112,940.906684
N·m²/C
898,755.1788
N/C
898.755179
kV/m
1
0.125664
m²
112,940.906684
N·m²/C
898,755.1788
N/C
898.755179
kV/m
1
0.125664
m²
The Gauss's Law Calculator applies one of the four Maxwell equations to compute electric flux and electric field strength for symmetric charge distributions. Gauss's Law is a cornerstone of electrostatics that relates the total electric flux through a closed surface to the charge enclosed within that surface:
$$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$
This elegant relationship, first formulated by Carl Friedrich Gauss, provides a powerful shortcut for calculating electric fields when the charge distribution has sufficient symmetry—spherical, planar, or cylindrical. Instead of integrating Coulomb's Law over every charge element, Gauss's Law lets you determine the field from the enclosed charge and the geometry of the Gaussian surface.
This calculator supports four modes. The basic mode computes the total flux from any enclosed charge. The sphere mode calculates the electric field at distance \(r\) from a point charge or uniformly charged sphere using \(E = Q / (4\pi\varepsilon_0 r^2)\). The infinite plane mode finds the field from a sheet of charge with surface charge density \(\sigma\), yielding \(E = \sigma / (2\varepsilon_0)\)—a uniform field independent of distance. The cylinder mode computes the field at distance \(r\) from a long line charge with linear charge density \(\lambda\), giving \(E = \lambda / (2\pi\varepsilon_0 r)\).
Gauss's Law is not merely a computational tool—it embodies a deep physical principle. It tells us that electric field lines originate on positive charges and terminate on negative charges, and that the total flux through any closed surface depends only on the net charge inside, regardless of charges outside. This principle underlies the operation of Faraday cages, capacitors, and electromagnetic shielding.
The permittivity of free space \(\varepsilon_0 = 8.854 \times 10^{-12}\) F/m is the fundamental constant that sets the strength of the electrostatic interaction in vacuum. In materials, \(\varepsilon_0\) is replaced by \(\varepsilon = \kappa\varepsilon_0\), where \(\kappa\) is the dielectric constant. This calculator assumes vacuum (or air, which is very close to vacuum).
Gauss's Law (general form):
$$\Phi_E = \frac{Q_{\text{enc}}}{\varepsilon_0}$$
where \(\varepsilon_0 = 8.854 \times 10^{-12}\) F/m.
Sphere (point charge or uniform sphere):
Gaussian surface: sphere of radius \(r\). By symmetry, \(E\) is constant on the surface:
$$E \cdot 4\pi r^2 = \frac{Q}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{Q}{4\pi\varepsilon_0 r^2}$$
Infinite plane (sheet charge density \(\sigma\)):
Gaussian surface: pillbox straddling the plane. Field is perpendicular and uniform:
$$2EA = \frac{\sigma A}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\sigma}{2\varepsilon_0}$$
Infinite cylinder (linear charge density \(\lambda\)):
Gaussian surface: coaxial cylinder of radius \(r\) and length \(L\):
$$E \cdot 2\pi r L = \frac{\lambda L}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2\pi\varepsilon_0 r}$$
The flux output shows the total electric flux through the Gaussian surface in N·m²/C. Positive flux means net outward field (positive enclosed charge); negative flux means net inward field (negative charge). The electric field is the magnitude at the specified distance or for the given geometry. For the plane, the field is constant everywhere—independent of distance. For the sphere and cylinder, the field decreases with distance (as \(1/r^2\) and \(1/r\) respectively).
Inputs
Results
A +1 µC point charge produces 112,941 N·m²/C of total flux. At 10 cm distance, the electric field is about 899.5 kV/m—strong enough to cause air breakdown near the charge.
Inputs
Results
A plane with σ = 1 µC/m² creates a uniform field of about 56.5 kV/m on each side. This field is the same whether you are 1 mm or 1 km from the plane—a remarkable result of Gauss's Law.
A Gaussian surface is an imaginary closed surface used to apply Gauss's Law. You choose its shape to exploit the symmetry of the charge distribution—a sphere for point charges, a cylinder for line charges, or a pillbox for plane charges. The surface does not need to be a physical object; it is a mathematical construction that makes the flux integral easy to evaluate.
Gauss's Law is always true—it holds for any charge distribution and any closed surface. However, it is only useful for directly calculating the electric field when the distribution has sufficient symmetry (spherical, cylindrical, or planar). For arbitrary distributions, the law still constrains the total flux but does not uniquely determine the field at every point.
The permittivity of free space \(\varepsilon_0 = 8.854187817 \times 10^{-12}\) F/m (farads per meter) is a fundamental constant that characterizes the strength of electrostatic interactions in vacuum. It appears in Coulomb's Law, Gauss's Law, and the capacitance formula for parallel plates. In a dielectric medium, the effective permittivity is \(\varepsilon = \kappa\varepsilon_0\).
By symmetry, the field from an infinite plane must be perpendicular to the surface and uniform at any distance. As you move farther away, each small patch of charge subtends a smaller solid angle, but there are more patches contributing—the two effects exactly cancel. Mathematically, the Gaussian pillbox analysis shows \(E = \sigma / (2\varepsilon_0)\) with no distance dependence.
Yes. Negative enclosed charge produces negative total flux, meaning the net electric field points inward through the Gaussian surface. The calculator handles both positive and negative charges and indicates the flux direction in the output. The electric field magnitude is always reported as a positive value.
For a point charge, applying Gauss's Law with a spherical Gaussian surface of radius \(r\) directly yields Coulomb's Law: \(E = Q / (4\pi\varepsilon_0 r^2)\). The two laws are mathematically equivalent for electrostatics—Gauss's Law is the integral form, while Coulomb's Law gives the field from a single point charge. Gauss's Law is more general because it also applies to continuous charge distributions.
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