2) electric flux - Gauss' Law

define electric flux Φ

through a specific surface, measure for the density of field lines passing through a surface

formula for Φ

when is Φ = 0

E is parallel to surface and hence perpendicular to dA

where does dA point for a closed surface

outwards

when does E not contribute to Φ?

doesn’t originate from surface nor end in it

give Φ for a homogenous E and flat surface

EAcosθ

define Gauss’ law

when is it appropriate to use Gauss’ law

high symmetry and/or closed surface

why is Gauss’ law derivation valid for arbitrary closed surfaces

the charge is still enclosed and the difference in strength and area even out

E field of a charged hollow sphere: surface

charge sits on spherical surface

E has same magnitude at any point - take out of integral

E is perpendicular to surface everywhere = parallel to dA - cosθ = 1

A is 4πr²

=> same expression as for point charge

E field of a charged hollow sphere: inside

E has same magnitude at any point - take out of integral

E is perpendicular to surface everywhere

E = 0 and q_enclosed = 0

=> same result for massive conducting sphere bcuz charges want to sit as far from each other as possible

E field of non-conducting charged sphere: inside

charges can distribute over entire volume since they can’t move - non-conducting

draw Gaussian surface: (r < r₀) where r₀ is radius of sphere and r is inner radius of Gaussian surface

E has same magnitude on surface (bcuz at same distances from center)

E perpendicular to surface everywhere

E field of non-conducting charged sphere: outside

draw Gaussian surface: (r > r₀) where r₀ is radius of sphere and r is inner radius of Gaussian surface

E has same magnitude on surface (bcuz at same distances from center)

E perpendicular to surface everywhere

=> same field as conducting material FOR EVENLY DISTRIBUTED CHARGES

E field of charged plate

total charge density per unit area is σ

determine E near plate: since field is same everywhere and infinitely big plate, distance from plate does not matter

draw Gaussian cylinder

E upper and lower surface is parallel to dA due to symmetry

flux through side is 0 bcuz E is parallel to surface there

E field at surface of charged conductor

charge density per unit area is σ’ (for one side of conductor!!! **)

field lines always perpendicular to surface of conductor: if not, there would be an E_parallel which by definition means force moving charges along the conductor (this is electrostatics!)

draw Gauss cylinder with upper and lower surface parallel to conductor surface

E is 0 on lower surface - sits inside conductor so Φ = 0

=> only upper surface remains: see pic

=> same as plate but /2 !!!! (reflect why **)

what is the point effect (corona)?

since field lines are always perpendicular to conductor surface, net effect for sharp edge makes a huge field: charges are pushed to the tip due to repulsion, making large charge density

smaller radius of curvature → larger field strength

if field larger than air breakdown strength = spark bcuz easy charge transfer

E field of long charged conductor

draw bigger Gauss cylinder around it

charge density per unit length is λ

Φ through side surfaces is 0

** not closed bcuz took side surfaces into account