric current is and how it originates microscopically
🟢 Laws governing current flow (Ohm’s law, Kirchhoff’s laws)
🔴 Dependence of resistance on material and temperature
🟡 Circuit analysis using Wheatstone bridge, meter bridge, and potentiometer
⚡ Conversion of galvanometer into ammeter/voltmeter

🟢 1. Electric Current

Definition: Current is the rate of flow of charge through a conductor.
➡️ Formula: I = dq/dt

✔️ Unit: Ampere (A) → 1 A = 1 C/s.

💡 Concept: Current is scalar, but has direction along flow of positive charge.

🔵 Conventional current = positive charge motion.
🔴 Actual conduction in metals = electrons moving opposite.

🟡 2. Current Density (J)

Current per unit area perpendicular to flow.
➡️ Formula: J = I/A
➡️ Vector form: J = σE

✔️ Unit: A m⁻²

💡 Concept: Current density shows “intensity” of current inside conductor.

✏️ Note: If A increases, J decreases for constant I.

🔴 3. Drift Velocity & Mobility

Inside conductors:

Free electrons move randomly → average velocity = 0.

When field E applied → electrons gain drift velocity (vd).

➡️ Formula: vd = (eEτ)/m
➡️ Current relation: I = neAvd

🔵 n = number density of electrons
🟢 A = cross-sectional area
🔴 τ = relaxation time
🟡 e = electron charge

✔️ Mobility (μ): μ = vd/E = eτ/m

💡 Concept: Larger mobility → higher conductivity.

✏️ Note: Typical drift velocity is ~10⁻⁴ m/s, very small compared to random electron speed (~10⁶ m/s).

🟢 4. Ohm’s Law

Statement: V ∝ I (constant T, physical state).
➡️ Formula: V = IR

✔️ Graph → straight line for metals (ohmic).

🔵 Ohmic conductors → copper, aluminium.
🔴 Non-ohmic → diode, filament lamp.

✏️ Note: Ohm’s law fails when material properties change with temperature, electric field, etc.

🔵 5. Resistance & Resistivity

Resistance: R = ρ (L/A)

ρ = resistivity

L = length

A = cross-sectional area

✔️ Resistivity (ρ): property of material.
✔️ Conductivity (σ): reciprocal of ρ.

💡 Concept: Resistance depends on size/shape; resistivity depends only on nature.

🟡 6. Temperature Dependence

Metals: R ↑ with T
Formula: R = R₀(1 + αΔT)

Semiconductors: R ↓ with T

✔️ α = temperature coefficient of resistance.

✏️ Note: Thermistors use negative temperature coefficient (NTC).

🔴 7. Electric Power & Heating

Work done: W = qV

Power: P = VI = I²R = V²/R

Heat: H = I²Rt (Joule’s law).

💡 Concept: Electrical energy always partly converted into heat due to collisions.

Applications:
🔵 Fuses
🟢 Electric heaters
🔴 Bulbs
🟡 Electric irons

🟢 8. Resistor Combinations

Series: Rs = R₁ + R₂ + …
➡️ Same current flows, voltages add.

Parallel: 1/Rp = 1/R₁ + 1/R₂ + …
➡️ Same voltage, currents add.

✏️ Note: Effective resistance in parallel is always smaller than smallest resistor.

🔵 9. Cells, EMF & Internal Resistance

EMF (ε) = work done per unit charge by source.

For cell of emf ε, internal resistance r, external R:
ε = I(R + r)

Combination of cells:
🔵 Series: ε = ε₁ + ε₂ + … , r = r₁ + r₂ + …

🟢 Parallel: ε same, 1/r = 1/r₁ + 1/r₂ …

💡 Concept: Internal resistance increases with age of cell.

🟡 10. Kirchhoff’s Laws

KCL: Σ incoming = Σ outgoing (charge conservation).

KVL: Σ potential changes in loop = 0 (energy conservation).

✏️ Note: Used when circuits are too complex for Ohm’s law.

🔴 11. Wheatstone Bridge

Four resistances form quadrilateral.

Balance condition: R₁/R₂ = R₃/R₄.

✔️ Used for measuring unknown resistances.

💡 Concept: Balance point is independent of emf of cell.

🟢 12. Meter Bridge

Practical Wheatstone bridge with uniform resistance wire.

Unknown resistance found by balancing length method.

✏️ Note: Accuracy increases with longer bridge wire.

🔵 13. Potentiometer

Long uniform wire connected to voltage source.

Principle: Potential drop ∝ length of wire.

Applications:
🔵 Compare emf of two cells.
🟢 Measure internal resistance.
🔴 Measure small potential differences.

💡 Concept: More accurate than voltmeter because it does not draw current from the circuit.

🟡 14. Instruments

Galvanometer: Detects small current.

Ammeter: Galvanometer + low resistance in parallel.

Voltmeter: Galvanometer + high resistance in series.

✏️ Note: Ideal ammeter has zero resistance; ideal voltmeter has infinite resistance.

⚡ 15. Applications in Life

Household circuits

Electrical appliances

Lab measurements (meter bridge, potentiometer)

Electrical power transmission

✅ Summary (~300 words)

🔵 Current: Flow of charges, I = dq/dt.
🟢 Drift velocity: vd = eEτ/m, mobility μ = eτ/m.
🔴 Ohm’s law: V = IR, ohmic vs non-ohmic.
🟡 Resistance: R = ρL/A; ρ material-dependent.
⚡ Temperature effect: Metals ↑ R with T, semiconductors ↓ R with T.
✔️ Heating effect: H = I²Rt, basis of appliances.
🔵 Resistor combinations: Series (add), Parallel (reciprocals add).
🟢 Cells: ε = I(R + r). Series and parallel combinations.
🔴 Kirchhoff’s Laws: Charge and energy conservation.
🟡 Wheatstone Bridge: Condition R₁/R₂ = R₃/R₄.
⚡ Meter Bridge: Practical Wheatstone bridge.
✔️ Potentiometer: Most accurate device, does not draw current.
🧠 Instruments: Galvanometer, Ammeter, Voltmeter.

This chapter builds the foundation of circuit theory, practical measurement, and microscopic electron motion.

📝 Quick Recap

⚡ Current = dq/dt.

🧠 Drift velocity & mobility explain microscopic origin.

🔵 Ohm’s law → V = IR.

🟢 R = ρL/A, resistivity material-specific.

🔴 Joule’s law: H = I²Rt.

🟡 Kirchhoff’s laws → circuit analysis.

✔️ Potentiometer > Voltmeter in accuracy.

⚡ Practical devices: Wheatstone bridge, meter bridge, galvanometer conversions.

————————————————————————————————————————————————————————————————————————————

QUESTIONS FROM TEXTBOOK

Question 3.1

The storage battery of a car has an emf of 12 V. If the internal resistance of the battery is 0.4 Ω, what is the maximum current that can be drawn from the battery?

Answer
➡️ Given: ε = 12 V, r = 0.4 Ω

Formula: I = ε / r

➡️ Substitution:
I = 12 / 0.4 = 30 A

🔴 Final Answer: Maximum current = 30 A

✏️ Note: Maximum current occurs when external resistance = 0.

Question 3.2

A battery of emf 10 V and internal resistance 3 Ω is connected to a resistor. If the current in the circuit is 0.5 A, what is the resistance of the resistor? Also, find the terminal voltage of the battery when the circuit is closed.

Answer
➡️ Given: ε = 10 V, r = 3 Ω, I = 0.5 A

1️⃣ Finding R:
ε = I(R + r)
10 = 0.5(R + 3)
R + 3 = 20
R = 17 Ω

2️⃣ Terminal voltage:
V = IR = 0.5 × 17 = 8.5 V

🔴 Final Answer:

Resistance of resistor = 17 Ω

Terminal voltage = 8.5 V

💡 Concept: Terminal voltage < emf due to internal resistance.

Question 3.3

At room temperature (27.0 °C) the resistance of a heating element is found to be 100 Ω. What is the temperature of the element if the resistance is found to be 117 Ω, given that the temperature coefficient of the material of the resistor is 1.70 × 10⁻⁴ °C⁻¹?

Answer
➡️ R₀ = 100 Ω, R = 117 Ω, T₀ = 27 °C, α = 1.7 × 10⁻⁴ °C⁻¹

Formula: R = R₀[1 + α(T – T₀)]

117 = 100[1 + 1.7 × 10⁻⁴(T – 27)]
1.17 = 1 + 1.7 × 10⁻⁴(T – 27)
0.17 = 1.7 × 10⁻⁴(T – 27)

T – 27 = 1000
T = 1027 °C

🔴 Final Answer: Temperature = 1027 °C

✏️ Note: High temperature rise shows why heating coils glow red-hot.

Question 3.4

A negligibly small current is passed through a wire of length 15 m and uniform cross-section 6.0 × 10⁻⁷ m², and its resistance is measured to be 5.0 Ω. What is the resistivity of the material at the temperature of the experiment?

Answer
➡️ R = 5 Ω, L = 15 m, A = 6.0 × 10⁻⁷ m²

ρ = RA / L = (5 × 6.0 × 10⁻⁷) / 15
ρ = 2.0 × 10⁻⁷ Ω·m

🔴 Final Answer: Resistivity = 2.0 × 10⁻⁷ Ω·m

💡 Concept: Resistivity is intrinsic to the material, independent of size.

Question 3.5

A silver wire has a resistance of 21 Ω at 27.5 °C and resistance of 21.7 Ω at 100 °C. Determine the temperature coefficient of resistivity of silver.

Answer
➡️ R₁ = 21 Ω, T₁ = 27.5 °C, R₂ = 21.7 Ω, T₂ = 100 °C

Formula: R₂ = R₁[1 + α(T₂ – T₁)]

21.7 = 21[1 + α(72.5)]
21.7 / 21 = 1 + 72.5α
1.0333 – 1 = 72.5α
0.0333 = 72.5α

α = 4.59 × 10⁻⁴ °C⁻¹

🔴 Final Answer: Temperature coefficient = 4.6 × 10⁻⁴ °C⁻¹

✏️ Note: Silver has high conductivity but also noticeable temperature dependence.

Question 3.6

A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to a steady value of 2.8 A. What is the steady temperature of the heating element if the room temperature is 27.0 °C? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10⁻⁴ °C⁻¹.

Answer
➡️ V = 230 V, I₁ = 3.2 A, I₂ = 2.8 A, T₀ = 27 °C, α = 1.7 × 10⁻⁴ °C⁻¹

1️⃣ Initial resistance: R₁ = V/I₁ = 230/3.2 = 71.9 Ω
2️⃣ Final resistance: R₂ = V/I₂ = 230/2.8 = 82.1 Ω
3️⃣ Relation: R₂ = R₁[1 + α(T – T₀)]

82.1 = 71.9[1 + 1.7 × 10⁻⁴(T – 27)]
82.1/71.9 = 1 + 1.7 × 10⁻⁴(T – 27)
1.142 = 1 + 1.7 × 10⁻⁴(T – 27)
0.142 = 1.7 × 10⁻⁴(T – 27)

T – 27 = 835
T = 862 °C

🔴 Final Answer: Steady temperature = 862 °C

💡 Concept: Resistance increases with temperature until equilibrium is reached.

Question 3.7

Determine the current in each branch of the network shown in Fig. 3.20.

Answer
➡️ The network has arms: (10 Ω, 5 Ω) in upper branch and (10 Ω, 5 Ω) in lower branch with 10 V batteries.

1️⃣ Check for balance:
Ratio 10/5 = 2, same for other branch → bridge balanced.
Therefore, current in galvanometer = 0.

2️⃣ Equivalent resistance of each arm:

Upper branch: R = 10 + 5 = 15 Ω

Lower branch: R = 10 + 5 = 15 Ω

3️⃣ Current distribution:
Total emf = 10 V + 10 V = 20 V across parallel 15 Ω and 15 Ω.

Equivalent R = (15 × 15)/(15 + 15) = 7.5 Ω

Total current = 20/7.5 = 2.67 A

Since arms are equal, current divides equally:

Current in upper branch = 1.33 A

Current in lower branch = 1.33 A

Current through galvanometer = 0 A

🔴 Final Answer:

Upper 10 Ω = 0.89 A, upper 5 Ω = 0.44 A

Lower 10 Ω = 0.89 A, lower 5 Ω = 0.44 A

Galvanometer = 0 A

✏️ Note: Balance condition simplifies current distribution; galvanometer is bypassed.

Question 3.8

A storage battery of emf 8.0 V and internal resistance 0.5 Ω is being charged by a 120 V DC supply using a series resistor of 15.5 Ω. What is the terminal voltage of the battery during charging? What is the purpose of having a series resistor in the charging circuit?

Answer
➡️ ε = 8 V, r = 0.5 Ω, V = 120 V, R = 15.5 Ω

1️⃣ Current: I = (V – ε)/(R + r) = (120 – 8)/(15.5 + 0.5) = 112/16 = 7 A
2️⃣ Terminal voltage: Vt = ε + Ir = 8 + 7 × 0.5 = 11.5 V
3️⃣ Purpose: Series resistor prevents excessive current → protects battery.

🔴 Final Answer:

Terminal voltage = 11.5 V

Series resistor ensures safe charging.

💡 Concept: During charging, terminal voltage is greater than emf.

Question 3.9

The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5 × 10²⁸ m⁻³. How long does an electron take to drift from one end of a wire 3.0 m long to its other end? The area of cross-section of the wire is 2.0 × 10⁻⁶ m² and it is carrying a current of 3.0 A.

Answer
➡️ n = 8.5 × 10²⁸ m⁻³, L = 3.0 m, A = 2.0 × 10⁻⁶ m², I = 3 A, e = 1.6 × 10⁻¹⁹ C

Formula: I = neAvd

vd = I / (neA)
= 3 / [(8.5 × 10²⁸)(1.6 × 10⁻¹⁹)(2 × 10⁻⁶)]
= 3 / (2.72 × 10⁴)
= 1.1 × 10⁻⁴ m/s

Time: t = L/vd = 3 / (1.1 × 10⁻⁴) = 2.7 × 10⁴ s (~7.5 h)

🔴 Final Answer: Drift time ≈ 2.7 × 10⁴ s (7.5 hours)

✏️ Note: Drift velocity is extremely small → current propagates due to electric field, not actual electron speed.

————————————————————————————————————————————————————————————————————————————

OTHER IMPORTANT QUESTIONS FOR EXAMS

(CBSE MODEL QUESTION PAPER)

ESPECIALLY MADE FROM THIS LESSON ONLY

Section A – MCQs (Q1–Q18)
Question 1

The SI unit of current density is:
🔵 (A) A·m
🟢 (B) A/m²
🟠 (C) A·m²
🔴 (D) A/m
Answer: (B) A/m²

Question 2

Which statement about drift velocity is correct?
🔵 (A) It is in the same direction as electric field.
🟢 (B) It is opposite to electric field.
🟠 (C) Equal to random velocity of electrons.
🔴 (D) Independent of relaxation time.
Answer: (B) Opposite to electric field

Question 3

If current is 2 A, 4 C of charge flows in how much time?
🔵 (A) 0.5 s
🟢 (B) 2 s
🟠 (C) 8 s
🔴 (D) 4 s
Answer: (A) 0.5 s

Question 4

Mobility of charge carriers is defined as:
🔵 (A) vd / E
🟢 (B) σ / ρ
🟠 (C) E / vd
🔴 (D) J / E
Answer: (A) vd / E

Question 5

The resistance of a conductor depends upon:
🔵 (A) length only
🟢 (B) area only
🟠 (C) material, length, and area
🔴 (D) emf applied
Answer: (C) Material, length, and area

Question 6

In a metal, resistivity with temperature:
🔵 (A) decreases
🟢 (B) increases
🟠 (C) remains constant
🔴 (D) first increases then decreases
Answer: (B) Increases

Question 7

When two resistors are connected in parallel, their equivalent resistance is:
🔵 (A) always greater than largest resistor
🟢 (B) always less than smallest resistor
🟠 (C) average of the two
🔴 (D) sum of the two
Answer: (B) Always less than smallest resistor

Question 8

Ohm’s law fails in:
🔵 (A) metallic conductors at constant temperature
🟢 (B) semiconductors and diodes
🟠 (C) alloys
🔴 (D) copper wire
Answer: (B) Semiconductors and diodes

Question 9

Power dissipated in resistor =
🔵 (A) I²R
🟢 (B) V²/R
🟠 (C) VI
🔴 (D) All of these
Answer: (D) All of these

Question 10

The SI unit of resistivity is:
🔵 (A) Ω·m
🟢 (B) Ω/m
🟠 (C) Ω·m²
🔴 (D) Ω/m²
Answer: (A) Ω·m

Question 11

For a cell, terminal voltage is:
🔵 (A) always greater than emf
🟢 (B) always less than emf
🟠 (C) may be less or greater depending on condition
🔴 (D) independent of internal resistance
Answer: (C) May be less or greater depending on condition

Question 12

Kirchhoff’s junction law is based on conservation of:
🔵 (A) Energy
🟢 (B) Charge
🟠 (C) Momentum
🔴 (D) Current density
Answer: (B) Charge

Question 13

In Wheatstone bridge, galvanometer shows no deflection when:
🔵 (A) R1R3 = R2R4
🟢 (B) R1/R2 = R3/R4
🟠 (C) R1+R3 = R2+R4
🔴 (D) Potential difference across galvanometer = ∞
Answer: (B) R1/R2 = R3/R4

Question 14

Potentiometer is preferred over voltmeter because:
🔵 (A) it is cheaper
🟢 (B) it measures resistance
🟠 (C) it does not draw current
🔴 (D) it is smaller
Answer: (C) It does not draw current

Question 15

The drift velocity is proportional to:
🔵 (A) Square of electric field
🟢 (B) Inverse of relaxation time
🟠 (C) Electric field
🔴 (D) Area of cross-section
Answer: (C) Electric field

Question 16

Unit of mobility is:
🔵 (A) m²/V·s
🟢 (B) V·s/m²
🟠 (C) A·m²/V
🔴 (D) m/s
Answer: (A) m²/V·s

Question 17

The heating effect of current is proportional to:
🔵 (A) I²
🟢 (B) R
🟠 (C) t
🔴 (D) All of these
Answer: (D) All of these

Question 18

A galvanometer can be converted to an ammeter by:
🔵 (A) Connecting low resistance in parallel
🟢 (B) Connecting high resistance in series
🟠 (C) Connecting low resistance in series
🔴 (D) Connecting high resistance in parallel
Answer: (A) Connecting low resistance in parallel

Section B – Short Answer (Q19–Q23)
Question 19

Define current density and write its relation with conductivity and electric field.

Answer

Current density (J) = current per unit area.
➡️ Formula: J = I/A
➡️ Relation: J = σE

σ = conductivity

E = electric field

💡 Concept: Current density gives “intensity” of flow of charges inside a conductor.

Question 20

State Ohm’s law. Give one example of a non-ohmic conductor.

Answer

Statement: At constant temperature, current through a conductor is directly proportional to potential difference across it.
➡️ V = IR

Non-ohmic conductor: Semiconductor diode / filament lamp.

✏️ Note: Ohm’s law is a linear relation but fails when material properties vary with temperature.

Question 21

Why is potentiometer more accurate than a voltmeter?

Answer
✔️ Potentiometer measures potential difference without drawing current.
✔️ Voltmeter draws small current → alters circuit conditions.

💡 Concept: Potentiometer uses null method → infinite resistance → highly accurate.

Question 22

State Kirchhoff’s junction law. Which conservation principle is it based on?

Answer

Law: Algebraic sum of currents at a junction = 0.
➡️ Σ I(in) = Σ I(out)

Conservation principle: Conservation of charge.

✏️ Note: This law assumes no charge accumulation at the junction.

Question 23

Two resistors 3 Ω and 6 Ω are connected in parallel. Find equivalent resistance.

Answer
Formula: 1/R = 1/3 + 1/6 = (2 + 1)/6 = 1/2
➡️ R = 2 Ω

🔴 Final Answer: Equivalent resistance = 2 Ω

Section C – Mid-Length (Q24–Q28)
Question 24

A wire of length 2 m and cross-sectional area 1 mm² has resistance 4 Ω. Find resistivity.

Answer
➡️ Given: L = 2 m, A = 1 × 10⁻⁶ m², R = 4 Ω
ρ = RA/L = (4 × 1 × 10⁻⁶)/2 = 2 × 10⁻⁶ Ω·m

💡 Concept: Resistivity is a material property, independent of length/area.

Question 25

State Joule’s law of heating. Write expression for heat produced.

Answer

Heat produced ∝
🔵 Square of current (I²)
🟢 Resistance (R)
🔴 Time (t)

Formula: H = I²Rt

✏️ Note: This effect is used in heaters, fuses, and electric bulbs.

Question 26

A cell of emf 2 V, internal resistance 0.1 Ω connected to 3.9 Ω resistor. Find current and terminal voltage.

Answer
I = ε / (R + r) = 2 / (3.9 + 0.1) = 2/4 = 0.5 A
Vt = IR = 0.5 × 3.9 = 1.95 V

🔴 Final Answer: Current = 0.5 A, Terminal voltage = 1.95 V

Question 27

What is the principle of Wheatstone bridge? Write its condition of balance.

Answer

Principle: Null deflection method → no current through galvanometer at balance.

Condition: R1/R2 = R3/R4

💡 Concept: Balance point is independent of source emf.

Question 28

A heater coil rated 1000 W, 220 V. Find resistance and current drawn.

Answer
R = V²/P = 220²/1000 = 48.4 Ω
I = V/R = 220/48.4 ≈ 4.55 A

🔴 Final Answer: R = 48.4 Ω, I = 4.55 A

Section D – Long Answer (Q29–Q31)
Question 29

Derive expression for drift velocity of electrons.

Answer
➡️ Consider conductor in electric field E:
1️⃣ Force on electron: F = –eE
2️⃣ Acceleration: a = F/m = –eE/m
3️⃣ Mean free time between collisions = τ
4️⃣ Average velocity gained in one collision time: v = aτ = –(eE/m)τ
5️⃣ This average velocity = drift velocity:
➡️ vd = –eEτ/m

💡 Concept:

Proportional to electric field.

Negative sign → opposite to E.

🔴 Final Expression: vd = –(eEτ)/m

Question 30

Explain principle & working of potentiometer for comparing emf.

Answer

Principle: Potential drop across uniform wire ∝ length.

Working:

Cell ε₁ gives balance length l₁, cell ε₂ gives l₂.

ε₁/ε₂ = l₁/l₂.

💡 Concept: Potentiometer is highly sensitive since no current is drawn.

Question 31

State and explain Kirchhoff’s laws with example.

Answer

KCL: Σ currents at junction = 0 (conservation of charge).

KVL: Σ potential differences in closed loop = 0 (conservation of energy).

✔️ Example: Used in multi-loop circuits with resistors and batteries to calculate unknown currents.

Section E – Case/Extended (Q32–Q33)
Question 32

Wire 1.5 m, A = 1 mm², R = 3 Ω, V = 1.5 V.
(a) Find resistivity. (b) Drift velocity for I = 0.5 A.

Answer
ρ = RA/L = (3 × 10⁻⁶)/1.5 = 2 × 10⁻⁶ Ω·m
vd = I / (neA) = 0.5 / [(8.5 × 10²⁸)(1.6 × 10⁻¹⁹)(1 × 10⁻⁶)] ≈ 3.7 × 10⁻⁵ m/s

🔴 Final Answer: ρ = 2 × 10⁻⁶ Ω·m, vd = 3.7 × 10⁻⁵ m/s

Question 33

Battery emf 12 V, r = 1 Ω, R = 5 Ω.
(a) Find current. (b) Terminal voltage. (c) If another 5 Ω in series, find new values.

Answer
(a) I = ε/(R + r) = 12/6 = 2 A
(b) Vt = IR = 2 × 5 = 10 V
(c) New R = 10 Ω, total = 11 Ω → I = 12/11 = 1.09 A, Vt ≈ 10.9 V

——————————————————————————————————————————————————————————————————————————————————–