Pick a symmetric distribution (solid or hollow sphere, cylinder, line, plane or capacitor) and get the electric field inside and outside with Gauss's law, with the Gaussian surface drawn, the enclosed charge and the E versus r chart.
The electric flux through a closed surface equals the enclosed charge over ε₀: ∮E·dA = Q/ε₀.
Computing the field of a charge distribution by adding up Coulomb's law point by point often requires long integrals. But if the distribution is highly symmetric (a sphere, a long cylinder, a plane), Gauss's law gives the field in a few lines. This guide explains the law and how to choose the Gaussian surface, and works the classic cases.
The electric flux through a surface measures how many field lines cross it: Φ = ∮ E · dA. Gauss's law says that the flux through any closed surface depends only on the charge it encloses:
∮ E · dA = Qenc / ε0, ε0 ≈ 8.854 × 10⁻¹² C²/(N·m²)
The law always holds, but it only lets you solve for E when the symmetry allows choosing a surface on which E is constant and perpendicular to it (or parallel, in which case it adds no flux). Then the integral reduces to E times the area.
1. Uniformly charged solid sphere. Q = 5 nC spread through a sphere of radius R = 10 cm.
2. Conducting sphere. With the same charge on a conducting sphere, all the charge sits on the surface and the field inside is zero; outside it is the same as in case 1.
3. Infinite line. λ = 2 nC/m, at r = 10 cm: E · 2πrL = λL/ε0, so E = λ/(2πε0r) = 2kλ/r ≈ 359.5 N/C. It falls off as 1/r, more slowly than a point charge.
4. Infinite plane. σ = 2 µC/m²: E · 2A = σA/ε0, so E = σ/(2ε0) ≈ 1.129 × 10⁵ N/C, the same at any distance from the plane.
The tool gives the same values in all four cases.
For a finite rod, a ring, an arc or a disk, Gauss's law is still true but does not let you solve for E, because the field is not constant over any simple surface. Then Coulomb's law is integrated over the distribution. The tool also solves those cases with the same interface.
R. A. Serway and J. W. Jewett, Physics for Scientists and Engineers (Cengage); H. D. Young and R. A. Freedman, University Physics (Pearson). Simuveliq is not affiliated with these authors or publishers.
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