Gauss's Law (Electric):

Gauss's Law (Electric): Definition and Mathematical Representation

Introduction

Gauss’s Law is a fundamental law in electrostatics that relates the electric flux through a closed surface to the total electric charge enclosed by that surface. It is one of the four equations in Maxwell's Equations and provides a powerful method for calculating electric fields, especially with high symmetry.

Mathematical Formulation

Integral Form

${\oint }_{\text{closed surface}}\overrightarrow{E}\cdot d\overrightarrow{A}=\frac{Q_{\text{enc}}}{{\epsilon }_0}$

Where:

• $\overrightarrow{E}$ is the electric field vector,
• $d\overrightarrow{A}$ is an infinitesimal area vector on the closed surface (pointing outward),
• $Q_{\text{enc}}$ is the total electric charge enclosed within the surface,
• ${\epsilon }_0$ is the vacuum permittivity (${\epsilon }_0\approx 8.854\times 10^{-12}{\text{C}}^2/{\text{N·m}}^2$).

Differential Form

Using the divergence theorem, Gauss's Law can also be expressed in differential form:

$\mathrm{\nabla }\cdot \overrightarrow{E}=\frac{\rho }{{\epsilon }_0}$

Where:

• $\mathrm{\nabla }\cdot \overrightarrow{E}$ is the divergence of the electric field,
• $\rho$ is the volume charge density (charge per unit volume).

Physical Meaning

Gauss's Law states that the total electric flux through a closed surface is proportional to the amount of electric charge enclosed within that surface. The law implies:

• A net outward flux occurs when positive charge is enclosed.
• A net inward flux occurs when negative charge is enclosed.
• If no charge is enclosed, the net electric flux is zero.

Applications of Gauss’s Law

Gauss’s Law is especially useful when dealing with problems involving symmetry:

1. Spherical Symmetry

For a point charge $q$ at the center of a spherical surface of radius $r$:

$E\cdot 4\pi r^2=\frac{q}{{\epsilon }_0}\Rightarrow E=\frac{1}{4\pi {\epsilon }_0}\cdot \frac{q}{r^2}$

2. Cylindrical Symmetry

For an infinite line charge with linear charge density $\lambda$:

$E\cdot (2\pi rL)=\frac{\lambda L}{{\epsilon }_0}\Rightarrow E=\frac{\lambda }{2\pi {\epsilon }_{0}r}$

3. Planar Symmetry

For an infinite plane of surface charge density $\sigma$:

$E=\frac{\sigma }{2{\epsilon }_0}$

Conditions for Application

• Symmetry is essential — spherical, cylindrical, or planar.
• The electric field must be constant in magnitude over the chosen Gaussian surface.
• The surface must be closed.

Relation to Coulomb's Law

Gauss’s Law is consistent with Coulomb’s Law and can be derived from it for point charges. Conversely, Coulomb’s Law can also be derived from Gauss’s Law under the assumption of spherical symmetry.

See Also

• Electric Field
• Electric Flux
• Coulomb's Law
• Maxwell's Equations
• Charge Density
• Vacuum Permittivity
• Electrostatics