AP Physics C E&M- 8.5 Electric Flux- Study Notes- New Syllabus

Electric Flux Through an Arbitrary Area or Geometric Shape

Electric flux is a measure of the number of electric field lines passing through a given surface. It links the electric field with the geometry and orientation of the surface.

General Formula:

\( \mathrm{\Phi_E = \vec{E} \cdot \vec{A} = EA \cos \theta} \)

• \(\mathrm{\Phi_E}\) = electric flux (\(\mathrm{N \, m^2 / C}\))
• \(\mathrm{\vec{E}}\) = electric field vector
• \(\mathrm{\vec{A}}\) = area vector (magnitude = area, direction = normal to surface)
• \(\mathrm{\theta}\) = angle between \(\mathrm{\vec{E}}\) and \(\mathrm{\vec{A}}\)

Interpretation:

• If \(\theta = 0^\circ\): field is perpendicular to surface → maximum flux, \( \mathrm{\Phi_E = EA} \).
• If \(\theta = 90^\circ\): field is parallel to surface → zero flux, \( \mathrm{\Phi_E = 0} \).

If field is non-uniform: flux must be computed using an integral.

Integral Form for Arbitrary Surfaces:

\( \mathrm{\Phi_E = \displaystyle \int \vec{E} \cdot d\vec{A}} \)

• \(\mathrm{d\vec{A}}\) = infinitesimal area element with direction normal to surface.

Valid for curved or irregular surfaces, where \(\vec{E}\) may vary across the area.

Closed Surface:

For a closed surface (like a sphere or cube), the total flux is:

\( \mathrm{\Phi_E =\displaystyle \oint \vec{E} \cdot d\vec{A}} \)

This is the integral form of Gauss’s law.