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 Flux FE 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) A q

Add up all contributions dF. 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

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

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. q dA E

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:

Select the mathematical surface (a.k.a. Gaussian Surface) 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

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)

Faraday “ice-pail” experiment 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

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

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

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

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

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 + -