Electric Flux & Gauss's Law Calculator

Calculate electric flux through surfaces and enclosed charge using Gauss's law.

0°=perpendicular to surface (max flux)
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Electric Flux Measures Field Passing Through a Surface

Electric flux combines electric-field strength, surface area, and orientation into a measure of how much field passes through a surface. For a uniform field crossing a flat area, ΦE=EAcosθ, where θ is measured between the field and the surface normal. A field parallel to the normal gives maximum positive flux; a field tangent to the surface gives zero flux.

Gauss’s law generalizes this to any closed surface: the net outward flux equals enclosed charge divided by ε0. The law is always true, but it becomes a practical field-solving tool only when symmetry lets E be treated directly over the Gaussian surface.

ΦE=∫E·dA,   ∮E·dA=Qenc/ε0
SymbolMeaningWhy it appears / units
ΦEElectric fluxN·m²/C; signed surface integral of the field.
EElectric fieldN/C or V/m.
AAream²; orientation enters through the area vector.
QencEnclosed chargeC; only charge inside the closed surface contributes to net flux.

External charges can produce strong local electric fields on a closed surface yet contribute zero net flux through it. Net flux depends only on enclosed charge, while the detailed field distribution can depend on both internal and external charges.

Gauss’s law depends on enclosed charge, not nearby external charge. External charges can change the local field pattern, but their net flux through a closed surface is zero. For a symmetric problem, confirm that the chosen Gaussian surface makes E·dA easy to integrate before treating E as constant.

Worked Examples

Example 1: Flux: E=1000 N/C, A=0.5m², θ=30°
Φ=1000×0.5×cos30°
Result: 433 N·m²/C
Flux through tilted surface
Example 2: Sphere: Φ=10⁶ N·m²/C
Q=ε₀×Φ=8.854e-12×1e6
Result: Q=8.85×10⁻⁶ C = 8.85 μC
Find charge from total flux
Example 3: Flat surface at 60°
E=500N/C, A=0.020m², θ=60°
Result: Φ=5.0N·m²/C
The cosine uses the angle to the surface normal, not the plane itself.
Example 4: Enclosed charge
Qenc=2nC
Result: net flux≈226N·m²/C
Gauss’s law gives total closed-surface flux without needing the surface shape.

Common Mistakes

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Measuring angle from the surface instead of its normal

Flux uses the angle between E and the area vector, which is perpendicular to the surface.

⚠️
Including outside charge in Qenc

Only enclosed charge appears on the right side of Gauss’s law, even though outside charges affect local field values.

⚠️
Assuming Gauss’s law always makes E easy to calculate

The law is universal, but extracting E directly requires enough symmetry to know its magnitude and direction over the chosen surface.

Frequently Asked Questions

Gauss's law power?
For symmetric charge distributions (spheres, cylinders, planes), Gauss's law gives E instantly without integration. Choose a Gaussian surface matching the symmetry of the charge distribution.
Why flux is zero inside conductor?
Charges redistribute on the surface until E=0 inside. Any enclosed Gaussian surface has Φ=0, so Q_enclosed=0 — all charge is on the surface.
Can electric flux be negative?
Yes. Flux is negative where the electric field points opposite the outward surface normal. A closed surface can have positive, negative, or zero net flux.
Why is flux zero for a field tangent to a surface?
The dot product E·dA contains cos90°=0, so no field component passes through the surface normal direction.
Does zero net flux mean zero electric field everywhere?
No. A closed surface can have strong field entering and leaving in equal amounts, producing zero net flux when enclosed charge is zero.
What makes a good Gaussian surface?
Choose a surface that matches the symmetry of the charge distribution so E is constant or zero on large parts and the dot product becomes easy to evaluate.
When can electric flux be written as EA?
Only when the electric field has uniform magnitude over the surface and is everywhere perpendicular to it does Φ=EA apply directly. In general use Φ=∫E·dA. Gauss’s law is especially powerful when symmetry makes E constant over selected parts of a closed Gaussian surface.

Formula Explorer connections

Interpretation: This formula links charge, voltage, current, resistance, capacitance, power or circuit time response. Assumption: Confirm DC versus AC conditions, RMS versus peak values, component topology and steady-state versus transient behavior. Ideal components may be assumed.

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