Electrostatics

Electric charge is an intrinsic scalar property of elementary particles that gives rise to all electrostatic interactions. Benjamin Franklin established the two-charge convention: positive (vitreous) and negative (resinous). The modern standard SI unit of charge is the Coulomb (C), dimensionally represented as [I·T] = [A·s].

Three cardinal conservation and structural principles govern all electric charges in the universe:

• Quantization of Electric Charge: Electric charge exists only in discrete integral multiples of the elementary charge quantum e:
  q = ± n·e,   where n ∈ ℤ⁺ and e = 1.602176634 × 10⁻¹⁹ C
  At macroscopic scales involving microcoulombs (10¹³ electrons), the discrete nature is grain-like and appears continuous.
• Conservation of Electric Charge: In any electrically isolated system, the algebraic sum of all positive and negative charges remains invariant over time. Charge cannot be created or destroyed, only transferred (e.g., pair production γ → e⁻ + e⁺ conserves net charge zero).
• Relativistic Invariance of Charge: Unlike mass, which increases with relativistic velocity, the magnitude of electric charge is strictly independent of the speed of the reference frame: q(rest) = q(motion).

In 1785, Charles-Augustin de Coulomb formulated the fundamental law governing the interaction between two stationary point charges in a vacuum using a torsion balance:

Here, ε₀ is the permittivity of free space: ε₀ = 8.8541878128 × 10⁻¹² C²/(N·m²) = F/m, with dimensional formula [M⁻¹ L⁻³ T⁴ A²].

Let position vectors of charges q₁ and q₂ be r₁ and r₂. The force exerted on q₂ by q₁ is:

This confirms that electrostatic forces obey Newton Third Law of Motion (action-reaction pair) and act strictly along the central line joining the two charge centers (central forces).

Principle of Superposition: The total electrostatic force experienced by any single charge due to a configuration of N discrete charges is the vector sum of the individual forces exerted by each charge independently:

The electric field at any point in space is defined as the electrostatic force experienced per unit positive infinitesimal test charge placed at that point:

For an isolated point charge q situated at the origin, the electric field at a position vector r is: E(r) = (1 / 4πε₀) · (q / r²) · r̂. The field points radially outwards for positive charges (q > 0) and radially inwards for negative charges (q < 0).

Michael Faraday introduced electric field lines as visual representations of force fields. Key physical properties tested in WBCHSE board examinations include:

• Field lines continuously originate from positive charges and terminate on negative charges (or extend to infinity).
• The tangent drawn to a field line at any point gives the instantaneous direction of the electric field vector E at that point.
• Two electric field lines can NEVER intersect. If they did, two distinct tangents could be drawn at the intersection point, implying two different directions of the electric field at a single point in space, which is physically impossible.
• Electric field lines never form closed loops because the electrostatic field is strictly conservative (∮ E · dr = 0).
• Field lines are always normal (perpendicular) to the equipotential surface of a conducting body in electrostatic equilibrium.
• The relative spatial density of lines is proportional to the local field magnitude (crowded lines denote strong fields; widely spaced lines denote weak fields).

An electric dipole consists of a pair of equal and opposite point charges (+q and -q) separated by an infinitesimal distance 2a. The electric dipole moment vector (p) is defined as:

Crucial Examination Relationship: For a short dipole at equal distances r, |E_axial| = 2 · |E_equatorial|, and both fields decay rapidly as 1/r³ (faster than a point charge 1/r²).

• Net Translatory Force: F_net = (+qE) + (-qE) = 0. An electric dipole experiences ZERO net translatory force in a uniform electric field.
• Torque (τ): τ = p × E ⟹ |τ| = p E sinθ. Torque tends to align the dipole moment vector parallel to the external field.
• Electrostatic Potential Energy (U): Work done in rotating the dipole from angle θ₁ to θ₂:
  W = ∫_{θ₁}^{θ₂} τ dθ = pE(cosθ₁ - cosθ₂) ⟹ U = - p · E = - p E cosθ
  At θ = 0° (p ∥ E), U = -pE (Stable Equilibrium, minimum energy). At θ = 180° (p ∦ E), U = +pE (Unstable Equilibrium, maximum energy).

Electric flux measures the total number of electric field lines passing normally through a given geometric area vector dA (where dA = n̂ dA, directed normal outward):

Theorem Statement: The total electric flux through any arbitrary closed imaginary surface (Gaussian surface) in free space is equal to 1/ε₀ times the net algebraic charge enclosed within that closed boundary:

Key Board Insight: If a dipole (charges +q and -q) is enclosed inside a closed surface, net enclosed charge Q_enc = +q - q = 0, hence total flux through the surface is identically zero (even though electric field E ≠ 0 on the surface).

WBCHSE examinations routinely ask for the rigorous derivation of electric fields for three fundamental symmetric geometries:

1. Infinitely Long Straight Uniformly Charged Wire:

• Let linear charge density be λ = q/L (C/m).
• Construct a coaxial cylindrical Gaussian surface of radius r and length l.
• Flux through flat circular ends: E ⊥ dA ⟹ cos 90° = 0 (Zero flux).
• Flux through curved cylindrical surface: E ∥ dA ⟹ ∬ E dA = E · (2πrl).
• Applying Gauss Law: E · (2πrl) = (λl / ε₀) ⟹ E = λ / (2πε₀r) = (2kλ / r).
• Field direction is radially outward (for λ > 0) and falls off as E ∝ 1/r.

2. Uniformly Charged Infinite Thin Non-Conducting Plane Sheet:

• Let surface charge density be σ = q/A (C/m²).
• Construct a pillbox cylinder of cross-sectional area A piercing normally through the sheet.
• Flux through cylindrical curved sides is zero since E is parallel to sheet.
• Flux through both circular ends: Φ = E·A + E·A = 2EA.
• Applying Gauss Law: 2EA = (σA / ε₀) ⟹ E = σ / (2ε₀).
• Critical Takeaway: The electric field of an infinite plane sheet is uniform and completely independent of the distance r from the sheet. For a charged conducting sheet, induced charge on both faces yields E = σ / ε₀.

3. Uniformly Charged Thin Spherical Shell of Radius R (Total Charge Q):

The electrostatic field is conservative. The electric potential (V) at any point is defined as the work done per unit positive test charge by an external agent against the electrostatic force in bringing it without acceleration from infinity to that point:

For an isolated point charge q at distance r: V(r) = (1 / 4πε₀) · (q / r). Electric potential is positive for positive charges and negative for negative charges.

The electric field is the negative spatial gradient of electric scalar potential:

Two Physical Implications:

1. The negative sign indicates that the electric field vector always points in the direction of maximum rate of decrease of electric potential.
2. The magnitude of electric field equals the potential change per unit displacement normal to the equipotential surface (potential gradient).

At a general point with polar coordinates (r, θ) relative to the dipole center (where θ is angle with dipole moment p):

• Axial Line (θ = 0° or 180°): V_axial = ± (1 / 4πε₀) · (p / r²)
• Equatorial Line (θ = 90°): V_equatorial = (1 / 4πε₀) · (p cos 90° / r²) = 0. Although electric potential is ZERO everywhere on the equatorial plane, the electric field is NON-ZERO: E_eq = kp/r³!

An equipotential surface is any geometric locus of points in an electric field that possess identical electric potential values (V = constant).

• Zero Work Done: W_{A→B} = q₀(V_B - V_A) = 0. No work is done in moving any charge along an equipotential surface.
• Perpendicularity of Field Lines: Since dW = F · dr = q₀(E · dr) = 0, the electric field vector E is strictly normal (perpendicular) to the equipotential surface at every point.
• Spatial Spacing: From dr = -dV/E, equipotential surfaces are closely spaced where electric fields are intense and widely separated where fields are weak.
• Equipotential surfaces never intersect (intersecting surfaces would imply two different potentials at the same point).

For a system of two point charges q₁ and q₂ separated by distance r₁₂:

For an N-charge system, total potential energy is the sum over all unique interacting pairs: U_total = ∑_{i < j} (1 / 4πε₀) · (qᵢqⱼ / rᵢⱼ).

Metallic conductors contain abundant free electrons that redistribute instantaneously under electrostatic conditions. Key equilibrium properties include:

• Inside a conductor, E = 0: Mobile electrons shift until internal induced field perfectly cancels external applied field.
• Net electrostatic charge inside a conductor is zero: By Gauss law (∮ E·dA = Q/ε₀ = 0), all excess charge resides exclusively on the outer boundary surface.
• Electrostatic potential is constant throughout the interior and surface: Since E = -dV/dr = 0 inside, V = constant. A conductor is an equipotential volume.
• Electric field just outside a charged conductor: E = (σ / ε₀) n̂, directed normal to surface.
• Electrostatic Shielding (Faraday Cage): The electric field inside an empty metallic cavity within a conductor is identically ZERO, regardless of external electric fields or charges on the conductor.

Dielectrics are insulating materials that do not conduct electricity but permit transmission of electrostatic forces via localized molecular charge displacement.

• Non-polar Dielectrics (e.g., O₂, N₂, CO₂): Centers of positive and negative charges coincide; individual molecules have zero permanent dipole moment. An external field stretches electron clouds, creating induced dipole moments.
• Polar Dielectrics (e.g., H₂O, HCl, NH₃): Possess permanent dipole moments due to asymmetric molecular geometry. In external fields, thermal agitation is overcome, aligning dipoles along E₀.
• Electric Polarization Vector (P): Dipole moment per unit volume: P = χ_e · ε₀ · E, where χ_e is electric susceptibility.
• Reduced Electric Field inside Dielectric: Bound surface charges create an internal opposing field E_p, reducing net field: E = E₀ - E_p = E₀ / K, where K (or εᵣ) is the dielectric constant (K > 1).

Capacitance (C) is the ability of a conductor or system of conductors to store electric charge and energy for a given electric potential:

For an isolated spherical conductor of radius R in vacuum: C = 4πε₀R. (To realize 1 Farad, an isolated sphere would require radius R ≈ 9 × 10⁹ m, 1400 times Earth radius!).

Parallel Plate Capacitor in Vacuum:

Consider two parallel conducting plates of cross-sectional area A separated by distance d with surface charge densities +σ and -σ:

• Electric field between plates: E = σ / ε₀ = Q / (ε₀A).
• Potential difference between plates: V = E · d = (Qd) / (ε₀A).
• Capacitance: C₀ = Q / V = (ε₀A / d).

Parallel Plate Capacitor with Dielectric Slab (Thickness t < d, Constant K):

When the dielectric completely fills the plate gap (t = d): C = K · (ε₀A / d) = K · C₀. Insertion of a dielectric always amplifies capacitance by factor K.

Charging a capacitor requires an external agent to transfer incremental charges dq against the accumulating potential v = q/C. Integrating from 0 to total charge Q:

Crucial Battery Work Comparison: The total work done by a battery of voltage V supplying charge Q is W_battery = Q·V. Since stored energy is only U = (1/2)QV, exactly 50% of the energy supplied by the battery is inevitably dissipated as Joule heat in the connecting leads and electromagnetic radiation during charging!

Electrostatic energy is physically stored inside the electric field pervading the space between plates. Dividing total energy by volume (A·d):

In a dielectric medium of permittivity ε = Kε₀, the energy density generalizes to: u_E = (1/2) ε E² = (1/2) K ε₀ E².

When two isolated conductors or capacitors (C₁, V₁) and (C₂, V₂) are connected by a conducting wire, charges flow until they reach a shared common potential (V_c):

Energy Dissipated during Redistribution (ΔU):

Since (V₁ - V₂)² ≥ 0, ΔU is strictly positive whenever V₁ ≠ V₂. This lost electrostatic energy appears as spark discharge, thermal Joule heat in the wire, and transient electromagnetic radiation.

The Van de Graaff generator is an electrostatic machine designed by Robert J. Van de Graaff (1929) to produce voltages of several million volts (MV) to accelerate subatomic particles. It operates on two foundational principles:

1. Discharge Action of Sharp Points (Corona Discharge): Near sharply pointed metal combs, high electric fields (E = σ/ε₀) ionize surrounding air, spraying charge onto a moving insulating belt.
2. Complete Transfer of Charge to Outer Shell: When a charged conductor is connected internally to an outer hollow conducting sphere, ALL charge transfers instantaneously to the outer sphere regardless of the shell potential.