1. Steps to Applying Gauss’ Law
To find the E field produced by a charge distribution at a point of distance r from the center
Decide which type of symmetry best complements the problem
Draw a Gaussian surface (mathematical not real) reflecting the symmetry you chose around the charge distribution at a distance of r from the center
Using Gauss’s law obtain the magnitude of E

3. Gauss’s Law

4. Applications of the Gauss’s Law
Remember – electric field lines must start and must end on charges!
If no charge is enclosed within Gaussian surface – flux is zero!
Electric flux is proportional to the algebraic number of lines leaving
the surface, outgoing lines have positive sign, incoming - negative

5. 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 the
surface

6. Field of a charged conducting sphere

7. Field of a thin, uniformly charged conducting wire
Field outside the wire can only point
radially outward, and, therefore, may
only depend on the distance from the wire
l- linear density of charge

8. Field of the uniformly charged sphere
Uniform charge within a sphere of radius r
Q - total charge
- volume density of charge
Field of the infinitely large conducting plate
s- uniform surface charge density

9. 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

10. 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

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

12. Properties of a Conductor in Electrostatic Equilibrium
The E field is zero everywhere inside the conductor
If an isolated conductor carries a charge, the charge resides on its surface
The electric field just outside a charged conductor is perpendicular to the surface and has the magnitude given above
On an irregularly shaped conductor, the surface charge density is greatest at locations where the radius of curvature of the surface is smallest

13. Charges on Conductors
Field within conductor
E=0

14. Experimental Testing of the Gauss’s Law

15. Stable equilibrium with other constraints
Earnshaw’s theorem
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)
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

16. Electric Potential Energy
Concepts of work, potential energy
and conservation of energy
For a conservative force, work can always
be expressed in terms of potential energy difference
Energy Theorem
For conservative forces in play,
total energy of the system is conserved

17. 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

18. Electric Potential Energy of Two Point Charges

19. Electric potential energy of two point charges

20. Example: Conservation of energy with electric forces
A positron moves away from an a – particle
a-particle
positron
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?
Conservation of energy principle

21. Electric Potential Energy of the System of Charges
Potential energy of a test charge q0
in the presence of other charges
Potential energy of the system of charges
(energy required to assembly them together)
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