Electric Charges and Fields — Physics Class 12 Notes (CBSE & HBSE)
Free NCERT Physics notes for Electric Charges and Fields (Class 12) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Electric Charges and Fields (CBSE & HBSE)
CBSE emphasizes Gauss's law derivations, superposition principle, and electric field for symmetric charge distributions with analytical problems. HBSE focuses on Coulomb's law numericals, properties of electric charge, and field line diagrams with direct definition-based questions.
Coulomb's Law and Electric Charge
Electric Charge
Electric charge is a fundamental scalar property of matter that causes it to experience a force in an electromagnetic field. It exists in two types: positive (protons) and negative (electrons). The SI unit is Coulomb (C).
Properties of Electric Charge
| Property | Description |
|---|---|
| Quantization | q = ne, where n = ±1, ±2, ... and e = 1.6×10⁻¹⁹ C |
| Conservation | Total charge in an isolated system remains constant |
| Additivity | Total charge = algebraic sum of all individual charges |
| Invariance | Charge is independent of velocity (unlike mass) |
Coulomb's Law
The electrostatic force between two point charges q₁ and q₂ separated by distance r in vacuum:
F = kq₁q₂/r²
where k = 1/(4πε₀) = 9×10⁹ N·m²/C²
ε₀ = permittivity of free space = 8.854×10⁻¹² C²·N⁻¹·m⁻²
Vector form: F⃗₁₂ = (q₁q₂/4πε₀r²) r̂₁₂
- Positive F → repulsion; Negative F → attraction
Coulomb's Law in a Medium
In a medium with dielectric constant K:
F_medium = kq₁q₂/(Kr²) = q₁q₂/(4πε₀εᵣr²)
where εᵣ = K = ε/ε₀ is the relative permittivity of the medium.
Comparison: Coulomb vs Gravitational Force
| Feature | Coulomb Force | Gravitational Force |
|---|---|---|
| Source | Electric charge | Mass |
| Nature | Attractive or Repulsive | Always Attractive |
| Constant | k = 9×10⁹ N·m²/C² | G = 6.67×10⁻¹¹ N·m²/kg² |
| Ratio Fc/Fg (H atom) | ~2.27×10³⁹ | 1 |
| Medium effect | Reduced by factor K | Not affected |
| Range | Long range (1/r²) | Long range (1/r²) |
Coulomb force is approximately 10³⁹ times stronger than gravity.
Superposition Principle
The net electric force on a charge due to multiple charges is the vector sum of all individual Coulomb forces:
F⃗_net = F⃗₁ + F⃗₂ + F⃗₃ + ...
Each pair interaction is independent of the presence of other charges.
Continuous Charge Distributions
| Type | Symbol | Formula | Unit |
|---|---|---|---|
| Linear | λ | λ = dq/dl | C/m |
| Surface | σ | σ = dq/dA | C/m² |
| Volume | ρ | ρ = dq/dV | C/m³ |
Charging Methods
- Friction: Electrons transfer between rubbed materials (e.g., glass rod + silk)
- Conduction: Direct contact with charged body
- Induction: Nearby charged body redistributes charges without contact
Diagram Indicator: [Two point charges q₁ and q₂ separated by distance r; arrows showing F₁₂ (force on q₂ due to q₁) and F₂₁ (force on q₁ due to q₂) for both attraction and repulsion cases. Unit vector r̂₁₂ labeled.]
Electric Field and Field Lines
Electric Field
The electric field at a point in space is defined as the force experienced per unit positive test charge placed at that point, without disturbing the source charges:
E⃗ = F⃗/q₀
Unit: N/C or V/m. Electric field is a vector quantity.
Electric Field due to a Point Charge
For point charge Q at distance r:
E = kQ/r² = Q/(4πε₀r²)
- Direction: radially outward for +Q; radially inward for −Q
Superposition of Electric Fields
Net electric field due to multiple charges:
E⃗_net = E⃗₁ + E⃗₂ + E⃗₃ + ...
Vectors are added taking direction into account.
Electric Field Lines
An electric field line is an imaginary smooth curve in space such that the tangent at any point gives the direction of the electric field at that point.
Properties of Electric Field Lines
| Property | Details |
|---|---|
| Origin/End | Start from +ve charges, terminate at −ve charges |
| Direction | Tangent at any point = direction of E⃗ |
| Density | More crowded lines → stronger field |
| Non-intersecting | Two lines never cross (E unique at each point) |
| Perpendicular | Always perpendicular to equipotential surfaces |
| No closed loops | Cannot form closed curves in free space |
Electric Field for Special Geometries
| Geometry | Formula | Remarks |
|---|---|---|
| Point charge Q | E = kQ/r² | Spherically symmetric |
| Infinite line charge λ | E = λ/(2πε₀r) | Cylindrically symmetric |
| Infinite plane sheet σ | E = σ/(2ε₀) | Uniform, perpendicular |
| Spherical shell (outside) | E = Q/(4πε₀r²) | Like point charge |
| Spherical shell (inside) | E = 0 | No field inside |
Electric Dipole
An electric dipole = two equal and opposite charges ±q separated by distance 2a.
Dipole moment: p = q·2a, direction from −q to +q. Unit: C·m
Field on axial line (at distance r from center, r >> a): E_ax = 2kp/r³
Field on equatorial line (at distance r from center, r >> a): E_eq = kp/r³
Ratio: E_ax : E_eq = 2 : 1 (at same distance)
Torque on Dipole in Uniform Electric Field
τ = p⃗ × E⃗ |τ| = pE sinθ
- θ = 0° (parallel): τ = 0, stable equilibrium
- θ = 90°: τ = pE (maximum)
- θ = 180° (antiparallel): τ = 0, unstable equilibrium
Potential Energy of Dipole
U = −pE cosθ = −p⃗·E⃗
Diagram Indicator: [Field lines from a positive charge and between a dipole (±q). Arrows showing direction, with denser lines near charges indicating stronger field. Perpendicular bisector and axial line labeled for dipole.]
Gauss's Law and Applications
Electric Flux
Electric flux through a surface measures the 'flow' of electric field lines through it.
φ = E⃗·A⃗ = EA cosθ (uniform field, flat surface)
For non-uniform field or curved surface: φ = ∮E⃗·dA⃗
Unit: N·m²/C or V·m φ > 0: lines exit; φ < 0: lines enter; φ = 0: equal entry and exit
Gauss's Law
The total electric flux through any closed Gaussian surface equals the net charge enclosed divided by ε₀:
∮E⃗·dA⃗ = q_enc/ε₀
This is one of Maxwell's four fundamental equations of electromagnetism.
Choosing Gaussian Surfaces
| Geometry | Best Gaussian Surface | Condition Used |
|---|---|---|
| Spherical charge distribution | Concentric sphere | E·dA = E·4πr² |
| Cylindrical distribution | Coaxial cylinder | E·dA = E·2πrL |
| Planar distribution | Pillbox (cylinder) | E·dA = 2EA |
Application 1: Field of Infinitely Long Line Charge (λ)
Gaussian cylinder of radius r, length L:
φ_curved = E·2πrL; φ_ends = 0
By Gauss's law: E·2πrL = λL/ε₀
E = λ/(2πε₀r) directed radially outward
Application 2: Field of Infinite Plane Sheet (σ)
Pillbox with area A through sheet:
φ = 2EA = σA/ε₀
E = σ/(2ε₀) (on each side, uniform, perpendicular to sheet)
For two parallel plates (±σ): E_between = σ/ε₀; E_outside = 0
Application 3: Uniformly Charged Thin Spherical Shell
| Region | Enclosed charge | Electric Field E |
|---|---|---|
| r > R (outside) | Q | Q/(4πε₀r²) = kQ/r² |
| r = R (surface) | Q | Q/(4πε₀R²) = σ/ε₀ |
| r < R (inside) | 0 | E = 0 |
Application 4: Solid Insulating Sphere (volume charge density ρ, radius R)
- Outside (r ≥ R): E = Q/(4πε₀r²) (same as point charge)
- Inside (r < R): q_enc = ρ·(4/3)πr³ = Q(r/R)³
E = Qr/(4πε₀R³) = ρr/(3ε₀) (linear in r)
Properties of Conductors in Electrostatic Equilibrium
- E = 0 everywhere inside a conductor
- Net charge resides only on the outer surface
- E at surface = σ/ε₀, perpendicular to surface
- The conductor surface is an equipotential
- Electrostatic shielding: hollow conductor shields interior from external fields (Faraday cage)
Diagram Indicator: [Plot of E vs r for a uniformly charged spherical shell: E=0 for r<R, E rising to max at r=R, then falling as 1/r² for r>R. Also diagram of Gaussian surface inside and outside shell.]
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Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Electric Charges and Fields.