Gauss Law

lecture 5

flux usually refers to

flow

electric fields do not

flow, but they do share some things in common that do flow

field lines act as a

flow

field lines: they are produced on

positive charges → this is like water faucets 

field lines: they end on

negative charges → this is like water drains

field lines:they can cross

a surface → like water flowing

electric flux can be thought of as

“the number of field lines” passing though a given area

flux is directly proportional to

density of flow

flux varies by

how the boundary faces the direction of flow

flux is also directly proportional to

area within the boundary

electric flux is a measure of

the amount of electric field passing through a surface of a given area

flux can be 

positive or negative 

if the field lines come out of the surface,

then the flux is positive

if the field lines go into the surface,

then the flux is negative

for a closed surface, field lines leaving the bounded region contribute

a positive flux

for a closed surface, field lines entering the bounded region contribute 

a negative flux 

if the same number of field lines leave a region as enter the region, the net flux is 

zero 

If more field lines leave then enter, the net flux is

positive

if more field lines enter than leave, the net flux is

negative

if more positive charge than negative charge,

positive flux

if there is more negative than positive charge

negative flux

gauss’ law related

the net flux leaving or entering a closed surface to the net amount of charge contained in that surface

the net electric flux through a closed surface is proportional to

the net charge contained within the closed surface

the closed surface does not need to be a

physical thing or boundary between physical things

the flux does not depend on

the shape of the gaussian surface

the flux depends on the

net charge enclosed (inside) the gaussian’ surface

uniform means that

the charge density is the same everywhere in or on the sphere

charge distribution = spherically symmetric

electric field = spherically symmetric

• can only depend on distance from center of sphere (not on direction)

• most point radially inward or radially outward 

for outside a insulator sphere where charge Q

on this surface, the electric field must be constant in magnitude and be perpendicular to the surface (either pointing out or in)

inside the charged sphere 

depends on how the charge is distributed in or on the sphere 

isolated means

not attached to anything

we are interested in the equilibrium state of such conductor

this means nothing is changing or moving

any excess charge placed on an isolate conductor will

reside on the outer surface of the conductor

the electric field within an isolated conductor is

zero 

the electric field just outside an isolate conductor will be 

• perpendicular to the surface of the conductor

• will be constant and proportional to the surface charge density of the conductor

a conductor, by definition,

has charges that can move

if a charge that can move experiences

an electric field, electric field would accelerate the charge into motion, and it will move 

if there are charges moving within the conductor,

it is not in equilbrium

therefore, there can not be

electric field within an isolated conductor

if some region within the conductor had an excess charge, you could draw

a gaussian surface around it and there would be a non-zero flux through it

if there is a flux though a surface,

the electric field can’t be zero everywhere on the surface

so an excess charge within the conductor would produce

an electric field within the conductor, but that can’t happen, so that excess charge can only be on the surface of the conductor

if the conductor has an excess charge

• that excess charge lives on the srface

• there will be an electric field just outside the surface of the conductor

if that electric field has a component that was parallel to the surface, the electrons

would accelerate into motion then the charges on the surface would move; the field must be entirely perpendicular to the surface 

in equilibrium - in a isolated conductors - the electric field 

is always perpendicular to the surface outside of a conductor - again, if it were not, charges would move 

in equilibrium - for nonconductor or insulators, which does not have free electrons, a static electric field 

can exist inside it and electric field outside it is not necessary perpendicular to the surface