Understanding Gauss’s Law: Electric Field and Flux Relationships

1. A q q Chapter 23: Gauss’s Law • Gauss’s Law is an alternative formulation of the relation between an electric field and the sources of that field in terms of electric flux. • Electric FluxFE through an area A • ~ Number of Field Lines which pierce the area • depends upon • geometry (orientation and size of area, direction of E) • electric field strength (|E| ~ density of field lines)

2. Gauss’s law relates to total electric flux through a closed surface to the total enclosed charge. • Start with single point charge enclosed within an arbitrary closed surface. • Add up all contributions dF. q

3. q • intermediate steps: charge at the center of a spherical surface • two patches of area subtending the same solid angle Adding up the flux over the surface of one of the spheres

4. dA E q For a charge in an arbitrary surface Project area increment onto “nearest sphere”: Flux through area = flux through area increment on “nearest sphere” with same solid angle. Flux through “nearest sphere” area increment = flux through area increment on a common sphere for same solid angle. Add up over all solid angles => over entire surface of common sphere => simple sphere results.

5. For charges located outside the closed surface number of field lines exiting the surface (+FE) = number of field lines entering the surface (-FE) => no net contribution to FE Gauss’s Law:

6. Using Gauss’s Law • Select the mathematical surface (a.k.a. Gaussian Surface) • to determine the field at a particular point, that point must lie on the surface • Gaussian surface need not be a real physical surface in empty space, partially or totally embedded in a solid body • Gaussian surface should have the same symmetries as charge distribution. • concentric sphere, coaxial cylinder, etc. • Closed Gaussian surface can be thought of as several separate areas over which the integral is (relatively) easy to evaluate. • e.g. coaxial cylinder = cylinder walls + caps • If E is perpendicular to the surface (E parallel to dA) and has constant magnitude then • If E is tangent (parallel) to the surface (E perpendicular to dA) then

7. Conductors and Electric Fields in Electrostatics • Conductors contain charges which are free to move • Electrostatics: no charges are moving • F = q E • => for a conductor under static conditions, the electric field within the conductor is zero. E = 0 • For any point within a conductor, and all Gaussian surfaces completely imbedded within the conductor q = 0 within bulk conductor => all (excess) charge lies on the surface! (for a conductor under static conditions)

8. Conductor with void: all charge lies on outer surface unless there is an isolated charge within void. • Faraday “ice-pail” experiment • charged conducting ball lowered to interior of “ice-pail” • ball touches pail => part of interior of conductor Ball comes out uncharged => verifies Gauss’s Law => Coulomb’s Law Modern versions establish exponent in Coulomb’s = 2 to 16 decimal places

9. Field of a conducting sphere, with total charge q and radius R Spherical symmetry => spherical Gaussian surfaces E constant on surface, E perpendicular to surface E = 0 on interior exterior: R r E r r=R

10. Field of a uniform ball of charge, with total charge q and radius R Spherical symmetry => spherical Gaussian surfaces E constant on surface, E perpendicular to surface exterior: interior: R r r E r r=R

11. Line of charge (infinite), charge per unit length l • cylindrical symmetry, E is radially outward (for positive l) • Gaussian surface: finite cylinder, length l and radius r Caps: E parallel to surface, F = 0 Cylinder: E perpendicular to the surface r l

12. Field of an infinite sheet of charge, charge per area s • infinite plane, E is perpendicular to the plane (for positive s) with reflection symmetry • Gaussian surface: finite cylinder, length 2x centered on plane, caps with area A Tube: E parallel to surface, F = 0 2 Caps: E perpendicular to the surfaces A E E l x x

13. Two oppositely charged infinite conducting plates (+/-s) • planar geometry, E is perpendicular to the plane • Gaussian surfaces: finite cylinder, length l centered on plane, caps with area A Tube: E parallel to surface, F = 0 Caps: E perpendicular to the surfaces + + + + + + + + + + + - - - - - - - - - - -