Linear Charge Density Converter

Coulomb/Meter (C/m)

The SI unit of linear charge density. It represents the amount of electric charge per unit length. λ = Q / L.

Common uses: International standards, charged wires, line charge analysis, electromagnetic theory.

Coulomb/Centimeter (C/cm)

Linear charge density using centimeters. 1 C/cm = 0.01 C/m. Convenient for smaller dimensions.

Common uses: Smaller charged objects, laboratory measurements, microscale systems.

Coulomb/Inch (C/in)

Linear charge density using inches. 1 C/in ≈ 0.0254 C/m. Used in some engineering contexts.

Common uses: US engineering, legacy systems, certain industrial applications.

Abcoulomb/Meter (abC/m)

Linear charge density in CGS electromagnetic units. 1 abC/m ≈ 10 C/m. Used in older physics texts.

Note: Obsolete in modern SI applications, but found in historical literature.

Linear Charge Density Definition

Linear charge density is the charge per unit length on a one-dimensional object:

• Definition: λ = Q / L (charge / length)
• SI unit: Coulomb/meter (C/m)
• Can be: Positive or negative depending on type of charge
• Example: A 2-meter wire with 0.5 C has λ = 0.25 C/m

Electric Field from a Charged Wire

Linear charge density is used to calculate electric field around charged wires:

• Infinite line charge: E = λ / (2πε₀r)
• Where: ε₀ = 8.85 × 10⁻¹² F/m (permittivity of free space)
• r: Perpendicular distance from the wire
• Application: Coaxial cables, power transmission lines, capacitors

Typical Linear Charge Density Values

• Moderately charged wire: 10⁻⁹ to 10⁻⁶ C/m (nanofarad range)
• Highly charged wire: 10⁻⁶ to 10⁻³ C/m
• Power transmission line: ~10⁻⁵ to 10⁻³ C/m
• Charged polymer fiber: ~10⁻⁸ to 10⁻⁵ C/m
• Ion beam: Highly variable, 10⁻¹² to 10⁻¹ C/m
• DNA molecule: ~1.3 × 10⁻⁶ C/m (due to phosphate groups)

Related Charge Densities

Charge can also be described in other dimensions:

• Linear charge density (λ): Charge per unit length (C/m)
• Surface charge density (σ): Charge per unit area (C/m²)
• Volume charge density (ρ): Charge per unit volume (C/m³)
• Relationship: λ = σ × A (where A is cross-sectional area)

Gauss's Law and Linear Charge

Gauss's law relates charge to electric field:

• Law: ∮ E·dA = Q_enclosed / ε₀
• For line charge: E × 2πr × L = λ × L / ε₀
• Simplifies to: E = λ / (2πε₀r)
• Application: Fundamental tool for calculating electric fields

Electric Potential from a Line Charge

The electric potential around a charged wire:

• Formula: V = -λ/(2πε₀) × ln(r/r₀)
• λ: Linear charge density
• r: Distance from wire
• r₀: Reference distance (often 1 meter)
• Used in: Capacitor design, coaxial cable analysis

Common Applications

Linear charge density is essential in:

• Coaxial Cables: Center conductor charge distribution
• Cylindrical Capacitors: Charge on inner and outer cylinders
• Power Transmission: High-voltage line analysis and safety
• Electrostatics: Theoretical problem solving, field calculations
• Plasma Physics: Ion beams and charged particles
• Molecular Physics: Charge distribution on molecules and fibers
• Electronics: Wire charging effects, EMI considerations

Force Between Parallel Charged Wires

Two parallel charged wires experience electrostatic forces:

• Force per unit length: F/L = λ₁ × λ₂ / (2πε₀d)
• λ₁, λ₂: Linear charge densities of the two wires
• d: Distance between wire centers
• Direction: Attractive if opposite sign, repulsive if same sign

Energy Associated with Line Charge

Energy calculations for charged wires:

• Energy per unit length: u = ½ × λ × V
• Total energy: U = ½ × Q × V = ½ × λ × L × V
• Energy density: u/L = ½ × (λ²/(2πε₀)) × ln(r₂/r₁)
• Application: Capacitor energy storage, transmission line energy