Gauss's Law Calculator — flux, enclosed charge and field

The Gauss's law calculator above applies the first of Maxwell's equations in the form most people actually use it: total electric flux out of a closed surface equals the charge enclosed divided by the permittivity. From that one statement it derives the field for the four symmetric distributions where the derivation is clean, and it works the relation backwards to recover a charge from a measured flux.

Arb Digital builds free physics calculators that each own one job. The electric field calculator works from the field definition and superposition; the Coulomb's law calculator gives the force between two point charges. This page is the flux route: the one that turns a symmetry argument into a field without any integration.

What This Gauss's Law Calculator Does

Electric flux measures how much field passes through a surface. For a uniform field crossing a flat surface it is simply the field times the area times the cosine of the angle between them. For a closed surface in any field it is the surface integral of the field's outward component, and Gauss's law says that integral equals the enclosed charge divided by the permittivity.

The tool reports that flux, the field magnitude at the distance you specify, the area of the natural Gaussian surface for the chosen symmetry, and the electric displacement D at the same point. It also echoes the enclosed charge, which is trivial in forward mode and is the answer in reverse mode.

The four distributions cover the standard cases. A point charge and a sphere both give an inverse-square field outside; the sphere additionally has an interior solution that rises linearly with radius. An infinite line gives a field falling as one over distance. An infinite sheet gives a field that does not fall off at all, which is the most counter-intuitive result in elementary electrostatics and the reason a parallel-plate capacitor has a uniform gap field.

How to Use It

1. Choose the distribution first. It decides which of the charge inputs matters and which Gaussian surface the tool builds around it.
2. Enter the charge, line density or surface density. Negative values are fine and produce a negative flux, meaning field lines entering the surface rather than leaving it.
3. Set the distance to the field point. For the sphere, compare it against the sphere radius: inside and outside are genuinely different solutions.
4. Change the relative permittivity for a dielectric. It divides both the flux and the field, which is exactly how a dielectric weakens the field for a given free charge.
5. Switch to reverse mode to recover a charge. Enter a measured flux and the tool returns the charge that must be inside the surface producing it.

The Formula: How Gauss's Law Is Applied

The law states that the closed-surface integral of E dotted with the outward normal equals q_enc divided by ε. OpenStax University Physics Volume 2, section 6.2 on explaining Gauss's law, gives it in exactly that form and makes the key point that it holds regardless of the shape or size of the surface, as long as the enclosed charge is the same. Section 6.1 on electric flux defines the flux itself as the dot product of the field with the area for a uniform field on a flat surface, and as a surface integral otherwise.

Applying it needs a surface on which the field magnitude is constant and the field is either perpendicular or parallel to the surface everywhere. For a point charge or a sphere that is a concentric sphere of area 4πr², giving E = q ÷ (4πεr²). For a line it is a coaxial cylinder of area 2πrL, giving E = λ ÷ (2πεr). For a sheet it is a pillbox crossing the plane, giving E = σ ÷ (2ε) with no distance dependence at all. Georgia State University's HyperPhysics page on Gauss's law sets out each of these derivations and stresses that the law's practical value is precisely this simplification for symmetric distributions.

Inside a uniformly charged solid sphere the enclosed charge grows as the cube of radius while the surface area grows as the square, so the field rises linearly: E = qr ÷ (4πεR³). Inside a conducting shell, by contrast, all the charge sits on the outer surface and the enclosed charge is zero, so the field is exactly zero everywhere inside — the basis of electrostatic shielding.

Work the defaults by hand. A point charge of 1 µC in vacuum: the flux is 10⁻⁶ ÷ 8.8541878 × 10⁻¹² = 112,941 N·m²/C, independent of any distance. The field at 0.1 m is 8.98755 × 10⁹ × 10⁻⁶ ÷ 0.01 = 8.9876 × 10⁵ V/m. Cross-check: the Gaussian sphere at that radius has an area of 4π × 0.01 = 0.12566 m², and 8.9876 × 10⁵ × 0.12566 = 112,941, which is the flux again.

Why the Surface Shape Does Not Matter

The result that catches students out is that a cube, a sphere and a wildly irregular blob all give the same flux if they enclose the same charge. It looks like it should depend on distance, because the field certainly does.

It does not, and the reason is a cancellation. Moving the surface further out weakens the field as one over distance squared while growing the area as distance squared. The two effects cancel exactly, and only because the field of a point charge is exactly inverse-square. Gauss's law in this form is therefore equivalent to the inverse-square law; experimental tests of one are tests of the other.

Charges outside the surface contribute nothing either, and that is a separate cancellation. Field lines from an external charge enter the surface somewhere and leave it somewhere else, so their inward and outward contributions cancel exactly. Only lines that begin or end inside — that is, lines from enclosed charge — produce net flux.

The Infinite Sheet and Why the Field Does Not Fall Off

Of the four cases, the infinite sheet is the one people distrust. The field is σ divided by twice the permittivity, and r does not appear. Move a metre away and the field is unchanged; move a kilometre away and it is still unchanged.

The resolution is geometric. As you retreat from a point charge you see it shrink; as you retreat from an infinite plane you see more of it, and the extra area contributing to your field exactly compensates for the greater distance. Nothing is infinite in practice, of course, which is why the result holds only while your distance from the sheet is small compared with the sheet's own dimensions.

That condition is easy to meet in a capacitor, where the plate separation is tiny compared with the plate size, and it is why the field between capacitor plates is uniform. Two oppositely charged sheets give σ/ε between them by superposition and zero outside — the standard parallel-plate result. Our capacitance calculator takes it from there, and the field energy density calculator gives the energy that uniform field is storing.

What Gauss's Law Cannot Do for You

Gauss's law is always true. It is not always useful. It gives a field only when you can find a surface on which E has constant magnitude and a fixed orientation relative to the surface, and that requires the charge distribution to have spherical, cylindrical or planar symmetry.

Two point charges side by side have no such symmetry. Neither does a finite rod, a charged disc, a dipole, or any real object with edges. For those you must go back to integrating Coulomb's law over the distribution, or superpose the fields of individual charges. Gauss's law still holds for them — the flux out of any surface still equals the enclosed charge over epsilon — but that single equation cannot be unwrapped to give E at a point.

A related subtlety: the law constrains the total flux, not the field at any individual location. A surface can have a large flux with a weak field spread over a big area, or a small flux with an intense field concentrated on part of it. Only symmetry lets you conclude anything pointwise. For the potential rather than the field, use the electric potential calculator, and for charge unit arithmetic the electric charge converter.

Common Mistakes to Avoid

• Counting charge outside the surface — only enclosed charge contributes to net flux. External charges affect the field at every point on the surface but contribute nothing to the total.
• Assuming zero flux means zero field — a surface with equal positive and negative charge inside has zero net flux while the field on it is nowhere near zero.
• Using Gauss's law on an unsymmetric distribution — the law still holds, but it cannot be solved for E. Two point charges or a finite rod need integration instead.
• Confusing a solid sphere with a conducting shell — the field inside a uniformly charged solid sphere rises linearly with radius; inside a conductor it is exactly zero.
• Applying the infinite-sheet result too far away — the distance-independent field only holds while you are much closer to the sheet than the sheet is wide.

Related Free Tools From Arb Digital

For fields from arbitrary charges use the electric field calculator, and for the force between two of them the Coulomb's law calculator. Potential is handled by the electric potential calculator, unit arithmetic by the electric charge converter, and the parallel-plate case by the capacitance calculator. The energy that a field stores is covered by the field energy density calculator. Everything Arb Digital publishes is listed on the free online tools hub.

Frequently Asked Questions

This tool is provided for educational and study use. It assumes idealised static charge distributions in a uniform linear medium, with truly infinite lines and sheets where those are selected, so treat its output as a physics teaching result rather than a measurement of a real object.