Lecture on Electric Flux

Introduction

• Electric Flux: Measures the quantity of the electric field passing through a surface.
• Surface Area (A): Area of the surface through which the electric field passes.
• Electric Field (E): Field emanating perpendicular to the surface.
• Formula: Electric Flux (Φ) = E * A
• Angle (φ): Angle between the normal line to the surface and the electric field vector.

Key Concepts

1. Perpendicular Electric Field:

   • When E is perpendicular to the surface, φ=0 and cos(0)=1.
   • Φ = E * A

2. Non-Perpendicular Electric Field:

   • When E is at an angle φ to the normal line.
   • Φ = E * A * cos(φ)
   • Maximum value when φ=0.
   • If E is parallel to the surface (φ=90), cos(90)=0 and Φ=0.

Example Problems

1. Sphere Example:

   • Given: Sphere of radius 4m, charge of +50 μC and -50 μC.
   • Electric field perpendicular to the surface.
   • Formula: Φ = E * A, A=4πR².
   • Simplified with Gauss's Law: Φ = Q/ε₀.
   • Calculation: Q = 50 x 10⁻⁶ C, ε₀ = 8.85 x 10⁻¹², Φ ≈ 5.65 x 10⁶ N·m²/C.
   • Result: Positive for outward flux; negative for inward flux.

2. Disk Example:

   • Given: Horizontal disk with radius 3m, E=100 N/C, angle θ=30°.
   • Calculate Flux: φ=60° (complementary angle), Φ=E * πr² * cos(φ).
   • Calculation: E=100, r²=9, cos(60)=0.5, Φ ≈ 450π N·m²/C.
   • Result: Φ = 450π.

Gauss's Law Applications

1. Cube with Center Charge:

   • Given: Positive charge at cube center.
   • Calculate Total Flux: Use Gauss's Law, Φ = Q/ε₀.
   • Result: Φ ≈ 3.39 x 10⁶ N·m²/C.
   • Through Each Face: Divide total flux by 6, Φ ≈ 5.65 x 10⁵ N·m²/C per face.

2. Cube with No Enclosed Charge:

   • Given: Cube with no charge inside, E field from bottom to top.
   • Top and Bottom Faces: Top has positive Ea, bottom has negative -Ea.
   • Total Flux: Sum of top and bottom flux, Φ = 0 (inward flux equals outward flux).
   • Gauss's Law: With no enclosed charge, Q=0, so Φ=0.

Summary

• Electric Flux: Depends on the orientation of the electric field relative to the surface.
• Gauss's Law: Provides a simplified way to calculate flux, especially for symmetric shapes.
• Practical Applications: Using these principles to solve problems involving various geometries and charges.