Electric Flux Calculator

Electric flux measures how many electric field lines pass through a surface. Gauss law states that total flux through a closed surface equals the enclosed charge divided by epsilon_0.

⚡ Electrostatics📐 Φ = EA cosθ📐 Gauss Law
Electric Field E (V/m)
Area A (m²)
Angle θ (deg, between E and surface normal)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Electric FluxΦΦ = E A cosθV·m (or N·m²/C)
Enclosed ChargeQencQenc = ε0ΦC
E from ChargeEE = Q/(4πε0r²)V/m
Epsilon_0ε08.854 × 10⁻¹² C²/N·m²C²/N·m²

Step-by-Step Examples

Example 1
Uniform Field Through Surface

E=500 V/m, A=0.05 m², theta=30°.

  • Φ = E A cosθ = 500 × 0.05 × cos30°
  • Φ = 500 × 0.05 × 0.866 = 21.65 V·m
✓ 21.65 V·m electric flux
Example 2
Gauss Law: Charge from Flux

Sphere encloses Q. Surface area 4πr² = 1.257 m², E = 9000 V/m on surface.

  • Φ = E × A = 9000 × 1.257 = 11,313 V·m
  • Q = ε0Φ = 8.854×10⁻¹² × 11313 = 1×10⁻⁷ C
✓ Enclosed charge = 100 nC
Example 3
Parallel Field (Zero Flux)

E parallel to surface (theta=90°).

  • Φ = E A cos90° = 0
✓ Zero flux when field is parallel to surface

Real-World Applications

⚡
Capacitor Analysis
Gauss law with electric flux determines field between capacitor plates.
📡
Antenna Theory
Flux through surfaces around antennas relates to radiated power via Poynting vector.
🔬
Charge Distribution
Gauss law simplifies E-field calculation for spherical, cylindrical, and planar symmetry.
🧲
Dielectrics
Electric flux through dielectric surfaces relates to free and bound charge distributions.

Common Mistakes to Avoid

⚠️
Confusing flux with field

Electric field E has units V/m. Flux Phi = EA has units V·m. They are different quantities.

⚠️
Forgetting the cosine factor

Only the component of E perpendicular to the surface contributes to flux. Field parallel to surface gives zero flux.

⚠️
Applying Gauss law to non-symmetric charge

Gauss law always holds, but only gives simple field calculation for highly symmetric charge distributions.

Frequently Asked Questions

What is Gauss law in words? ▾
Total electric flux through any closed surface equals the total enclosed charge divided by epsilon_0. It is one of Maxwell four equations.
Why does flux through closed surface equal Q/eps0? ▾
Each charge contributes field lines that must all eventually pass through any surrounding surface. The total number of lines is proportional to charge.
When does Gauss law simplify E calculation? ▾
When charge distribution has spherical, cylindrical, or planar symmetry. The field is constant on the chosen Gaussian surface, so Phi = E * A.
What is permittivity? ▾
epsilon_0 = 8.854e-12 C^2/N*m^2, the permittivity of free space. It relates electric flux to enclosed charge and appears in Coulomb law.
Is flux additive? ▾
Yes. Total flux through a surface equals the sum of fluxes from each individual field. This is the principle of superposition.
What is D-field flux? ▾
For dielectrics, displacement flux Phi_D = Q_free (no epsilon_0 needed). D = epsilon_0*E + P includes polarization P.
How does flux relate to capacitance? ▾
For a parallel plate capacitor: Phi = Q/epsilon_0, E = Q/(epsilon_0 * A), V = E*d, C = Q/V = epsilon_0*A/d.
What is flux tubes? ▾
Electric field lines that pass through a small area element form a flux tube. The product E*A is constant along the tube in free space.

Related Physics Calculators

⚡ Electric Field⚡ Electric Potential⚡ Coulomb Law🧲 Magnetic Flux🔋 Capacitance

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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