AcademyGauss's Law
Academy
Computing Electric Flux
Level 1 - Physics topic page in Gauss's Law.
Principle
Flux calculations become manageable when the surface is split into pieces with simple normal components.
Notation
\(\Phi_E\)
electric flux
\(\mathrm{N\,m^{2}\,C^{-1}}\)
\(\vec E\)
electric field
\(\mathrm{N\,C^{-1}}\)
\(E_n\)
field component normal to the surface
\(\mathrm{N\,C^{-1}}\)
\(dA\)
small scalar area element
\(\mathrm{m^{2}}\)
\(A_i\)
area of surface piece \(i\)
\(\mathrm{m^{2}}\)
\(\Delta\vec A_i\)
area vector for surface piece \(i\)
\(\mathrm{m^{2}}\)
Method
Derivation 1: Project the field onto the normal
Only the normal component of the field contributes to flux. Tangential components skim along the surface.
Area element
\[d\vec A=\hat n\,dA\]
Normal component
\[E_n=\vec E\cdot\hat n\]
Flux element
\[d\Phi_E=E_n\,dA\]
Derivation 2: Sum flat pieces
For a surface made from flat panels in a uniform field, each panel has one area vector.
Panel flux
\[\Phi_i=\vec E\cdot\Delta\vec A_i\]
Total flux
\[\Phi_E=\sum_i\Phi_i\]
Closed flat surface
\[\Phi_E=\sum_i\vec E_i\cdot\Delta\vec A_i\]
Derivation 3: Use cancellation before arithmetic
Uniform fields through closed boxes have equal inward and outward flux through opposite faces. Curved sides can contribute zero when the field is tangent everywhere.
Opposite faces
\[\Delta\vec A_R=-\Delta\vec A_L\]
Uniform-field cancellation
\[\vec E\cdot\Delta\vec A_R+\vec E\cdot\Delta\vec A_L=0\]
Tangential field
\[\vec E\perp d\vec A\Rightarrow d\Phi_E=0\]
Rules
These are the practical computation forms.
Normal component
\[E_n=\vec E\cdot\hat n\]
Flux element
\[d\Phi_E=E_n\,dA\]
Panel sum
\[\Phi_E=\sum_i\vec E_i\cdot\Delta\vec A_i\]
Surface integral
\[\Phi_E=\int_S E_n\,dA\]
Closed integral
\[\Phi_E=\oint_S\vec E\cdot d\vec A\]
Examples
Question
A
\[0.20\,\mathrm{m^2}\]
panel has outward normal \[\hat\imath\]
The field is \[(30\hat\imath+40\hat\jmath)\,\mathrm{N\,C^{-1}}\]
Find the flux.Answer
Only the \(x\)-component is normal.
\[\Phi_E=E_xA=(30)(0.20)=6.0\,\mathrm{N\,m^2\,C^{-1}}\]
Checks
- Compute with the normal component, not the total field magnitude.
- For closed surfaces, use outward area vectors.
- Opposite faces can cancel only when the relevant field values match.
- A zero net flux does not mean the field is zero everywhere.