Overview: Gauss' Theorem

The arc of a circle subtends an angle at the centre of the circle. This angle is called a 'plane angle'.

“1 radian is the angle which an arc of length equal to the radius of a circle subtends at the centre of the circle.”

“1 steradian is the solid angle subtended by a part of the surface of a sphere at the centre of the sphere, when the area of the part is equal to the square of the radius of the sphere.”

The electric flux is a measure of the number of lines of force passing through some surface held in the electric field. It is denoted by Ф_E.

OR

The electric flux through a surface is the dot product of the electric field and the area vector of the surface, is called electric flux.

The electric flux linked with a surface in an electric field may be defined as the surface integral of the normal component of the electric field over that surface.

In an electric field, the ratio of electric flux through a surface to the area A of the surface is called the 'electric flux density' at the location of the surface.

Mathematical Definition:

Electric flux density = \[\frac {Φ_E}{A}\]

For a plane surface normal to the electric field:

Electric flux density = \[\frac {E A}{A}\] = E

\[\Phi_{E}=\int_{A}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}\]

where,
∫_A = is the (surface) integral over the entire surface
Φ_E =  is positive when lines leave the surface, negative when they enter.

OR

Electric flux through a small area element:

dΦ = E ⋅ dS = E dS cos θ

Total electric flux through a surface:

Φ = \[\int_S\vec{E}\cdot d\vec{S}\]

Ф_Е = E A cos θ

• If the plane surface is normal to the electric field (θ = 0):
  ФЕ = ЕА cos 0 = ЕА
• If the plane surface is parallel to the electric field (θ = 90°):
  Or = E A cos 90° = 0
• For field lines entering the plane surface normally (θ = 180°):
  ФЕ = ЕА cos 180° = -EA

Statement

Gauss’s theorem in electrostatics states that the total electric flux through any closed surface (called a Gaussian surface) is equal to \[\frac {1}{ε_0}\] times the net charge enclosed by the surface, irrespective of the shape and size of the surface.

Mathematical Form

Φ_E = \[\oint\vec{E}\cdot d\vec{A}=\frac{q_{\mathrm{enc}}}{\varepsilon_0}\]​​

where

• \[\vec E\] = electric field intensity
• d\[\vec{A}\] = outward normal area element
• q_enc = net charge enclosed
• ε_0​ = permittivity of free space

Proof (Outline)

Consider a point charge +q placed inside a closed surface.
The electric field at a point on the surface is

E = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\]​

The flux through a small area element dA is

dΦ_E = \[\vec{E}\cdot d\vec{A}=EdA\cos\theta\]

Since dAcos⁡θ = r²dΩ,

dΦ_E = \[\frac{q}{4\pi\varepsilon_0}d\Omega\]

Integrating over the entire closed surface,

Φ_E = \[\frac{q}{4\pi\varepsilon_0}\int d\Omega\]

But the total solid angle subtended by a closed surface is 4π,

Φ_E = \[\frac  {q}{ε_0}\]​

Hence proved.

Key Note

Gauss’s law is most useful for symmetric charge distributions (spherical, cylindrical, planar) and is valid for all inverse-square law fields.

• Gauss’ theorem establishes a connection between the electric flux through a closed surface and the charge enclosed, and is especially useful for highly symmetric charge distributions.
• An area can be treated as a vector quantity, with both magnitude (area) and direction (the outward normal to the surface).
• A solid angle is the three-dimensional analogue of a plane angle and describes how a surface appears from a point.
• The total solid angle subtended at a point by a closed surface, irrespective of its shape, is always the same.
• The solid angle subtended by an area element depends on its orientation and position relative to the point, being maximum when the area faces the point directly.

• SI unit of Electric flux = N·m²·C⁻¹ or V·m (since E = N / C = V/m).
• Dimensions of electric flux: [Φ_E] = [E] [A] = [ML³T⁻³A⁻¹]

• A Gaussian surface is an imaginary closed surface used to calculate the electric flux of a vector field.
• It must be a closed surface (e.g., a sphere, cylinder, or cube); open surfaces such as discs or squares are not valid.
• The shape of the Gaussian surface should match the symmetry of the charge distribution so that the electric field is uniform or normal to the surface.
• The surface must not pass through any discrete charge, though it may pass through a continuous charge distribution.
• Electric flux through a Gaussian surface depends only on the charges enclosed, even though the electric field on the surface is due to both internal and external charges.

• Point Charge:
  Using a spherical Gaussian surface, the electric field due to a point charge is
  E = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\]​
  which directly leads to Coulomb’s law.
• Infinite Line of Charge:
  For a uniformly charged infinite wire with linear charge density λ,
  E = \[\frac{\lambda}{2\pi\varepsilon_0r}\]​
  The field is radial and varies inversely with distance r.
• Infinite Plane Sheet of Charge:
  For a sheet with surface charge density σ,
  E = \[\frac{\sigma}{2\varepsilon_0}\]​
  The field is independent of distance from the sheet.
• Two Parallel Charged Sheets:
  The electric field is uniform between the sheets and zero outside when the sheets carry equal and opposite charges.
• Charged Conductor:
  The electric field inside a conductor is zero, and just outside the surface,
  E = \[\frac{\sigma}{\varepsilon_0}\]​
  where σ is surface charge density.
• Uniformly Charged Spherical Shell / Conducting Sphere:
  Outside the shell: behaves like a point charge at the centre
  Inside the shell: the electric field is zero
• Uniformly Charged Non-conducting Sphere:
  Outside: E ∝ \[\frac {1}{r^2}\]​
  Inside: E ∝ r, increasing linearly from centre to surface