Overview
This lecture covers how to calculate the electric field produced by an infinitely long line of charge using Gauss’s Law, emphasizing symmetry and deriving the formula for the field.
Line Charge and Symmetry
- A line charge is a straight line with uniform charge density λ (lambda), measured in coulombs per meter.
- The electric field generated by an infinitely long line charge points radially outward by symmetry.
- Symmetry ensures the electric field has no horizontal component, only perpendicular (radial) to the line.
Applying Gauss’s Law
- Gauss’s Law states: ∮E·dA = Q_enclosed / ε₀, where ε₀ is the permittivity of free space.
- Choose a cylindrical Gaussian surface (can shape) around the line to match the symmetry.
- The electric field E and area vector dA are parallel on the curved surface (sleeve) of the cylinder.
Calculating the Electric Field
- On the sleeve, E·dA simplifies to E dA because the angle is 0° (cos(0°) = 1).
- The end caps of the cylinder don't contribute since E is perpendicular to dA there (cos(90°) = 0).
- The integral becomes E times the area of the sleeve: E × (2πs × l), with s as radius and l as length.
- Enclosed charge Q_enclosed = λ × l (charge density times length inside the cylinder).
- Gauss’s Law becomes: E(2πsl) = λl/ε₀.
- After cancelling l, E = (λ)/(2πε₀s).
Final Expression and Direction
- The electric field magnitude: E = (λ)/(2πε₀s).
- The electric field points radially outward (in the ŝ direction) from the line.
Key Terms & Definitions
- Line Charge — An infinitely long straight distribution of electric charge with uniform linear density λ.
- λ (Lambda) — Charge per unit length (C/m).
- Gauss’s Law — The net electric flux through a closed surface equals the enclosed charge divided by ε₀.
- Cylindrical Gaussian Surface — A cylinder used in applying Gauss’s Law to exploit cylindrical symmetry.
- ε₀ (Epsilon Naught) — The permittivity of free space, a universal constant.
Action Items / Next Steps
- Practice deriving the electric field for different charge distributions using Gauss’s Law.
- Review concepts of symmetry and their importance in choosing Gaussian surfaces.
- Complete assigned homework problems on electric fields of line charges.