Electric Charges and Fields
विद्युत आवेश तथा क्षेत्र | Class 12 Physics - Chapter 1 | NCERT + RBSE 2026
| Subject: | Physics |
| Class: | 12 (RBSE + NCERT) |
| Chapter: | 1 |
| Topic: | Electric Charges and Fields |
| Year: | 2025-26 |
| Status: | Complete & Verified |
• Elementary charge: e = 1.602 × 10⁻¹⁹ C
• Coulomb constant: k = 9 × 10⁹ N·m²/C²
• Permittivity of free space: ε₀ = 8.854 × 10⁻¹² C²/(N·m²)
• Relationship: k = 1/(4πε₀)
Learning Objectives
After completing this chapter, students will be able to:
- Understand the fundamental nature of electric charge and its three basic properties
- Distinguish between conductors and insulators based on their electrical properties
- Apply Coulomb's law to calculate forces between point charges
- Use the principle of superposition for systems with multiple charges
- Define and calculate electric field for various charge configurations
- Draw and interpret electric field lines
- Calculate electric flux through surfaces
- Analyze electric dipoles and their behavior in external fields
- Apply Gauss's law to find electric fields for symmetric charge distributions
- Solve problems involving continuous charge distributions
1. Introduction
600 BCE — Thales: Amber (elektron) attracts light objects when rubbed
↓
1600 — Gilbert: Coined term "electric", distinguished from magnetism
↓
1785 — Coulomb: Established inverse square law for electric forces
↓
1909 — Millikan: Proved charge quantisation (Q = ne)
↓
Modern Era — Quantum Electrodynamics (complete theory)
Electricity is one of the most fundamental interactions in nature. The study of electrostatics deals with electric charges at rest and the forces, fields, and potentials associated with them.
This chapter forms the foundation for understanding electric potential, capacitors, current electricity, electromagnetic induction, and modern electronics.
2. Electric Charge
2.1 Definition
Electric charge is a fundamental property of certain subatomic particles that causes them to experience electromagnetic forces. Charge is a scalar quantity denoted by q or Q.
- Positive charge (+): Carried by protons
- Negative charge (−): Carried by electrons
Interaction Rule: Like charges repel, unlike charges attract.
LIKE CHARGES (Repulsion):
(+) ←→ (+) | (−) ←→ (−)
UNLIKE CHARGES (Attraction):
(+) →← (−)
2.2 Atomic Origin
NUCLEUS (center):
• Protons: +e = +1.602×10⁻¹⁹ C
• Neutrons: 0 C
ELECTRONS (orbiting):
• Charge: −e = −1.602×10⁻¹⁹ C
• Mass: 9.1×10⁻³¹ kg
Neutral Atom: Protons = Electrons
In ordinary processes, only electrons transfer between objects. Protons remain in the nucleus.
- Negatively charged: Body gains electrons
- Positively charged: Body loses electrons
2.3 Unit of Charge
One coulomb is the charge transported by a current of 1 ampere in 1 second.
Practical units: 1 μC = 10⁻⁶ C, 1 nC = 10⁻⁹ C
3. Conductors and Insulators
3.1 Conductors
Materials in which electric charges (electrons) can move freely.
Metallic lattice with:
• Fixed positive ions
• Free electrons (mobile charge carriers)
• Electrons can move throughout material
Examples: Copper, Aluminum, Silver, Gold, Iron
- Electric field inside is zero (E = 0)
- Excess charge resides only on the surface
- All points are at the same potential (equipotential)
- Electric field just outside is perpendicular to surface
3.2 Insulators
Materials in which electric charges are tightly bound and cannot move freely.
Molecular structure with:
• Electrons bound to atoms/molecules
• No free charge carriers
• Charge remains localized
Examples: Glass, Rubber, Plastic, Wood, Mica
3.3 Comparison
| Property | Conductors | Insulators |
|---|---|---|
| Free electrons | Present (mobile) | Absent (bound) |
| Charge distribution | Redistributes to surface | Remains where placed |
| Internal E field | Zero (equilibrium) | Can be non-zero |
| Resistivity | 10⁻⁸ to 10⁻⁵ Ω·m | 10⁸ to 10¹⁷ Ω·m |
4. Basic Properties of Electric Charge
4.1 Additivity of Charges
Total charge is the algebraic sum of individual charges (scalar addition).
4.2 Conservation of Charge
The total electric charge in an isolated system remains constant. Charge can neither be created nor destroyed; it can only be transferred.
BEFORE: Glass (0) + Silk (0) = 0
↓ (rubbing transfers electrons)
AFTER: Glass (+5nC) + Silk (−5nC) = 0
Total charge conserved!
4.3 Quantisation of Charge
where:
- Q = total charge
- n = integer (0, ±1, ±2, ±3, ...)
- e = 1.602 × 10⁻¹⁹ C (elementary charge)
Charged oil droplets suspended in electric field
By adjusting E to balance gravity:
• Measured charge on droplets
• All charges were INTEGER MULTIPLES of e
• Proved: Q = ne
Problem: A body has charge 4.8 × 10⁻¹⁸ C. How many electrons were removed?
Solution:
Q = ne
n = Q/e = (4.8 × 10⁻¹⁸)/(1.602 × 10⁻¹⁹) = 30
Answer: 30 electrons were removed (positive charge means electron deficit).
5. Coulomb's Law
Coulomb's law describes the electrostatic force between two point charges.
Two charged spheres at distance r
Force measured by twist in suspension wire
Discovered: F ∝ q₁q₂/r²
Or equivalently:
where:
- F = electrostatic force (N)
- q₁, q₂ = charges (C)
- r = distance between charges (m)
- k = 9 × 10⁹ N·m²/C²
- ε₀ = 8.854 × 10⁻¹² C²/(N·m²)
Like charges: Force is repulsive (pushes apart)
Unlike charges: Force is attractive (pulls together)
Force acts along the line joining the charges
Force on q₂ due to q₁:
where r̂₂₁ is unit vector from q₁ to q₂
Problem: Two charges +2μC and −3μC are 30cm apart. Find the force.
Solution:
F = k|q₁q₂|/r²
F = (9×10⁹) × |2×10⁻⁶ × (−3)×10⁻⁶| / (0.3)²
F = (9×10⁹) × (6×10⁻¹²) / 0.09
F = 0.6 N
Answer: 0.6 N (attractive, as charges are opposite)
6. Forces Between Multiple Charges
The net force on a charge due to multiple charges is the vector sum of individual forces:
Each force is calculated independently using Coulomb's law.
To find force on charge q₀ due to q₁, q₂, q₃:
1. Calculate F₁ (force due to q₁ alone)
2. Calculate F₂ (force due to q₂ alone)
3. Calculate F₃ (force due to q₃ alone)
4. Fnet = F₁ + F₂ + F₃ (vector sum)
7. Electric Field
7.1 Definition
Electric field at a point is the force per unit positive test charge.
SI Unit: N/C or V/m
Electric field represents the condition of space around charges. It exists independently of any test charge and mediates the interaction between charges (no action at a distance).
7.2 Field Due to Point Charge
Vector form:
Direction:
- If Q > 0: radially outward from charge
- If Q < 0: radially inward toward charge
Positive charge (+Q):
Field lines radiate outward (E points away)
Negative charge (−Q):
Field lines converge inward (E points toward)
8. Electric Field Lines
- Originate from positive charges, terminate on negative charges
- Tangent at any point gives field direction
- Never intersect each other
- Density indicates field strength (closer = stronger)
- Never form closed loops in electrostatics
- Perpendicular to conductor surface
Single positive charge: Radial lines outward
Single negative charge: Radial lines inward
Electric dipole: Lines from + to −, curved
Two positive charges: Lines repel, null point at center
Why field lines never intersect: If they did, there would be two field directions at one point, which is impossible (field is unique at each point).
9. Electric Flux
For small area:
For extended surface:
SI Unit: N·m²/C or V·m
θ = 0° (perpendicular): φ = EA (maximum)
θ = 60°: φ = EA cos60° = EA/2
θ = 90° (parallel): φ = 0
Sign Convention (closed surface):
- Outward flux: Positive
- Inward flux: Negative
10. Electric Dipole
An electric dipole consists of two equal and opposite charges (+q and −q) separated by distance 2a.
Direction: from −q to +q
SI Unit: C·m
(−q) ←—2a—→ (+q)
Dipole moment p⃗ points from − to +
10.1 Field on Axial Line
For a point on the axis at distance r from center:
Direction: Along dipole axis (from − to +)
10.2 Field on Equatorial Line
For a point on perpendicular bisector at distance r:
Direction: Opposite to dipole moment (from + to −)
Important: Eaxial = 2 × Eequatorial (at same distance)
10.3 Physical Significance of Dipoles
Electric dipoles are important because:
- Many molecules (H₂O, HCl) are permanent dipoles
- Neutral atoms/molecules become dipoles in external fields (polarization)
- Dipole interactions explain many chemical and biological phenomena
- Radio antennas work on dipole principles
11. Dipole in Uniform External Field
Vector form: τ⃗ = p⃗ × E⃗
Torque tends to align dipole with field (θ → 0)
Vector form: U = −p⃗ · E⃗
Stable equilibrium: θ = 0° (minimum U)
Unstable equilibrium: θ = 180° (maximum U)
12. Continuous Charge Distribution
For continuous charge distributions, we use charge density:
| Type | Charge Density | Formula |
|---|---|---|
| Linear | λ (C/m) | dq = λ dl |
| Surface | σ (C/m²) | dq = σ dA |
| Volume | ρ (C/m³) | dq = ρ dV |
Electric field is found by integration: E⃗ = ∫ dE⃗
13. Gauss's Law
The total electric flux through any closed surface equals the enclosed charge divided by ε₀.
- Applies to any closed surface (Gaussian surface)
- Flux depends only on enclosed charge, not on surface shape
- External charges don't contribute to flux
- Equivalent to Coulomb's law but more powerful for symmetric cases
14. Applications of Gauss's Law
14.1 Field Due to Infinitely Long Straight Wire
Linear charge density: λ (C/m)
Gaussian surface: Cylindrical surface coaxial with wire
Result:
Direction: Radially outward from wire (if λ > 0)
14.2 Field Due to Infinite Plane Sheet
Surface charge density: σ (C/m²)
Gaussian surface: Cylindrical pillbox perpendicular to sheet
Result:
Direction: Perpendicular to sheet
Important: Independent of distance from sheet!
14.3 Field Due to Uniformly Charged Spherical Shell
Outside (r > R):
(Same as point charge at center)
Inside (r < R):
Electric field inside hollow shell is zero!
Practice Questions
Section A: Multiple Choice Questions (25 questions)
Q1. The SI unit of electric charge is:
(a) Newton
(b) Coulomb
(c) Ampere
(d) Volt
The SI unit is coulomb (C), defined as 1 A × 1 s.
Q2. Quantisation of charge means:
(a) Charge is conserved
(b) Charge exists in discrete multiples of e
(c) Charge can be transferred
(d) Charge is a scalar
Q = ne, where n is an integer and e = 1.602×10⁻¹⁹ C.
Q3. If distance between two charges is doubled, force becomes:
(a) F/2
(b) F/4
(c) 2F
(d) 4F
F ∝ 1/r². If r → 2r, then F → F/4.
Q4. Electric field inside a conductor in equilibrium is:
(a) Maximum
(b) Zero
(c) Variable
(d) Infinite
Free electrons redistribute to make E = 0 inside.
Q5. Electric flux through a closed surface depends on:
(a) Charge outside
(b) Charge enclosed
(c) Surface area
(d) Surface shape
By Gauss's law: φ = Qenc/ε₀.
Q6. Two electric field lines:
(a) Can intersect
(b) Never intersect
(c) Intersect at null point
(d) Form closed loops
Intersection would mean two field directions at one point, which is impossible.
Q7. Ratio of Eaxial to Eequatorial for dipole at same distance:
(a) 1:1
(b) 2:1
(c) 1:2
(d) 1:4
Eaxial = 2kp/r³, Eequatorial = kp/r³. Ratio = 2:1.
Q8. Elementary charge e equals:
(a) 1.602 × 10⁻¹⁹ C
(b) 1.602 × 10¹⁹ C
(c) 9 × 10⁹ C
(d) 8.854 × 10⁻¹² C
This is the charge on proton/electron (magnitude).
Q9. Electric field due to infinite plane sheet is:
(a) σ/ε₀
(b) σ/(2ε₀)
(c) σ/(4πε₀)
(d) Zero
From Gauss's law for infinite plane sheet.
Q10. Inside a hollow spherical shell, electric field is:
(a) Maximum
(b) Zero
(c) kQ/r²
(d) Uniform non-zero
By Gauss's law, E = 0 inside hollow shell.
Q11. When a glass rod is rubbed with silk:
(a) Electrons transfer from silk to glass
(b) Electrons transfer from glass to silk
(c) Protons transfer from glass to silk
(d) No charge transfer occurs
Glass becomes positive (loses electrons), silk becomes negative (gains electrons).
Q12. Coulomb's constant k equals:
(a) 9 × 10⁹ N·m²/C²
(b) 8.854 × 10⁻¹² C²/(N·m²)
(c) 1.602 × 10⁻¹⁹ C
(d) 6.67 × 10⁻¹¹ N·m²/kg²
k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C².
Q13. Total charge on isolated system:
(a) Increases
(b) Decreases
(c) Remains constant
(d) Becomes zero
Law of conservation of charge.
Q14. SI unit of electric field is:
(a) N/C
(b) C/N
(c) N·C
(d) C·m
E = F/q has unit Newton per coulomb (or V/m).
Q15. For dipole in uniform field, net force is:
(a) Maximum
(b) Zero
(c) pE
(d) 2pE
Forces on +q and −q are equal and opposite, net force = 0. But there is torque.
Q16. Electric field at distance r from infinite line charge varies as:
(a) 1/r
(b) 1/r²
(c) 1/r³
(d) Independent of r
E = λ/(2πε₀r) ∝ 1/r.
Q17. Two identical spheres with +3Q and −Q touch and separate. Final charge on each:
(a) +2Q, 0
(b) +Q, +Q
(c) +3Q, −Q
(d) +1.5Q, +0.5Q
Total charge = +2Q, distributes equally: each gets +Q.
Q18. Torque on dipole is maximum when angle with field is:
(a) 0°
(b) 45°
(c) 90°
(d) 180°
τ = pE sin θ is maximum when sin θ = 1, i.e., θ = 90°.
Q19. Electric field lines are closer where field is:
(a) Weaker
(b) Stronger
(c) Zero
(d) Uniform
Line density indicates field strength.
Q20. Dipole moment has SI unit:
(a) C·m
(b) C/m
(c) N·m
(d) J
p = q × distance has unit coulomb-meter.
Q21. Gauss's law is useful for charge distributions with:
(a) Any shape
(b) High symmetry
(c) Low charge
(d) Moving charges
Gauss's law is most useful for spherical, cylindrical, or planar symmetry.
Q22. Potential energy of dipole is minimum when:
(a) θ = 0°
(b) θ = 90°
(c) θ = 180°
(d) θ = 45°
U = −pE cos θ is minimum when cos θ = 1 (θ = 0°).
Q23. Electric flux has SI unit:
(a) N·m²/C
(b) N/C
(c) C/m²
(d) V
φ = EA has unit (N/C) × m² = N·m²/C or V·m.
Q24. Field outside spherical shell of charge Q at distance r:
(a) kQ/r
(b) kQ/r²
(c) kQ/r³
(d) Zero
Same as point charge at center for r > R.
Q25. Which is a vector quantity?
(a) Charge
(b) Electric field
(c) Electric flux
(d) Potential energy
E⃗ is a vector; charge, flux, and PE are scalars.
Formula Sheet
| Concept | Formula | Unit |
|---|---|---|
| Elementary Charge | e = 1.602 × 10⁻¹⁹ C | C |
| Quantisation | Q = ne | C |
| Coulomb's Law | F = k|q₁q₂|/r² | N |
| Coulomb Constant | k = 9 × 10⁹ N·m²/C² | - |
| Electric Field | E⃗ = F⃗/q | N/C |
| Point Charge Field | E = kQ/r² | N/C |
| Electric Flux | φ = EA cos θ | N·m²/C |
| Dipole Moment | p = q(2a) | C·m |
| Dipole Field (Axial) | E = 2kp/r³ | N/C |
| Dipole Field (Equatorial) | E = kp/r³ | N/C |
| Torque on Dipole | τ = pE sin θ | N·m |
| Dipole PE | U = −pE cos θ | J |
| Gauss's Law | ∮ E⃗·dA⃗ = Qenc/ε₀ | - |
| Infinite Line Charge | E = λ/(2πε₀r) | N/C |
| Infinite Plane | E = σ/(2ε₀) | N/C |
| Spherical Shell (outside) | E = kQ/r² | N/C |
| Spherical Shell (inside) | E = 0 | N/C |
Previous Chapter: None (This is Chapter 1)
Current Chapter: Electric Charges and Fields (Coulomb's law, Gauss's law, electric field)
Next Chapter: Electrostatic Potential and Capacitance (Electric potential, capacitors)
Understanding electric charges and fields provides the foundation for electrostatic potential energy, capacitors, and current electricity in subsequent chapters.
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