Authored by subject matter experts. Content strictly validated against latest NCERT rationalized curriculum and official Board Marking Schemes.
📑 In This Chapter Guide (Table of Contents)
1. Coulomb’s Law & Electric Dipole Derivations
Coulomb’s Inverse Square Law: Electrostatic force between two stationary point charges q₁ and q₂ separated by distance r in vacuum is:
F = [ 1 / (4πε₀) ] · [ |q₁ · q₂| / r² ]
(where 1/(4πε₀) = 9 × 10⁹ N·m²/C², and ε₀ = 8.854 × 10⁻¹² C²·N⁻¹·m⁻²)
Electric Field of an Electric Dipole (Dipole moment p = q · 2a):
- At Axial Point (End-on position, distance r from center where r >> a):
E_axial = [ 1 / (4πε₀) ] · [ 2p / r³ ](Points in the direction of dipole moment p). - At Equatorial Point (Broadside-on position, r >> a):
E_equatorial = [ 1 / (4πε₀) ] · [ p / r³ ](Points opposite to the direction of dipole moment p). - Ratio:
E_axial = 2 · E_equatorial. - Torque on Dipole in Uniform Electric Field:
τ = p × E = pE · sin(θ). Potential energy stored:U = -p · E = -pE · cos(θ).
2. Gauss’s Theorem & High-Yield Applications
Gauss’s Law Statement: The total electric flux Φ_E through any closed Gaussian surface enclosing net charge q_enclosed is equal to 1/ε₀ times the enclosed charge:
Φ_E = ∮ E · dA = q_enclosed / ε₀
Crucial Applications (Frequently Tested in 3 & 5 Marks):
- Infinitely Long Straight Charged Wire (Linear charge density λ = q/L):
Construct a coaxial cylindrical Gaussian surface of radius r and length l:E · (2πrl) = (λl) / ε₀ ⟹ E = λ / (2πε₀r) - Uniformly Charged Infinite Plane Sheet (Surface charge density σ = q/A):
Cylindrical Gaussian pillbox intersecting sheet:2 · E · A = (σA) / ε₀ ⟹ E = σ / (2ε₀)(Independent of distance r from sheet).
3. Parallel Plate Capacitor with Dielectric Slab
Capacitance is the ratio of charge to potential: C = Q / V (Unit: Farad, F).
1. Capacitor with Air/Vacuum: C₀ = (ε₀ · A) / d
2. With Dielectric Slab of thickness t (t < d) and Dielectric Constant K:
Electric field in vacuum = E₀ = σ/ε₀; Electric field inside dielectric = E = E₀/K.
Potential difference: V = E₀(d - t) + E·t = E₀[ (d - t) + t/K ] = (Q / ε₀A)[ (d - t) + t/K ].
Capacitance: C = Q / V = (ε₀ · A) / [ (d - t) + (t / K) ].
If dielectric fills the entire space (t = d): C = K · C₀ (Capacitance increases K times).
Energy Stored in Capacitor: U = ½ CV² = ½ (Q² / C) = ½ QV. Energy density: u = ½ ε₀E².
💡 Frequently Asked Questions (FAQ)
❓ Why is the electric field inside a hollow charged spherical conductor zero?
By Gauss law, ∮ E·dA = q_enclosed / ε₀. Inside a hollow spherical conductor, all electric charges reside exclusively on the outer surface of the conductor (q_enclosed = 0). Therefore, the electric field inside is identically zero (E = 0). This phenomenon is utilized in Electrostatic Shielding.
❓ What happens to the capacitance, charge, and energy when a dielectric is inserted with battery disconnected?
With battery disconnected: (1) Charge Q remains constant; (2) Capacitance increases: C = K·C₀; (3) Potential difference decreases: V = V₀/K; (4) Energy stored decreases: U = U₀/K (work done by attractive electrostatic forces pulling the slab inside).
📚 Related Study Guides & Notes
About Vidya Topper Academic Research Team
Our educational publishing team consists of experienced CBSE educators, state board toppers, and IIT/NIT alumni dedicated to providing 100% free, high-yield study materials, formula handbooks, and step-by-step NCERT solutions for students across India.
🚀 Test Your Mastery with Free Interactive MCQs
Solve chapter-wise quizzes, track your All-India percentile, and get instant explanations on Vidya Topper Web & Android Apps.
Open Free Web App 💻