Gauss's Law Calculator

Gauss's Law relates the electric flux through a closed surface to the electric charge enclosed within it — a fundamental principle in electromagnetism. Using the Gauss's Law Calculator, select your calculation method: enter the electric charge (Q) and your preferred charge unit to calculate flux directly, or enter the electric field magnitude, surface area, and angle between E and the surface normal (θ) to derive it geometrically. The calculator returns the electric flux (Φ) in N·m²/C, along with its scientific notation form, the charge used, and the permittivity of free space (ε₀).

Calculation Method *

nC

Total electric charge enclosed by the surface (in nanocoulombs).

V/m

Magnitude of the uniform electric field in volts per metre.

m²

Area of the closed surface in square metres.

°

Angle between the electric field vector and the outward normal to the surface.

Results

Electric Flux (ϕ)

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Electric Flux (Scientific Notation)

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Charge Used (Q)

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Permittivity of Free Space (ε₀)

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Frequently Asked Questions

What is Gauss's Law?

Gauss's Law states that the total electric flux through any closed surface is equal to the net electric charge enclosed divided by the permittivity of free space (ε₀ ≈ 8.854 × 10⁻¹² C²/N·m²). Mathematically, ϕ = Q / ε₀. It is one of Maxwell's four fundamental equations of electromagnetism.

What is electric flux?

Electric flux (ϕ) is a measure of the number of electric field lines passing through a given surface. It depends on the electric field strength, the area of the surface, and the angle between the field lines and the surface normal. A larger charge or a larger surface area results in greater flux.

How do you calculate electric flux using Gauss's Law?

Using the charge method, the formula is ϕ = Q / ε₀, where Q is the total enclosed charge in coulombs and ε₀ = 8.854 × 10⁻¹² C²/(N·m²). If you know the field instead, use ϕ = E × A × cos(θ), where E is the field magnitude, A is the surface area, and θ is the angle between the field vector and the surface normal.

Does the electric flux depend on the shape or size of the closed surface?

No — according to Gauss's Law, the total electric flux through a closed surface depends only on the net charge enclosed inside it, not on the shape or size of the surface. Whether you use a sphere, a cube, or any other closed surface, the flux remains ϕ = Q / ε₀ as long as the enclosed charge is the same.

Is electric flux a scalar or a vector quantity?

Electric flux is a scalar quantity. Although it is derived from the dot product of the electric field vector and the area vector, the result is a single signed number representing the net flow of field lines outward through the surface.

What are the two types of electric flux?

Electric flux can be positive (outward flux) when field lines exit the closed surface, indicating a net positive charge inside. It is negative (inward flux) when field lines enter the surface, indicating a net negative charge inside. Zero flux means the enclosed positive and negative charges cancel out.

What is the permittivity of free space (ε₀)?

The permittivity of free space, ε₀, is a physical constant with a value of approximately 8.854 × 10⁻¹² C²/(N·m²) or equivalently F/m. It appears in Gauss's Law as the factor relating enclosed charge to electric flux, and it measures how easily an electric field can permeate a vacuum.

When should I use the field-based method instead of the charge-based method?

Use the field-based method (ϕ = E × A × cos θ) when you know the electric field magnitude, surface area, and angle between the field and the surface normal — for example, for a flat surface in a uniform field. Use the charge-based method (ϕ = Q / ε₀) when you know the total enclosed charge, which is the more common form of Gauss's Law.