AP Physics C: Electric Flux and Gauss’ Law

Introduction to Electric Flux

• Flux Definition: Effect appearing to pass through a surface; does not need to move.
• Electric Flux (Φₑ): Measure of the electric field passing through a defined area.
  • Equation: Φₑ = E ⋅ A = EAcosθ
    • E = Electric field (vector)
    • A = Area (vector)
    • θ = Angle between E and A
  • Electric flux is a scalar quantity.
  • Units: Newton-meters squared per coulomb (N·m²/C).
  • Similar form to work equation: Work = Force · Displacement

Electric Flux through a Closed Surface

• Commonly determined through closed surfaces.
• Example of Calculation: Right triangular box.
  • Determine electric flux through all sides, sum them.
  • Sides Identified:
    • Side 1: Rectangle (back, left direction)
    • Side 2: Bottom (down direction)
    • Side 3: Triangle (closest, outward direction)
    • Side 4: Triangle (farthest, inward direction)
    • Side 5: Top (hypotenuse, outward direction)
  • Calculations:
    • Side 1: Negative flux
    • Sides 2, 3, 4: Zero flux (electric field parallel)
    • Side 5: Positive flux
  • Net Flux: Sum equals zero.

Gauss’ Law

• Definition: Total electric flux through a closed surface equals the charge enclosed divided by the permittivity of free space.
• Equation: Φ = ∮ E · dA = Q_enc/ε₀
  • Q_enc = Charge enclosed
  • ε₀ = Permittivity of free space
  • Integral over a closed surface (∮)
• Application requires symmetrical charge distributions for easy computation.

Examples of Gauss’ Law

• Sphere Surrounding a Point Charge:

  • Electric field constant on sphere surface.
  • Simplified with integral, E can be factored out.
  • Results in Q_enc/ε₀ for electric flux.

• Infinite Plane of Charges:

  • Electric field uniform, determined using cylindrical Gaussian surface.
  • E = σ/2ε₀
  • Adding a parallel plane with opposite charge cancels external fields, doubles internal field.

Relationship with Maxwell’s Equations

• Gauss’ law is the first of Maxwell’s equations.
• Discussed usage in spherically, cylindrically, and planarly symmetric situations.

Additional Notes

• Charged sphere behaves as a point charge outside its surface.
• Electric field inside conductors is zero; depends on distribution in insulators.
• Encouraged understanding through imagination and practice.

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