Electric flux through a surface is measure of electric field passing through it.

$\Phi_E = \int \vec{E} \cdot d\vec{A} = \int E\,dA\cos\theta$

Where:

• $\vec{E}$ = electric field
• $d\vec{A}$ = area element (vector perpendicular to surface)
• $\theta$ = angle between E and area normal

Units: N - m2/C or V - m

Uniform field, flat surface: $\Phi_E = \vec{E} \cdot \vec{A} = EA\cos\theta$

The dot product $\vec{E} \cdot d\vec{A}$ projects E onto surface normal.

$\vec{E} \cdot d\vec{A} = |\vec{E}||d\vec{A}|\cos\theta$

Physical meaning:

• Component of E perpendicular to surface
• Parallel component contributes zero to flux

Closed surface flux (integral over entire closed surface):

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

Outward normal convention:

• Outward flux (field leaving): positive
• Inward flux (field entering): negative

Net flux = outward - inward

The net electric flux through any closed surface equals the total charge enclosed divided by epsilon0.

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

Where:

• $\oint$ = closed surface integral
• $Q_{enc}$ = total charge enclosed by surface
• $\varepsilon_0$ = 8.85 × 10-12 C2/(N - m2)

Physical meaning:

• Electric field lines originate from charges
• Number of field lines leaving surface ∝ enclosed charge

Choose surface to match symmetry of problem.

Guidelines:

• Surface should be closed
• Choose surface where E is constant on each part or parallel to surface
• Match charge distribution symmetry

Symmetry types:

• Spherical: spherical Gaussian surface
• Cylindrical: cylindrical Gaussian surface
• Planar: pillbox Gaussian surface

$Q_{enc} = \varepsilon_0 \oint \vec{E} \cdot d\vec{A}$

Key insight:

• Flux depends only on enclosed charge
• External charges produce zero net flux (field lines enter and exit)
• Allows finding E without integrating charge distribution

Uniformly charged solid sphere (radius R, total charge Q):

Outside ($r \ge R$): $E = \frac{k_e Q}{r^2}$ (like point charge at center)

Inside ($r < R$): $E = \frac{k_e Q r}{R^3}$ (proportional to r)

Uniformly charged spherical shell (radius R, total charge Q):

Outside ($r > R$): $$E = \frac{k_e Q}{r^2}$ (like point charge at center)

Inside ($r < R$): $$E = 0$ (shielding effect)

Procedure:

1. Concentric spherical Gaussian surface at radius r
2. $E$ constant on surface (spherical symmetry)
3. $\oint \vec{E} \cdot d\vec{A} = E \cdot 4\pi r^2$

Infinite line charge (linear density λ):

At distance r from line: $E = \frac{\lambda}{2\pi\varepsilon_0 r} = \frac{2k_e\lambda}{r}$

Uniformly charged infinite cylinder:

Outside ($r \ge R$): $$E = \frac{\lambda}{2\pi\varepsilon_0 r}$

Inside ($r < R$): $$E = \frac{\lambda r}{2\pi\varepsilon_0 R^2}$

Procedure:

1. Coaxial cylindrical Gaussian surface
2. $E$ constant on curved surface
3. No flux through ends (E ⟂ ends)

Infinite plane (surface density σ):

$E = \frac{\sigma}{2\varepsilon_0}$

Direction: perpendicular to plane

Parallel plates (one +σ, one -σ):

Between plates: $E = \frac{\sigma}{\varepsilon_0}$

Outside plates: $E = 0$ (fields cancel)

Procedure:

1. Pillbox Gaussian surface (cylinder crossing plane)
2. Flux only through ends (curved surface: E ∥ surface)
3. Both ends contribute if field passes through

For uniformly charged insulating sphere (radius R):

Inside ($r < R$): $E = \frac{k_e Q r}{R^3}$

Derived using Gauss's law with $Q_{enc(r)} = Q(r/R)^3$

Spherical symmetry:

• $E$ depends only on r (distance from center)
• $E$ radial (either inward or outward)
• Gaussian surface: sphere centered at symmetry point

Cylindrical symmetry:

• $E$ depends only on r (distance from axis)
• $E$ radial (perpendicular to axis)
• Gaussian surface: coaxial cylinder

Planar symmetry:

• $E$ constant magnitude, perpendicular to plane
• Same on both sides of plane
• Gaussian surface: pillbox crossing plane