1. Enter the Values
2. Calculated Result
Net Electric Flux (Φ_E)
Enter values and click Calculate.
3. Step-by-Step Solution
Enter values above to generate the complete solution.
4. Solid Angle & Spherical Surface Visualization (Fig 3.4)
5. Gauss's Law Graphs
Flux vs Enclosed Charge (Φ vs Q)
Flux increases linearly with the enclosed charge Q.
Electric Field vs Radius (E vs r)
Electric field obeys the inverse-square law with respect to radius.
6. Important Relationships
Flux depends entirely on the enclosed charge.
Flux is independent of Gaussian surface geometry.
Field intensity drops with the square of distance.
Charges outside the surface yield net zero flux.
7. Theory
What is Gauss's Law?
Gauss's Law states that the net electric flux passing through any closed surface is equal to the net charge enclosed by that surface divided by the vacuum permittivity (ε₀).
Integral Form & Solid Angle
dΩ = (u_r · n̂ ds) / r² = ds' / (r')²
Φ_E = ∮_S E · n̂ ds = Q_enc / ε₀
Where:
Φ_E = total electric flux (N·m²/C or V·m)
E = electric field vector (N/C)
ds = differential area element on arbitrary surface S
ds' = projected differential area element on concentric sphere S' of radius r'
Q_enc = net electric charge inside the surface (C)
ε₀ = 8.854 × 10⁻¹² C²/(N·m²)