Lecture Notes: Electric Flux and Gauss's Law

Learning Objectives

• Calculate Electric Flux
• Use Gauss's Law to determine electric field due to uniformly distributed charges on:
  • Wires
  • Spheres
  • Large plates

Recap of Previous Topics

Electric Field Lines

• Electric Field: Space around an electric charge.
  • Positive Charge: Electric field lines directed outward.
  • Negative Charge: Electric field lines directed toward the charge.
• Interactions:
  • Opposite charges attract.
  • Like charges repel.
• Electric Field Magnitude:
  • Equation: ( E = \frac{F}{q} )
  • Derived from Coulomb’s Law: ( E = k \frac{q}{r^2} )

Main Topic: Electric Flux

Definition

• Electric Flux: The rate of flow of electric field through a given surface.
• Denoted by ( \Phi )
• Equation: ( \Phi = \mathbf{E} \cdot \mathbf{A} )
  • Expanded: ( \Phi = EA \cos \theta )
  • Theta (( \theta )): Angle between electric field ( E ) and area vector ( A )

Units

• Electric Flux Unit: Newton meters squared per coulomb (N·m²/C)

Calculating Electric Flux

• Orientation 1: Area vector parallel to electric field vector (( \theta = 0 ))
  • Maximum flux: ( \Phi = EA )
• Orientation 2: Area vector perpendicular to electric field vector (( \theta = 90^\circ ))
  • Zero flux: ( \Phi = 0 )
• Orientation 3: Area vector at angle less than 90 degrees
  • Calculate using ( \Phi = EA \cos \theta )

Special Cases

• Curved Surfaces: Divide surface into small regions and integrate.
  • ( d\Phi = E \cdot dA )
  • Integrate: ( \Phi = \int E \cdot dA )

Gauss's Law

Definition

• Gauss's Law: Relates net electric flux through a closed surface to the charge enclosed by the surface.
• Equation: ( \Phi = \frac{Q_{\text{enclosed}}}{\varepsilon_0} )_

Concepts

• Gaussian Surface: A symmetrical closed surface.
  • Sphere Example: Electric field ( E ) and area vector ( dA ) are parallel.

Important Constants

• Permittivity of Free Space: ( \varepsilon_0 = 8.85 \times 10^{-12} ) N·m²/C²

Applications

• Single Charge: Calculate using ( \Phi = \frac{q}{\varepsilon_0} )
• Multiple Charges: Calculate total flux using ( \Phi = \frac{\sum q}{\varepsilon_0} )

Summary

• Gauss's Law: The electric flux through a closed surface is the sum of all charges inside the surface divided by the permittivity of free space.
• Key Concept: When calculating total electric flux, consider the orientation of the surface and the position of charges.

Inspirational Quote

"What is a soul? It is like electricity. We don't really know what it is, but it is a force that can light up." - Unknown

These notes summarize the key concepts from the lecture on electric flux and Gauss's Law, providing a foundational understanding of how electric fields interact with surfaces and how to calculate related phenomena.