Few examples on calculating the electric flux

32 10 [ / ]E N C

Find electric flux

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' r

q Qa

Q - total charge

Q

V - volume density of charge

Field of the infinitely large conducting plate

- uniform surface charge densityQ

A

02E

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

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

€

me = 9.1 ×10−31kg

mα = 7000me

qα = 2e

r0 =10−10m

v0 = 3 ×106m /s

Electric Potential Energy of the System of Charges

Potential energy of a test charge q0

in the presence of other charges0

04i

ii

q qU

r

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

0

1

4i 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

Electric Potential Energy of System

• The potential energy of a system of two point charges

• If more than two charges are present, sum the energies of every pair of two charges that are present to get the total potential energy

12

2112 r

qqkVqU e

ji ij

jietotal r

qqkU

,

23

32

13

31

12

21

rqq

rqq

rqq

kU etotal

Reading assignment: 23.3 – 23.5