AcademyGauss's Law
Academy
Charge and Electric Flux
Level 1 - Physics topic page in Gauss's Law.
Principle
Electric flux measures the normal component of electric field passing through an oriented surface.
Notation
\(\Phi_E\)
electric flux through a surface
\(\mathrm{N\,m^{2}\,C^{-1}}\)
\(\vec E\)
electric field
\(\mathrm{N\,C^{-1}}\)
\(\vec A\)
area vector for a flat surface
\(\mathrm{m^{2}}\)
\(d\vec A\)
small outward area vector
\(\mathrm{m^{2}}\)
\(\hat n\)
unit normal to the surface
1
\(\theta\)
angle between \(\vec E\) and the area vector
\(\mathrm{rad}\)
Method
Derivation 1: Attach direction to area
Flux depends on how a surface is oriented. A flat surface therefore uses an area vector normal to the surface.
Normal direction
\[\vec A=A\hat n\]
Perpendicular part
\[E_\perp=E\cos\theta\]
Uniform flux
\[\Phi_E=E_\perp A=EA\cos\theta\]
Derivation 2: Write flux as a dot product
The dot product selects the field component along the surface normal.
Dot product
\[\vec E\cdot\vec A=EA\cos\theta\]
Flux form
\[\Phi_E=\vec E\cdot\vec A\]
Sign
\[\Phi_E>0\ \text{when field points with the chosen normal}\]
Derivation 3: Generalize to curved surfaces
For a curved or nonuniform case, divide the surface into small pieces and add their flux contributions.
Small area vector
\[d\vec A=\hat n\,dA\]
Small flux
\[d\Phi_E=\vec E\cdot d\vec A\]
Surface integral
\[\Phi_E=\int_S\vec E\cdot d\vec A\]
Rules
These are the flux definitions used before Gauss's law.
Area vector
\[\vec A=A\hat n\]
Uniform flux
\[\Phi_E=EA\cos\theta\]
Dot-product flux
\[\Phi_E=\vec E\cdot\vec A\]
General flux
\[\Phi_E=\int_S\vec E\cdot d\vec A\]
Closed surface
\[\Phi_E=\oint_S\vec E\cdot d\vec A\]
Examples
Question
A uniform field of
\[80\,\mathrm{N\,C^{-1}}\]
passes normally through a \[0.30\,\mathrm{m^2}\]
flat surface. Find the flux.Answer
The field is aligned with the area vector, so
\[\theta=0\]
\[\Phi_E=EA=(80)(0.30)=24\,\mathrm{N\,m^2\,C^{-1}}\]
Checks
- Flux is scalar; it can be positive, negative, or zero.
- The area vector is normal to the surface, not along the surface.
- Tangential field contributes no flux through that surface.
- Closed surfaces use outward normals by convention.