Electrostatics

ELECTRIC CHARGE

• Two types: positive, negative. Like repels, unlike attracts.
• Quantized: q = ne (e = 1.6 × 10⁻¹⁹ C).
• Conserved in isolated systems.
• Additive (scalars).

COULOMB'S LAW

F = (1/4πε₀) · q₁q₂/r²

• k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C² (in vacuum).
• In medium of permittivity ε: divide by εᵣ (dielectric constant).

ELECTRIC FIELD (E)

E = F/q₀ = (1/4πε₀)·Q/r² (point charge).

• Vector field. Lines: from + to −, never cross.
• Continuous charge distribution: dE = kdq/r².

Field of common charges:

• Point charge: kQ/r².
• Infinite line of charge: λ/(2πε₀r).
• Infinite plane sheet: σ/(2ε₀).
• Between parallel plates: σ/ε₀.
• Dipole on axis (far): 2kp/r³; on equator: kp/r³.
• Uniformly charged ring axis: kQx/(R²+x²)^(3/2).

ELECTRIC POTENTIAL (V)

V = work done per unit positive charge in bringing it from ∞ to that point.
V = (1/4πε₀)·Q/r (point charge).

• Scalar — easier to handle.
• E = −dV/dr (gradient).

Potential difference: V_A − V_B = W_AB/q.

ELECTRIC DIPOLE

• Two equal and opposite charges separated by 2a.
• Dipole moment: p = q · 2a, directed from −q to +q.
• Torque in field: τ = p × E. (|τ| = pE sin θ.)
• Potential energy: U = −p·E (= −pE cos θ).
• Min U at θ = 0 (stable); max at θ = π (unstable).

GAUSS'S LAW

∮ E · dA = Q_enc / ε₀

Total electric flux through closed surface = (1/ε₀) × charge enclosed.

Applications (use symmetry):

• Spherical shell: outside acts as point charge at center; inside E = 0.
• Solid sphere of uniform charge: E_outside = kQ/r²; E_inside = kQr/R³.
• Infinite line: E = λ/(2πε₀r).
• Infinite plane sheet: E = σ/(2ε₀).
• Spherical capacitor.

CONDUCTORS IN ELECTROSTATIC EQUILIBRIUM

1. E = 0 inside conductor.
2. Charge resides on surface.
3. E just outside ⊥ surface, magnitude σ/ε₀.
4. Potential constant inside and on surface.
5. Sharp points have higher charge density (corona discharge).

CAPACITORS

C = Q/V.

• Parallel plate (vacuum): C = ε₀A/d.
• With dielectric: C = εᵣ ε₀ A/d.
• Series: 1/C = 1/C₁ + 1/C₂ + ...
• Parallel: C = C₁ + C₂ + ...
• Energy: U = ½CV² = ½QV = Q²/2C.

Dielectric: insulator that polarizes in field. Reduces E inside → increases C.

EXAM HOOKS:

• E from a uniformly charged sphere outside acts as if all charge at center.
• For a conductor, inside E = 0 (always).
• Gauss law: use symmetry (spherical/cylindrical/planar).
• Force on dipole in uniform field = 0; torque ≠ 0.
• Equipotential surfaces perpendicular to E at every point.

Coulomb's law for the force on q₁ due to q₂:

F⃗ = (1/4πε₀) · (q₁q₂ / r²) · r̂

where r̂ is the unit vector from q₂ towards q₁. The constant 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C².

Electric field at a point P due to a point charge Q at distance r:

E⃗ = (1/4πε₀) · (Q / r²) · r̂

The field points away from a positive charge and towards a negative charge. Force on a test charge q in field E⃗:

F⃗ = qE⃗

Superposition principle: total field at a point = vector sum of fields from each source charge. There's no shielding by other charges in the region between.

Common pitfalls:

1. Distance, not distance squared, in r̂. The unit vector has magnitude 1; the r² is in the denominator.
2. Sign of q₁q₂. If both same sign, F is repulsive (positive component along r̂). If opposite signs, F is attractive (negative).
3. Forces are pairwise. To find total force on q₁ in a system of charges, sum the force from each other charge separately.

ELECTRIC CHARGE

• Two types: positive, negative. Like repels, unlike attracts.
• Quantized: q = ne (e = 1.6 × 10⁻¹⁹ C).
• Conserved in isolated systems.
• Additive (scalars).

COULOMB'S LAW

F = (1/4πε₀) · q₁q₂/r²

• k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C² (in vacuum).
• In medium of permittivity ε: divide by εᵣ (dielectric constant).

ELECTRIC FIELD (E)

E = F/q₀ = (1/4πε₀)·Q/r² (point charge).

• Vector field. Lines: from + to −, never cross.
• Continuous charge distribution: dE = kdq/r².

Field of common charges:

• Point charge: kQ/r².
• Infinite line of charge: λ/(2πε₀r).
• Infinite plane sheet: σ/(2ε₀).
• Between parallel plates: σ/ε₀.
• Dipole on axis (far): 2kp/r³; on equator: kp/r³.
• Uniformly charged ring axis: kQx/(R²+x²)^(3/2).

ELECTRIC POTENTIAL (V)

V = work done per unit positive charge in bringing it from ∞ to that point.
V = (1/4πε₀)·Q/r (point charge).

• Scalar — easier to handle.
• E = −dV/dr (gradient).

Potential difference: V_A − V_B = W_AB/q.

ELECTRIC DIPOLE

• Two equal and opposite charges separated by 2a.
• Dipole moment: p = q · 2a, directed from −q to +q.
• Torque in field: τ = p × E. (|τ| = pE sin θ.)
• Potential energy: U = −p·E (= −pE cos θ).
• Min U at θ = 0 (stable); max at θ = π (unstable).

GAUSS'S LAW

∮ E · dA = Q_enc / ε₀

Total electric flux through closed surface = (1/ε₀) × charge enclosed.

Applications (use symmetry):

• Spherical shell: outside acts as point charge at center; inside E = 0.
• Solid sphere of uniform charge: E_outside = kQ/r²; E_inside = kQr/R³.
• Infinite line: E = λ/(2πε₀r).
• Infinite plane sheet: E = σ/(2ε₀).
• Spherical capacitor.

CONDUCTORS IN ELECTROSTATIC EQUILIBRIUM

1. E = 0 inside conductor.
2. Charge resides on surface.
3. E just outside ⊥ surface, magnitude σ/ε₀.
4. Potential constant inside and on surface.
5. Sharp points have higher charge density (corona discharge).

CAPACITORS

C = Q/V.

• Parallel plate (vacuum): C = ε₀A/d.
• With dielectric: C = εᵣ ε₀ A/d.
• Series: 1/C = 1/C₁ + 1/C₂ + ...
• Parallel: C = C₁ + C₂ + ...
• Energy: U = ½CV² = ½QV = Q²/2C.

Dielectric: insulator that polarizes in field. Reduces E inside → increases C.

EXAM HOOKS:

• E from a uniformly charged sphere outside acts as if all charge at center.
• For a conductor, inside E = 0 (always).
• Gauss law: use symmetry (spherical/cylindrical/planar).
• Force on dipole in uniform field = 0; torque ≠ 0.
• Equipotential surfaces perpendicular to E at every point.

ELECTRIC CHARGE

• Two types: positive, negative. Like repels, unlike attracts.
• Quantized: q = ne (e = 1.6 × 10⁻¹⁹ C).
• Conserved in isolated systems.
• Additive (scalars).

COULOMB'S LAW

F = (1/4πε₀) · q₁q₂/r²

• k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C² (in vacuum).
• In medium of permittivity ε: divide by εᵣ (dielectric constant).

ELECTRIC FIELD (E)

E = F/q₀ = (1/4πε₀)·Q/r² (point charge).

• Vector field. Lines: from + to −, never cross.
• Continuous charge distribution: dE = kdq/r².

Field of common charges:

• Point charge: kQ/r².
• Infinite line of charge: λ/(2πε₀r).
• Infinite plane sheet: σ/(2ε₀).
• Between parallel plates: σ/ε₀.
• Dipole on axis (far): 2kp/r³; on equator: kp/r³.
• Uniformly charged ring axis: kQx/(R²+x²)^(3/2).

ELECTRIC POTENTIAL (V)

V = work done per unit positive charge in bringing it from ∞ to that point.
V = (1/4πε₀)·Q/r (point charge).

• Scalar — easier to handle.
• E = −dV/dr (gradient).

Potential difference: V_A − V_B = W_AB/q.

ELECTRIC DIPOLE

• Two equal and opposite charges separated by 2a.
• Dipole moment: p = q · 2a, directed from −q to +q.
• Torque in field: τ = p × E. (|τ| = pE sin θ.)
• Potential energy: U = −p·E (= −pE cos θ).
• Min U at θ = 0 (stable); max at θ = π (unstable).

GAUSS'S LAW

∮ E · dA = Q_enc / ε₀

Total electric flux through closed surface = (1/ε₀) × charge enclosed.

Applications (use symmetry):

• Spherical shell: outside acts as point charge at center; inside E = 0.
• Solid sphere of uniform charge: E_outside = kQ/r²; E_inside = kQr/R³.
• Infinite line: E = λ/(2πε₀r).
• Infinite plane sheet: E = σ/(2ε₀).
• Spherical capacitor.

CONDUCTORS IN ELECTROSTATIC EQUILIBRIUM

1. E = 0 inside conductor.
2. Charge resides on surface.
3. E just outside ⊥ surface, magnitude σ/ε₀.
4. Potential constant inside and on surface.
5. Sharp points have higher charge density (corona discharge).

CAPACITORS

C = Q/V.

• Parallel plate (vacuum): C = ε₀A/d.
• With dielectric: C = εᵣ ε₀ A/d.
• Series: 1/C = 1/C₁ + 1/C₂ + ...
• Parallel: C = C₁ + C₂ + ...
• Energy: U = ½CV² = ½QV = Q²/2C.

Dielectric: insulator that polarizes in field. Reduces E inside → increases C.

EXAM HOOKS:

• E from a uniformly charged sphere outside acts as if all charge at center.
• For a conductor, inside E = 0 (always).
• Gauss law: use symmetry (spherical/cylindrical/planar).
• Force on dipole in uniform field = 0; torque ≠ 0.
• Equipotential surfaces perpendicular to E at every point.

Capacitors store charge: Q = CV, where C is capacitance in farads.

Series combination:

• Same charge Q on each capacitor
• Voltages add: V = V₁ + V₂ + ...
• 1/C_eq = 1/C₁ + 1/C₂ + ... (reciprocals add — like resistors in parallel)

Parallel combination:

• Same voltage V across each
• Charges add: Q = Q₁ + Q₂ + ...
• C_eq = C₁ + C₂ + ... (capacitances add directly — like resistors in series)

This is opposite to resistors! Common exam trap. Memorize:

• Resistors: series adds; parallel reciprocates.
• Capacitors: parallel adds; series reciprocates.

Energy stored: U = ½CV² = ½Q²/C = ½QV

Effect of dielectric: if a dielectric of constant K fills the gap, capacitance becomes C′ = KC (always larger). Energy and field change depending on whether the battery is connected (constant V) or disconnected (constant Q).

ELECTRIC CHARGE

• Two types: positive, negative. Like repels, unlike attracts.
• Quantized: q = ne (e = 1.6 × 10⁻¹⁹ C).
• Conserved in isolated systems.
• Additive (scalars).

COULOMB'S LAW

F = (1/4πε₀) · q₁q₂/r²

• k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C² (in vacuum).
• In medium of permittivity ε: divide by εᵣ (dielectric constant).

ELECTRIC FIELD (E)

E = F/q₀ = (1/4πε₀)·Q/r² (point charge).

• Vector field. Lines: from + to −, never cross.
• Continuous charge distribution: dE = kdq/r².

Field of common charges:

• Point charge: kQ/r².
• Infinite line of charge: λ/(2πε₀r).
• Infinite plane sheet: σ/(2ε₀).
• Between parallel plates: σ/ε₀.
• Dipole on axis (far): 2kp/r³; on equator: kp/r³.
• Uniformly charged ring axis: kQx/(R²+x²)^(3/2).

ELECTRIC POTENTIAL (V)

V = work done per unit positive charge in bringing it from ∞ to that point.
V = (1/4πε₀)·Q/r (point charge).

• Scalar — easier to handle.
• E = −dV/dr (gradient).

Potential difference: V_A − V_B = W_AB/q.

ELECTRIC DIPOLE

• Two equal and opposite charges separated by 2a.
• Dipole moment: p = q · 2a, directed from −q to +q.
• Torque in field: τ = p × E. (|τ| = pE sin θ.)
• Potential energy: U = −p·E (= −pE cos θ).
• Min U at θ = 0 (stable); max at θ = π (unstable).

GAUSS'S LAW

∮ E · dA = Q_enc / ε₀

Total electric flux through closed surface = (1/ε₀) × charge enclosed.

Applications (use symmetry):

• Spherical shell: outside acts as point charge at center; inside E = 0.
• Solid sphere of uniform charge: E_outside = kQ/r²; E_inside = kQr/R³.
• Infinite line: E = λ/(2πε₀r).
• Infinite plane sheet: E = σ/(2ε₀).
• Spherical capacitor.

CONDUCTORS IN ELECTROSTATIC EQUILIBRIUM

1. E = 0 inside conductor.
2. Charge resides on surface.
3. E just outside ⊥ surface, magnitude σ/ε₀.
4. Potential constant inside and on surface.
5. Sharp points have higher charge density (corona discharge).

CAPACITORS

C = Q/V.

• Parallel plate (vacuum): C = ε₀A/d.
• With dielectric: C = εᵣ ε₀ A/d.
• Series: 1/C = 1/C₁ + 1/C₂ + ...
• Parallel: C = C₁ + C₂ + ...
• Energy: U = ½CV² = ½QV = Q²/2C.

Dielectric: insulator that polarizes in field. Reduces E inside → increases C.

EXAM HOOKS:

• E from a uniformly charged sphere outside acts as if all charge at center.
• For a conductor, inside E = 0 (always).
• Gauss law: use symmetry (spherical/cylindrical/planar).
• Force on dipole in uniform field = 0; torque ≠ 0.
• Equipotential surfaces perpendicular to E at every point.