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Surface Charge Density Formula

Electricity and magnetism formulasScalarClass 12Updated 1 Nov 2026

σ=qA\sigma = \frac{q}{A}
SI unit
C/m²
Dimensions
[M⁰ L⁻² T¹ I¹]
where
  • σ surface charge density C/m²
  • q charge on the surface C
  • A area over which the charge is spread m²

The surface charge density formula is σ=qA\sigma = \frac{q}{A}: total charge on a surface divided by the area it covers. The Greek letter sigma stands for the result, in coulombs per square metre.

Charge on a conductor never sits inside it; it spreads over the outer skin. So for metal plates, spheres and shells, σ is the natural way to describe how much charge is where. Once you know σ you can write the field near the surface in one line, which is why Class 12 Gauss's law problems lean on it.

Quick facts

  • Formula

    σ=qA\sigma = \frac{q}{A}

  • SI unit

    coulomb per square metre (C/m²)

  • Dimensional formula

    [M⁰ L⁻² T¹ I¹]

  • Sphere of radius R

    σ=q4πR2\sigma = \frac{q}{4\pi R^2}

  • Field of a large sheet

    E=σ2ε0E = \frac{\sigma}{2\varepsilon_0}

  • Field just outside a conductor

    E=σε0E = \frac{\sigma}{\varepsilon_0}

Understanding the surface charge density formula

Symbols in σ = q / A
SymbolMeaningSI unit
σ\sigmaSurface charge densityC/m²
qqCharge on the surface, with its signC
AAArea of the charged surfacem²

A negatively charged plate has a negative σ. If the charge is spread unevenly, the formula gives the average value, and the local value is σ=dqdA\sigma = \frac{dq}{dA}. On an odd shaped conductor σ is largest at sharp points, which is why lightning conductors are pointed.

Area formulas you will need

Surface areas for common shapes
ShapeArea to use
Flat plate or sheetA=length×widthA = \text{length}\times\text{width}
Disc of radius r (one face)A=πr2A = \pi r^2
Solid or hollow sphere, radius RA=4πR2A = 4\pi R^2
Curved surface of a cylinderA=2πrLA = 2\pi rL
For a charged metal plate, check whether the question means charge on one face or on both. The total area of a thin plate counts both faces.

Worked examples

Example 1: a flat sheet

A sheet of area 0.20 m² carries 4.0 μC spread evenly. Find σ.

σ=4.0×10−60.20=2.0×10−5 C/m2\sigma = \frac{4.0\times10^{-6}}{0.20} = 2.0\times10^{-5}\ \text{C/m}^2

Example 2: a metal sphere in cm and nC

A metal sphere of radius 10 cm holds 6.0 nC. Find σ and the field just outside its surface.

A = 4π(0.10)² = 0.126 m².

σ=6.0×10−90.126≈4.8×10−8 C/m2\sigma = \frac{6.0\times10^{-9}}{0.126} \approx 4.8\times10^{-8}\ \text{C/m}^2

E=σε0=4.77×10−88.854×10−12≈5.4×103 N/CE = \frac{\sigma}{\varepsilon_0} = \frac{4.77\times10^{-8}}{8.854\times10^{-12}} \approx 5.4\times10^{3}\ \text{N/C}

Check with kq/R²: (8.99 × 10⁹)(6.0 × 10⁻⁹)/0.010 ≈ 5.4 × 10³ N/C. The two agree.

Example 3: charge from a known density

A disc of radius 5.0 cm has σ = 3.0 × 10⁻⁶ C/m² on one face. Find the charge.

A = π(0.050)² = 7.85 × 10⁻³ m².

q=σA=(3.0×10−6)(7.85×10−3)≈2.4×10−8 Cq = \sigma A = (3.0\times10^{-6})(7.85\times10^{-3}) \approx 2.4\times10^{-8}\ \text{C}

From σ to the electric field

Gauss's law turns σ into a field without any integration for flat or round shapes. For a large thin sheet of charge in empty space, field lines leave both faces, so E=σ2ε0E = \frac{\sigma}{2\varepsilon_0} on each side. For the surface of a conductor, the field inside is zero and all the flux leaves outward, giving E=σε0E = \frac{\sigma}{\varepsilon_0}.

Two oppositely charged parallel plates add their fields between them and cancel outside, giving E=σε0E = \frac{\sigma}{\varepsilon_0} in the gap. That is the start of the capacitance formula for plates, C=ε0AdC = \frac{\varepsilon_0A}{d}. More field results are on the electric field page.

Slips seen in answer scripts

  • Using πR² for a sphere instead of 4πR².
  • Leaving the radius in cm, which makes A too large by 10⁴.
  • Writing σ/ε₀ for an isolated sheet; that sheet needs σ/2ε₀.
  • Dropping the sign of the charge when the question asks about attraction.

Back to the full list of electrostatics and current formulas: electricity formulas.

Frequently asked questions

What is the formula for surface charge density?

σ = q / A, the charge on a surface divided by its area. For a sphere, σ = q / (4πR²).

In what units is surface charge density given?

Coulomb per square metre (C/m²), with dimensions [M⁰ L⁻² T¹ I¹].

Is surface charge density a vector?

No. It is a scalar that can be positive or negative depending on the sign of the charge.

Where is surface charge density highest on a conductor?

At the sharpest points, where the radius of curvature is smallest. Flat or gently curved parts have a lower σ.

What is the relation between surface charge density and electric field?

Just outside a charged conductor, E = σ / ε₀. Near a large isolated sheet of charge, E = σ / (2ε₀).