Electric Field of Line Charge

The electric field of an infinite line charge is a fundamental result in electrostatics that demonstrates how symmetry simplifies field calculations. This result appears directly in problems related to coaxial cables, cylindrical conductors, and transmission line analysis. In GATE, this formula and its derivation using Gauss's Law are frequently tested.

Core Concept Explanation

An infinite line charge is an idealised charge distribution where charge is distributed uniformly along an infinitely long straight line. The charge per unit length is called the linear charge density ρ_L with units of coulombs per metre (C/m). Although a truly infinite line does not exist physically, this model accurately describes the field of long cylindrical conductors when the distance of interest is much less than the length of the conductor.

Due to the symmetry of the infinite line charge, the electric field has no component along the axis of the line (z-direction) and no variation with azimuthal angle φ. The field is purely radial, pointing perpendicular to the line outward for a positive ρ_L. This cylindrical symmetry is the key that makes Gauss's Law the most efficient method for deriving the field.

Mathematical Expression

The Gauss's Law approach encloses the line charge in a cylindrical Gaussian surface of radius r and length L. The total enclosed charge is Q_enc = ρ_L × L. Since the field is purely radial, the flux through the top and bottom caps is zero, and the flux through the curved surface is E × (2πr L). Applying Gauss's Law: E × 2πr L = ρ_L L / ε₀. Simplifying gives the electric field magnitude as E = ρ_L / (2πε₀ r).

In vector form, E = (ρ_L / 2πε₀ r) ρ̂, where ρ̂ is the unit vector in the cylindrical radial direction pointing away from the line. If ρ_L is negative, the field points toward the line. The field decreases as 1/r (not 1/r² as for a point charge), because the line extends infinitely in both directions, giving a weaker distance dependence than a point source.

Practical Understanding

The 1/r dependence is physically important. A coaxial cable consists of an inner conductor (modelled as a line charge) surrounded by a cylindrical outer conductor. The electric field between them follows E = ρ_L / (2πε₀ r), which is the basis for calculating the capacitance per unit length of the coaxial cable. The result C = 2πε₀ / ln(b/a) per metre is derived by integrating this field.

In cylindrical coordinate systems, the field from a line charge serves as the Green's function basis for more complex cylindrical geometries. When the line charge is not at the origin but at some offset position, the field is computed using the perpendicular distance from the line to the field point.

Given:
ρ_L = 20 nC/m = 20×10⁻⁹ C/m
r = 0.5 m (perpendicular distance from line)
Medium: free space

Why this formula applies:
Infinite line charge with cylindrical symmetry — field derived from Gauss's Law gives E = ρ_L/(2πε₀r).

Formula:
E = ρ_L / (2πε₀ r)

Substitution:
E = (20×10⁻⁹) / (2π × 8.854×10⁻¹² × 0.5)

Calculation:
Denominator = 2 × 3.1416 × 8.854×10⁻¹² × 0.5
            = 2.781×10⁻¹¹
E = (20×10⁻⁹) / (2.781×10⁻¹¹)

Final Answer:
E ≈ 719.4 V/m  (radially outward from the line)

Exam Tip: The field of a line charge decreases as 1/r while a point charge decreases as 1/r². In GATE, questions sometimes swap the two formulas as a trap. Remember — for a line charge, no r² in denominator. If a medium is present, replace ε₀ with ε₀εr.

• Electric field of an infinite line charge is purely radial due to cylindrical symmetry — no axial or azimuthal component.
• The formula E = ρ_L / (2πε₀ r) is derived using Gauss's Law with a cylindrical Gaussian surface.
• Field decreases as 1/r, not 1/r², because the line source is extended in one dimension.
• In a medium with permittivity εr, replace ε₀ with ε₀εr in the denominator.
• Application: electric field between conductors of a coaxial cable follows this same 1/r law.

Quick Revision

• E = ρ_L / (2πε₀ r) ρ̂ — radially outward for positive line charge.
• Derived from Gauss's Law using cylindrical Gaussian surface of radius r and length L.
• 1/r dependence (not 1/r²) — a key distinguishing feature from point charge fields.
• End caps of Gaussian cylinder contribute zero flux because E is perpendicular to the cap normal.
• ρ_L units: C/m. E units: V/m.
• Trap: do not confuse ρ_L (line charge density) with ρ (radial distance in cylindrical coordinates).
• Coaxial cable capacitance = 2πε₀ / ln(b/a) per metre — derived from this field formula.