Electric Charges and Fields
'Electric charge is the fundamental property of matter that gives rise to electric forces — the forces that hold atoms and molecules together.'
1. Chapter Overview
Electrostatics is the study of electric charges at REST. This chapter begins with the concept of electric charge, its properties (additivity, quantization, conservation), and the fundamental LAW governing the force between two point charges — COULOMB'S LAW. The concept of the ELECTRIC FIELD (the region around a charge where its influence is felt) is introduced, followed by the ELECTRIC DIPOLE (two equal and opposite charges separated by a small distance). Finally, GAUSS'S LAW — one of the most powerful tools in electrostatics — is presented and applied to calculate fields for symmetric charge distributions.
2. Electric Charge — Basic Properties
- Charge: Quantized (q = ±ne, e = 1.6×10⁻¹⁹ C), CONSERVED (cannot be created or destroyed, only transferred), ADDITIVE (net charge = algebraic sum).
- Conductors: Charges move freely. Insulators: Charges are bound.
- Charging: By FRICTION (one body to another), by CONDUCTION (direct contact), by INDUCTION (without contact).
3. Coulomb's Law
- Force between two point charges: F = k(q₁q₂)/r² where k = 1/(4πε₀) = 9×10⁹ N·m²/C²
- Direction: ALONG the line joining the charges. LIKE charges REPEL, UNLIKE charges ATTRACT.
- Coulomb's law in vector form: F₁₂ = k(q₁q₂/r²)·r̂₁₂ — 'pointing from the charge causing the force towards the charge experiencing the force.'
- Principle of Superposition: Net force on a charge = VECTOR SUM of individual forces from all other charges.
Worked Example 1
Problem: Two point charges q₁ = +10 μC and q₂ = −5 μC are placed 20 cm apart. Find the magnitude and direction of the force on q₂. Solution: F = (9×10⁹)(10×10⁻⁶)(5×10⁻⁶)/(0.2)² = 9×10⁹×50×10⁻¹²/0.04 = 0.45/0.04 = 11.25 N. Force on q₂ is ATTRACTIVE towards q₁ (since q₂ is negative).
4. Electric Field (E)
- Definition: E = F/q₀ — force per unit positive test charge.
- Electric field due to a point charge: E = kq/r² (radially OUTWARD if q > 0, INWARD if q < 0).
- Superposition: Net E = VECTOR SUM of fields from individual charges.
Electric Lines of Force
- Imaginary lines showing the PATH a positive test charge would follow.
- Properties: (1) Start at + charges, end at − charges. (2) NEVER CROSS. (3) Tangent at any point gives direction of E. (4) Closer lines → stronger field. (5) Perpendicular to surface of a conductor.
5. Electric Dipole
Dipole Moment
- p⃗ = q × 2a⃗ (from −q to +q). SI unit: C·m.
Electric Field Due to a Dipole
- Axial line (end-on): E = 2kp/r³ (along the dipole axis, in the SAME direction as p⃗).
- Equatorial line (broadside-on): E = kp/r³ (OPPOSITE to p⃗).
Torque on a Dipole in a Uniform Field
- τ⃗ = p⃗ × E⃗. Magnitude: τ = pE sin θ. 'The dipole tries to ALIGN itself with the field.'
- Potential energy: U = −p⃗ · E⃗ = −pE cos θ.
6. Gauss's Law
- Total electric flux through a closed surface: Φ_E = ∮ E⃗ · dA⃗ = q_enc/ε₀.
- 'The flux through a closed surface depends ONLY on the charge enclosed — not on the shape of the surface or the distribution of charges outside.'
Applications of Gauss's Law
| Charge Distribution | Electric Field | Key |
|---|---|---|
| Infinite line charge | E = λ/(2πε₀r) | Cylindrical Gaussian surface |
| Infinite plane sheet | E = σ/(2ε₀) | Cylindrical Gaussian surface (pillbox) |
| Spherical shell (outside) | E = kQ/r² | Spherical Gaussian surface |
| Spherical shell (inside) | E = 0 | — |
| Solid sphere (outside) | E = kQ/r² | Spherical Gaussian surface |
| Solid sphere (inside) | E = kQr/R³ | Spherical Gaussian surface |
7. Comparison Table: Coulomb's Law vs Gauss's Law
| Aspect | Coulomb's Law | Gauss's Law |
|---|---|---|
| What it gives | Force between charges | Electric flux and field |
| Form | F = kq₁q₂/r² | Φ = q_enc/ε₀ |
| Best for | Point charges | Symmetric charge distributions |
| Vector or scalar? | Vector equation | Scalar equation (but yields vector E with symmetry) |
| Fundamental nature | Experimental law | Derived from Coulomb's law + superposition |
8. Common Mistakes
- Using k = 1/(4πε₀) without its value: MEMORISE: k = 9×10⁹ N·m²/C². ε₀ = 8.85×10⁻¹² C²/N·m².
- Direction of dipole moment: p⃗ points from −q to +q, NOT from +q to −q.
- Gaussian surface choice: Must pass through the point where E is desired and have the SAME symmetry as the charge distribution.
- Flux depends only on enclosed charge: Charge OUTSIDE the Gaussian surface contributes ZERO net flux — a very common confusion.
9. CBSE Exam Focus
- Coulomb's law — numerical problems with superposition principle
- Electric field due to point charges — superposition
- Electric dipole — field at axial and equatorial points, torque
- Gauss's law — flux calculation, field due to symmetric distributions
- Electric field due to infinite line charge, plane sheet, spherical shell
10. Self-Test
Q1: Two charges 2 μC and −2 μC are placed 1 cm apart. Find the electric field at a point on the perpendicular bisector at a distance of 50 cm from the centre. A1: This is a dipole with q=2×10⁻⁶, 2a=0.01, r=0.5. Since r >> a, use equatorial formula: E = kp/r³ = (9×10⁹)(2×10⁻⁶×0.01)/(0.5)³ = (9×10⁹)(2×10⁻⁸)/0.125 = 180/0.125 = 1440 N/C (opposite to p).
Q2: An infinite line charge has linear charge density 5 μC/m. Find the electric field at a distance of 10 cm from it. A2: E = λ/(2πε₀r) = (5×10⁻⁶)/(2π×8.85×10⁻¹²×0.1) = (5×10⁻⁶)/(5.56×10⁻¹²) ≈ 9×10⁵ N/C.
Q3: Find the flux through a cube of side 0.1 m if a charge of 10 μC is placed at its centre. A3: By Gauss, Φ = q_enc/ε₀ = (10×10⁻⁶)/(8.85×10⁻¹²) = 1.13×10⁶ N·m²/C.
Q4: A uniformly charged conducting sphere of radius 0.1 m has total charge 5 μC. Find E at r=0.05 m and r=0.3 m. A4: Inside conductor: E=0. Outside: E=kQ/r²=(9×10⁹)(5×10⁻⁶)/(0.3)²=45000/0.09=5×10⁵ N/C.
Q5: An electric dipole is placed at an angle of 30° with a uniform electric field of 2×10⁴ N/C. If the dipole moment is 10⁻⁷ C·m, find the torque and potential energy. A5: τ = pE sin θ = (10⁻⁷)(2×10⁴)(sin 30°) = 2×10⁻³×0.5 = 10⁻³ N·m. U = −pE cos θ = −(10⁻⁷)(2×10⁴)(cos 30°) = −2×10⁻³×0.866 = −1.732×10⁻³ J.
11. Conclusion
Electrostatics is the FOUNDATION of all electrical phenomena:
- COULOMB'S LAW: 'The force between charges decreases as the SQUARE of the distance.'
- ELECTRIC FIELD: 'A region of influence — charges alter the space around them.'
- DIPOLE: 'Two equal and opposite charges — the simplest neutral system with interesting properties.'
- GAUSS'S LAW: 'A powerful symmetry tool — flux depends only on enclosed charge.'
'Electrostatics teaches us that charges are the source of electric fields, and fields transmit forces across empty space.'
