Steps to Applying Gauss’ Law

To find the E field produced by a charge distribution at a point of distance r from the center

1. Decide which type of symmetry best complements the problem

2. Draw a Gaussian surface (mathematical not real) reflecting the symmetry you chose around the charge distribution at a distance of r from the center

3. Using Gauss’s law obtain the magnitude of E

Gauss’s Law

0

i

E

q

E d A

Applications of the Gauss’s Law

If no charge is enclosed within Gaussian surface – flux is zero!

Electric flux is proportional to the algebraic number of lines leavingthe surface, outgoing lines have positive sign, incoming - negative

Remember – electric field lines must start and must end on charges!

Examples of certain field configurations

Remember, Gauss’s law is equivalent to Coulomb’s law

However, you can employ it for certain symmetries to solve the reverse problem – find charge configuration from known E-field distribution.

Field within the conductor – zero(free charges screen the external field)

Any excess charge resides on thesurface

0S

E d A

Field of a charged conducting sphere

Field of a thin, uniformly charged conducting wire

Field outside the wire can only point radially outward, and, therefore, mayonly depend on the distance from the wire

0

QEd A

02E

r

- linear density of charge

Field of the uniformly charged sphere

rE03

Uniform charge within a sphere of radius r

3' rq Q

a

Q - total charge

Q

V - volume density of charge

Field of the infinitely large conducting plate

- uniform surface charge densityQ

A

02E

Charged Isolated Conductors

• In a charged isolated conductor all the charge moves to the surface

• The E field inside a conductor must be 0 otherwise a current would be set up

• The charges do not necessarily distribute themselves uniformly, they distribute themselves so the net force on each other is 0.

• This means the surface charge density varies over a nonspherical conductor

Charged Isolated Conductors cont

• On a conducting surface

• If there were a cavity in the isolated conductor, no charges would be on the surface of the cavity, they would stay on the surface of the conductor

o

E

Charge on solid conductor resides on surface.

Charge in cavity makes a equal but opposite charge reside on inner surface of conductor.

Properties of a Conductor in Electrostatic Equilibrium

1. The E field is zero everywhere inside the conductor

2. If an isolated conductor carries a charge, the charge resides on its surface

3. The electric field just outside a charged conductor is perpendicular to the surface and has the magnitude given above

4. On an irregularly shaped conductor, the surface charge density is greatest at locations where the radius of curvature of the surface is smallest

Charges on Conductors

Field within conductor E=0

Experimental Testing of the Gauss’s Law

A point charge cannot be in stable equilibrium in electrostatic field of other charges

(except right on top of another charge – e.g. in the middle of a distributed charge)

Earnshaw’s theorem

Stable equilibrium with other constraints

Atom – system of charges with only Coulombic forces in play.According to Earhshaw’s theorem, charges in atom must move

However, planetary model of atom doesn’t work

Only quantum mechanics explains the existence of an atom

Electric Potential Energy

Concepts of work, potential energy and conservation of energy

For a conservative force, work can alwaysbe expressed in terms of potential energy difference

( )b

a b b aa

W F d l U U U

Energy Theorem

For conservative forces in play,total energy of the system is conserved

a a b bK U K U

0a bW Fd q Ed 0U q Ey 0 ( )a b a bW U q E y y

Potential energy U increases as the test charge q0 moves in the direction opposite to the electric force : it decreases as it moves in the same direction as the force acting on the charge

0F q E

Electric Potential Energy of Two Point Charges

02

cosb

a

rb

a b ea r

qqW F d l k dl

r

01 1

a b ea b

W k qqr r

0eqq

U kr

Electric potential energy of two point charges

Example: Conservation of energy with electric forces

A positron moves away from an – particle

-particle

positron

0

31

100

60

9.1 10

7000

2

10

3 10 /

p

p

m kg

m m

q e

r m

V m s

What is the speed at the distance ?What is the speed at infinity?Suppose, we have an electron instead of positron. What kind of motion we would expect?

1002 2 10r r m

Conservation of energy principle

0 0 1 1K U K U

Electric Potential Energy of the System of Charges

Potential energy of a test charge q0in the presence of other charges

0

04i

ii

q qU

r

Potential energy of the system of charges(energy required to assembly them together)

04i j

iji j

q qU

r

Potential energy difference can be equivalently described as a work done by external force required to move charges into the certain geometry (closer or farther apart). External force now is opposite to the electrostatic force ( )a b b a extW U U F d l