Chapter 1: Electric Charges and Fields

ELECTRIC CHARGE

Properties of Electric Charge

  • Additivity: Net charge is algebraic sum of individual charges
  • Conservation: Total charge in an isolated system remains constant
  • Quantization: Charge exists in discrete units of ±e (e = 1.602 × 10⁻¹⁹ C)
  • Types: Positive and negative (like charges repel, unlike charges attract)

Conductors and Insulators

  • Conductors: Allow charge to flow freely (e.g., metals)
  • Insulators: Restrict charge movement (e.g., glass, rubber)
  • Semiconductors: Properties between conductors and insulators

Charging Methods

  • Friction: Transfer of electrons between materials
  • Conduction: Direct contact with charged object
  • Induction: Redistribution of charge without contact

COULOMB'S LAW

Statement

The electrostatic force between two point charges is directly proportional to the product of charges and inversely proportional to the square of distance between them.

\[ \vec{F} = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \hat{r} \]
  • \( \varepsilon_0 \) = permittivity of free space = 8.85 × 10⁻¹² C²/N·m²
  • \( k = \frac{1}{4\pi\varepsilon_0} \) = 9 × 10⁹ N·m²/C²
  • \( \hat{r} \) = unit vector from q₁ to q₂

Vector Nature

  • Force is a vector quantity (has magnitude and direction)
  • For multiple charges, use superposition principle

Comparison with Gravitational Force

  • Both follow inverse square law
  • Electric force can be attractive or repulsive
  • Electric force is much stronger than gravitational force

ELECTRIC FIELD

Definition

The electric field at a point is the force experienced by a unit positive charge placed at that point.

\[ \vec{E} = \frac{\vec{F}}{q_0} \]
  • Unit: Newton/Coulomb (N/C) or Volt/meter (V/m)
  • Vector quantity (has magnitude and direction)

Electric Field Due to Point Charge

\[ \vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r} \]
  • Direction: Away from positive charge, toward negative charge

Electric Field Lines

  • Start from positive charges, end on negative charges
  • Never intersect
  • Density proportional to field strength
  • Tangent to field line gives field direction

ELECTRIC DIPOLE

Definition

System of two equal and opposite charges separated by a small distance.

\[ \vec{p} = q \times 2\vec{a} \]
  • q = magnitude of each charge
  • 2a = separation between charges
  • Direction: From negative to positive charge

Electric Field of a Dipole

On axial line:

\[ E = \frac{1}{4\pi\varepsilon_0} \frac{2p}{r^3} \]

On equatorial line:

\[ E = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3} \]
  • Field decreases as 1/r³ (faster than point charge)

Torque on Dipole in Uniform Field

\[ \vec{\tau} = \vec{p} \times \vec{E} \]

Magnitude: τ = pE sin θ

  • Tends to align dipole with field
  • Maximum when perpendicular to field
  • Zero when aligned with field

Potential Energy

\[ U = -\vec{p} \cdot \vec{E} = -pE\cos\theta \]
  • Minimum when aligned with field (θ = 0°)
  • Maximum when anti-aligned with field (θ = 180°)

GAUSS'S LAW

Statement

The total electric flux through a closed surface is equal to 1/ε₀ times the total charge enclosed by the surface.

\[ \oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\varepsilon_0} \]

Electric Flux

\[ \Phi_E = \int \vec{E} \cdot d\vec{A} \]
  • Unit: N·m²/C
  • Measures number of field lines passing through a surface

Applications

Field due to:

  • Infinite charged plane: E = σ/(2ε₀) (independent of distance)
  • Charged spherical shell: E = 0 (inside), E = kQ/r² (outside)
  • Solid charged sphere: E = kQr/R³ (inside), E = kQ/r² (outside)
  • Infinite line charge: E = λ/(2πε₀r)

CONDUCTORS IN ELECTROSTATIC EQUILIBRIUM

Properties

  • Electric field inside is zero
  • Charge resides only on the surface
  • Electric field just outside is perpendicular to surface
  • Surface is an equipotential
  • Electric field just outside = σ/ε₀ (where σ is surface charge density)

Electrostatic Shielding

  • Hollow conductor shields interior from external fields
  • Faraday cage principle

KEY FORMULAS

  • Coulomb's Law: \( \vec{F} = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \hat{r} \)
  • Electric Field: \( \vec{E} = \frac{\vec{F}}{q_0} \)
  • Field due to point charge: \( \vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r} \)
  • Dipole moment: \( \vec{p} = q \times 2\vec{a} \)
  • Torque on dipole: \( \vec{\tau} = \vec{p} \times \vec{E} \)
  • Potential energy of dipole: \( U = -\vec{p} \cdot \vec{E} \)
  • Gauss's Law: \( \oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\varepsilon_0} \)
  • Field due to infinite plane: \( E = \frac{\sigma}{2\varepsilon_0} \)