5.9.1. Gauss's Law for Electricity

Table of Contents

Electric Flux Through a Closed Surface

Gauss's law for electricity connects electric charge to electric flux. The central idea is that electric charges act as sources or sinks of electric field, and the total electric flux through any closed surface depends only on how much charge is enclosed by that surface.

A closed surface is a surface that completely surrounds a volume, such as a sphere, cube, or cylinder with end caps. When electric field lines pass outward through such a surface, they contribute positive flux. When they pass inward, they contribute negative flux. Gauss's law tells us that if we add up the total flux over the entire closed surface, the result is determined by the enclosed charge.

For any closed surface,
$$
\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}
$$
This is Gauss's law for electricity.
Here, $\mathbf{E}$ is the electric field, $d\mathbf{A}$ is an outward area element of the closed surface, $Q_{\text{enc}}$ is the net charge inside the surface, and $\varepsilon_0$ is the permittivity of free space.

The symbol $\oint$ means that the integral is taken over a closed surface. The quantity $\mathbf{E} \cdot d\mathbf{A}$ is the component of the electric field perpendicular to the surface, multiplied by the small area element.

Meaning of the Law

Gauss's law does not say that electric field depends only on enclosed charge at every point. It says that the total flux through the whole closed surface depends only on the net charge inside. Charges outside the surface can affect the field on the surface, but their contributions to the total flux cancel out.

This is an important distinction. A charge outside a closed surface can produce field lines that enter the surface in one place and leave it in another. The inward and outward contributions balance, so the net flux from outside charges is zero.

Only the net charge enclosed by the closed surface appears in Gauss's law.
Charges outside the surface may change the electric field locally, but they do not change the total enclosed flux.

Surface Integral Form

The law is usually written in integral form as

$$
\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}
$$

If the surface is divided into many tiny patches, then each patch contributes a small flux

$$
d\Phi_E = \mathbf{E} \cdot d\mathbf{A}
$$

and the total electric flux is

$$
\Phi_E = \oint \mathbf{E} \cdot d\mathbf{A}
$$

If the electric field is perpendicular to the surface and has the same magnitude everywhere on it, then the integral becomes much simpler:

$$
\Phi_E = EA
$$

where $A$ is the total area of the surface.

Example with a Point Charge

Consider a point charge $q$ at the center of a sphere of radius $r$. By symmetry, the electric field has the same magnitude everywhere on the sphere and points radially outward.

The field magnitude is

$$
E = \frac{1}{4\pi \varepsilon_0}\frac{q}{r^2}
$$

The surface area of the sphere is

$$
A = 4\pi r^2
$$

Since $\mathbf{E}$ is parallel to $d\mathbf{A}$ everywhere,

$$
\Phi_E = \oint \mathbf{E}\cdot d\mathbf{A} = E(4\pi r^2)
$$

Substituting for $E$ gives

$$
\Phi_E = \left(\frac{1}{4\pi \varepsilon_0}\frac{q}{r^2}\right)(4\pi r^2) = \frac{q}{\varepsilon_0}
$$

This is exactly Gauss's law.

An important result appears here. The flux does not depend on the radius of the sphere. A larger sphere has weaker field, but larger area. These effects cancel.

Rendering scientific sketch…

Point charge inside a spherical Gaussian surface

Positive and Negative Enclosed Charge

If the enclosed charge is positive, the net flux is positive. More field lines leave the surface than enter it. If the enclosed charge is negative, the net flux is negative. More field lines enter the surface than leave it.

This matches the idea that positive charges are sources of electric field and negative charges are sinks of electric field.

Enclosed charge	Direction of net field line flow	Net flux
$Q_{\text{enc}} > 0$	Outward	Positive
$Q_{\text{enc}} < 0$	Inward	Negative
$Q_{\text{enc}} = 0$	Balanced inward and outward	Zero

When Gauss's Law Is Most Useful

Gauss's law is always true, but it is especially useful for calculating electric fields when there is a high degree of symmetry. Typical symmetric cases are spherical symmetry, cylindrical symmetry, and planar symmetry. In such cases, one can choose a Gaussian surface so that the electric field has constant magnitude over part or all of the surface, or is perpendicular or parallel to the area element in a simple way.

Without symmetry, Gauss's law may still be true but not very helpful for finding $\mathbf{E}$ directly.

Gauss's law is most powerful when symmetry allows the surface integral to simplify.
The usual strategy is to choose a Gaussian surface that matches the symmetry of the charge distribution.

Gaussian Surface

A Gaussian surface is an imaginary closed surface chosen for applying Gauss's law. It is not a physical object. Its shape is selected to make the flux integral easy to evaluate.

Common choices are shown below.

Symmetry of charge distribution	Useful Gaussian surface
Spherical	Sphere
Cylindrical	Cylinder
Planar	Pillbox, a short cylinder

The Gaussian surface should be chosen so that one or more of these conditions hold: the field has constant magnitude on a surface part, the field is perpendicular to the surface where it contributes flux, or the field is parallel to the surface where the flux becomes zero.

Outside Charges and Zero Net Contribution

Suppose a charge lies outside a closed surface. The field due to that charge can pass through the surface, but it cannot produce net flux through it. Any field line that enters the surface must also leave it.

Rendering scientific sketch…

External charge and zero net flux contribution

This is why Gauss's law involves only enclosed charge.

Several Charges Inside the Surface

If more than one charge lies inside the closed surface, the enclosed charge is the algebraic sum of all of them:

$$
Q_{\text{enc}} = q_1 + q_2 + q_3 + \cdots
$$

Then

$$
\oint \mathbf{E}\cdot d\mathbf{A} = \frac{q_1 + q_2 + q_3 + \cdots}{\varepsilon_0}
$$

A positive and a negative charge inside the same surface can partially or completely cancel in the total flux.

For example, if a surface encloses $+3 \text{ C}$ and $-1 \text{ C}$, then

$$
Q_{\text{enc}} = 2 \text{ C}
$$

so the net flux is

$$
\Phi_E = \frac{2}{\varepsilon_0}
$$

in SI units.

Differential Form

Gauss's law also has a local form, called the differential form:

$$
\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}
$$

Here, $\rho$ is the volume charge density, and $\nabla \cdot \mathbf{E}$ is the divergence of the electric field. This form says that the electric field diverges outward from regions of positive charge density and converges inward toward regions of negative charge density.

For beginners, the integral form is usually easier to understand first, because it directly relates total flux to enclosed charge.

Differential form of Gauss's law:
$$
\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}
$$
Integral form and differential form express the same physical law.

Vacuum Permittivity

The constant $\varepsilon_0$ is the permittivity of free space. Its approximate value is

$$
\varepsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2/(\text{N}\cdot \text{m}^2)
$$

It sets the scale for how electric charges produce electric field in vacuum.

A Simple Symmetry Application

For a point charge at the center of a spherical Gaussian surface, Gauss's law gives

$$
E(4\pi r^2) = \frac{q}{\varepsilon_0}
$$

$$
E = \frac{q}{4\pi \varepsilon_0 r^2}
$$

This reproduces the inverse square form of the electric field of a point charge. In more advanced situations, Gauss's law is often used in the reverse direction, starting from symmetry to find the electric field.

Common Misunderstandings

A common mistake is to think that zero net flux means zero electric field everywhere on the surface. That is not true. The field may be nonzero at many points, but the total inward and outward contributions can cancel.

Another common mistake is to include charges outside the Gaussian surface in $Q_{\text{enc}}$. Only charges inside the closed surface count.

A third mistake is to apply Gauss's law to an open surface. The standard form of Gauss's law applies to closed surfaces only.

Important cautions:
Zero net flux does not necessarily mean zero electric field.
Only enclosed charge contributes to $Q_{\text{enc}}$.
Gauss's law applies to closed surfaces.

Summary

Gauss's law for electricity is one of Maxwell's equations. It states that the total electric flux through any closed surface equals the net enclosed charge divided by $\varepsilon_0$: