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Linear Charge Density Formula

Electricity and magnetism formulasScalarClass 12Updated 1 Nov 2026

λ=qL\lambda = \frac{q}{L}
SI unit
C/m
Dimensions
[M⁰ L⁻¹ T¹ I¹]
where
  • λ linear charge density C/m
  • q charge on the line C
  • L length of the line m

The linear charge density formula is λ=qL\lambda = \frac{q}{L}, meaning the charge q on a thin object divided by its length L. Lambda tells you how many coulombs sit on every metre, so its unit is C/m.

It is the right measure whenever charge lies along something long and thin compared with its thickness: a charged thread, a straight wire, a rod or a ring. In Class 12 the main use is the field of an infinitely long line charge, E=λ2πε0rE = \frac{\lambda}{2\pi\varepsilon_0r}, one of the three standard Gauss's law results.

Quick facts

  • Formula

    λ=qL\lambda = \frac{q}{L}

  • SI unit

    coulomb per metre (C/m)

  • Dimensional formula

    [M⁰ L⁻¹ T¹ I¹]

  • Ring of radius R

    λ=q2πR\lambda = \frac{q}{2\pi R}

  • Field of a long line

    E=λ2πε0r=2kλrE = \frac{\lambda}{2\pi\varepsilon_0 r} = \frac{2k\lambda}{r}

  • Non uniform line

    q=∫λ dxq = \int \lambda\,dx

Terms in the linear charge density formula

Reading λ = q / L
SymbolNameSI unit
λ\lambdaCharge per metre of lengthC/m
qqTotal charge carried, sign includedC
LLLength along which it liesm

For a uniform rod λ is the same everywhere. If it varies, say λ=λ0x\lambda = \lambda_0 x, a small piece dx carries dq=λ dxdq = \lambda\,dx, and you add these pieces by integration to get the total charge or field. JEE problems use this often; board exams mostly stay with the uniform case.

Steps for a line charge problem

  1. Identify the shape: straight rod, arc or full ring.
  2. Find its length in metres. For a ring, use the circumference 2πR2\pi R; for an arc of angle θ in radians, use RθR\theta.
  3. Convert the charge to coulombs.
  4. Divide to get λ, or multiply λ by length to get q.

The calculator above does the last step once the length is known.

Worked examples

Example 1: a charged rod

A plastic rod 1.5 m long carries 6.0 μC spread evenly. Find λ.

λ=6.0×10−61.5=4.0×10−6 C/m\lambda = \frac{6.0\times10^{-6}}{1.5} = 4.0\times10^{-6}\ \text{C/m}

Example 2: a ring given its radius in cm

A thin ring of radius 20 cm carries 5.0 nC. Find the linear charge density.

L = 2π(0.20) = 1.257 m.

λ=5.0×10−91.257≈4.0×10−9 C/m\lambda = \frac{5.0\times10^{-9}}{1.257} \approx 4.0\times10^{-9}\ \text{C/m}

Example 3: field of a long wire

A very long straight wire has λ = 4.0 μC/m. Find the field 10 cm from it.

E=2kλr=2(8.99×109)(4.0×10−6)0.10≈7.2×105 N/CE = \frac{2k\lambda}{r} = \frac{2(8.99\times10^{9})(4.0\times10^{-6})}{0.10} \approx 7.2\times10^{5}\ \text{N/C}

The field points radially away from the wire because λ is positive.

Why the field falls as 1/r

Wrap an imaginary cylinder of radius r and length ℓ around the line. The charge inside is λℓ\lambda\ell, and the field crosses only the curved surface, of area 2πrℓ2\pi r\ell. Gauss's law gives

E(2πrℓ)=λℓε0⇒E=λ2πε0rE(2\pi r\ell) = \frac{\lambda\ell}{\varepsilon_0} \quad\Rightarrow\quad E = \frac{\lambda}{2\pi\varepsilon_0r}

Compare the point charge, where E falls as 1/r², and the flat sheet, where E does not fall at all. The flux through any closed surface is covered on the electric flux page.

The 1/r result needs a line much longer than the distance r. Near the ends of a short rod it is only approximate.

Linear, surface and volume charge density compared

The three charge densities
DensityFormulaUnitUsed for
Linearλ=q/L\lambda = q/LC/mWires, rods, rings
Surfaceσ=q/A\sigma = q/AC/m²Plates, shells, conductors
Volumeρ=q/V\rho = q/VC/m³Charged solids, clouds of charge

Each has its own page: surface charge density, volume charge density and the overview at charge density formula.

Points that trip students up

  • Using the diameter of a ring instead of its circumference.
  • Putting the arc angle in degrees when finding Rθ.
  • Writing 1/r² for a line charge field. It is 1/r.
  • Mixing λ for charge density with λ for wavelength in the same problem; label it clearly.

The electricity hub collects more formulas for this topic, from fields to capacitors.

Frequently asked questions

What is the formula of linear charge density?

λ = q / L, the charge divided by the length of the line it sits on.

What unit does linear charge density have?

Coulomb per metre (C/m). Because charge is current times time, the dimensions come to [M⁰ L⁻¹ T¹ I¹].

How do you find linear charge density of a ring?

Divide the ring's charge by its circumference: λ = q / (2πR).

What is the electric field of an infinite line charge?

E = λ / (2πε₀r), or 2kλ / r, pointing radially away from a positive line.

How does linear charge density differ from surface charge density?

Linear density is charge per unit length, in C/m. Surface density is charge per unit area, in C/m².