Lecture Notes: Electric Charges and Fields

Introduction

• Class: 12th Physics
• Chapter: Electric Charges and Field
• Importance: Fundamental for understanding Electrostatics.

Electric Charge

Key Concepts

• Definition: Intrinsic property of matter, represented by Q or q.
• Unit: Coulomb (C)
• Types: Positive and Negative
• Properties: Like charges repel, unlike charges attract.
• Conservation: Total charge in an isolated system remains constant.
• Quantization: Charge exists in integral multiples of the elementary charge e (1.6 x 10^-19 C).

Formulas

• Charge Conservation: Total charge stays constant in an isolated system.
• Quantization: q = ±ne, where n is an integer.

Methods of Charging

By Conduction

• Process: Involves direct contact.
• Example: Two identical spheres touch and share charge equally.

By Induction

• Process: Without direct contact, involves proximity.
• Example: Bringing a charged rod near a conductor induces charge separation.

By Friction

• Process: Rubbing two objects transfers electrons, causing charging.
• Example: Rubbing a balloon against hair.

Coulomb's Law

Key Concepts

• Definition: Force between two point charges q1 and q2 separated by distance r.
• Formula: F = k * q1 * q2 / r², where k = 1 / (4πϵ₀) in vacuum.
• Nature: Inverse square law, central force.
• Properties: Valid for point charges, conservative force.

Formulas

• Coulomb’s Law in Vector Form: F = k * q1 * q2 / r² * r̂, where r̂ is the unit vector.
• Medium Effect: F = k * q1 * q2 / (εr * r²).
• Max Force Condition: Equal charges maximize force when separated by a specific distance.

Electric Field

Key Concepts

• Definition: Region around a charge where its force can be felt by another charge.
• Formula: E = F / q.
• Unit: N/C (Newton per Coulomb) or V/m (Volt per Meter).
• Properties: Vector quantity, direction follows the force on a positive test charge.

Formulas

• Point Charge: E = k * q / r².
• Superposition Principle: E_net = E1 + E2 + ... + En for multiple charges.
• Symmetrical Charge Distribution: Field at the center is zero.*

Methods of Electric Field Calculation

Continuous Charge Distribution

• Linear Charge Density (λ): λ = Q / L (charge per unit length).
• Surface Charge Density (σ): σ = Q / A (charge per unit area).
• Volume Charge Density (ρ): ρ = Q / V (charge per unit volume).

Electric Field Due to Continuous Distributions

• Line Charge: E = 2kλ / r for infinite line charge.
• Ring Charge: E = (k * q * x) / (x² + R²)^(3/2) on the axis.
• Infinite Plane: E = σ / 2ϵ₀.

Motion of Charged Particles in Electric Fields

Key Concepts

• Force on Charge: F = qE.
• Motion: Can be analyzed using kinematic equations and work-energy theorem.

Formulas

• Time of Flight in Uniform Field: t = sqrt(2d / a) where a = qE / m.
• Kinetic Energy: KE = (1/2)mv².

Electric Dipole

Key Concepts

• Definition: Two equal and opposite charges separated by a small distance.
• Dipole Moment (p): p = q * 2a, directed from negative to positive charge.
• Unit: Coulomb-meter (Cm).*

Electric Field Due to Dipole

• On Axis: E = 2kp / r³ in the direction of dipole moment.
• On Equatorial Line: E = -kp / r³ opposite to the direction of dipole moment.

Torque on Dipole

• Formula: τ = pE sinθ or τ = p × E.
• Stable Equilibrium: Aligns with field direction (θ = 0).
• Unstable Equilibrium: Aligns opposite to field direction (θ = 180°).

Electric Flux and Gauss's Law

Key Concepts

• Electric Flux (Φ): Measure of electric field passing through a surface.
• Definition: Φ = E · A = EA cosθ.
• Gauss's Law: Φ = Q_enclosed / ϵ₀.
• Unit: Nm²/C.

Application of Gauss's Law

• Symmetry: Useful for high symmetry problems (spherical, cylindrical, planar).
• Spherical Symmetry: Field outside a spherical shell or solid sphere: E = kQ / r².
• Cylindrical Symmetry: Infinite line charge: E = λ / (2πϵ₀r).
• Planar Symmetry: Infinite plane sheet: E = σ / 2ϵ₀.

Conclusion

• Electric Charges and Fields: Fundamental principles required for understanding electrostatics and related phenomena.
• Key Takeaways: Conservation and quantization of charge, Coulomb's law, electric field calculations, motion of charges, dipoles, and Gauss's law applications.