21. Gauss

1
21. Gausss Law
The Prince of Mathematics Carl Friedrich
Gauss (1777 1855)
Wikemedia Commons
2
Topics

• Electric Field Lines
• Electric Flux
• Gausss Law
• Using Gausss Law
• Gausss Law and Conductors

3
Electric Field Lines
4
Electric Field Lines
A field line shows the direction of the
electric force on a positive point charge
5
Electric Field Lines
By using a convention for the number of
lines per unit charge, one can use field lines
to indicate the strength of an electric field.
6
Electric Flux
7
Electric Flux

• Flux is the flow of any quantity through a
  surface.
• For example, it could be sunlight through a
  window, or water through a hole.
• In particular, it can be electric field through a
  surface

8
The electric flux Df through a small
surface element DA is defined by
where
The unit vector is normal to the surface
element
9
The total electric flux f through a surface is
the sum of the individual fluxes Df
This is an example of a surface integral
10
Electric Flux A Closed Surface

• Lets compute the flux through a spherical
  surface about a point charge

We see that the flux is proportional to the
enclosed charge
11
Gausss Law
12
Gausss Law

• The electric flux through any closed surface is
  proportional to the net charge enclosed by the
  surface
• which is usually written as

-


-
-



where e0 (4pk)-1 8.85 x 10-12 C2/Nm2 is
called the permittivity constant
13
Gausss Law

• Gausss law is always true for any closed
  surface. However, it is most useful when the
• charge distribution and the enclosing
  surface have a high degree of
• symmetry.

-


-
-



a
14
Using Gausss Law
15
A Uniformly Charged Sphere
The enclosed charge is Q. Gausss law is
Because of the spherical symmetry of the
charge distribution, we can infer that the
magnitude of the electric field is constant on
any spherical surface enclosing the charge
16
A Uniformly Charged Sphere
The symmetry makes is easy to evaluate
the surface integral
The electric field of a spherically symmetric
charge distribution is like that of a point
charge
17
A Uniformly Charged Sphere
We can apply Gausss law within the sphere by
drawing a Gaussian surface of radius r. The
charge enclosed within this surface is
therefore,
18
A Uniformly Charged Sphere
Within the sphere the field varies linearly with
radius
Outside, the field looks like that of a point
charge
19
A Hollow Spherical Shell
The shell contains a net charge Q distributed
uniformly over its surface. Because of the
spherical symmetry, the field outside the shell
is like that of a point charge. But the field
inside is zero! Why? Because the field from
A cancels that from B.
20
Recap

• By convention, the electric field points away
  from a positive charge and towards a negative
  charge.
• Electric field lines can be used to visualize an
  electric field. By convention, the number of
  field lines is proportional to the charge.

21
Recap

• Electric field is additive the field at any
  point is the vector sum of the electric fields of
  all charges.
• Gausss law the net electric flux through any
  closed surface is proportional to the net
  enclosed charge

22
An Infinite Line of Charge

• By symmetry, the electric field is radial.
• Therefore, a suitable Gaussian surface is a
• cylinder of length L, radius r
• placed symmetrically
• about the line charge.
• The enclosed charge
• is q l L, where l is the
• charge per unit length

23
An Infinite Line of Charge

• From Gausss law, we deduce that the electric
  field of a long (strictly infinite) line charge is

24
A Sheet of Charge
For an infinite sheet of charge the field is
perpendicular to the sheet. The flux through a
cylindrical Gaussian surface is EA
EA. The enclosed charge is q
sA, where s is the charge per unit area.
Therefore, the field is
A
25
A Charged Disk
The electric field at a point P along the axis of
a disk is closely related to the field of a sheet
of charge
26
Gausss Law Conductors
27
Conductors
An applied electric field causes the free
positive and negative charges to separate until
the field they create exactly cancels the applied
field, at which point the charge migration stops.
The conductor is then in electrostatic equilibrium
.
28
Charged Conductors
Since like charges repel, all excess charge must
reside on the surface of a conductor. This is
also consistent with the fact that, in
equilibrium, the electric field within a
conductor is zero.
29
A Hollow Conductor
A conductor carries a net charge of 1 mC and
has a 2 mC charge in the internal cavity. The
charges must distribute themselves as shown in
order to be consistent with Gausss law.
30
A Hollow Spherical Conductor

• Consider a neutral spherical
• conductor in equilibrium with a
• cavity containing a net charge q.
• The charge on the inner surface of the cavity is
  q. Why?
• The charge on the outer surface of the conductor
  must therefore be q. Why?
• And this charge is uniformly distributed. Why?

31
Field at a Conductor Surface
The flux through a cylindrical Gaussian surface
is just EA since the field inside the conductor
is zero, in equilibrium. The enclosed charge
is q sA, therefore, the field at the
surface of a charged conductor is
E
E- 0 (inside conductor)
32
Applications
33
Electric Shielding

• The tendency of conductors to exclude electric
  fields from their bulk has many applications. For
  example
• Co-axial cables
• Lightning Safety
• Sensitive Compartmented Information Facility
  (SCIF)

34
Co-axial Cables

• Co-axial cables connect, for example, iPods to
  ear-phones.
• If the electric
• fields are too strong,
• the dielectric
• can suffer
• dielectric breakdown

Wikemedia Commons
35
Co-axial Cables and Dielectrics

• Some molecules, like H2O, have permanent dipole
  moments. Others can be distorted by an electric
  field, and become dipolar that is, acquire
  induced dipole moments. These materials are
  called dielectrics

36
Lightning Safety
http//www.lightningsafety.noaa.gov/lightning_map.
htm
37
SCIFs

• Wright-Patterson Air Force Base in Dayton, Ohio,
  is one of the major command posts of the U.S. Air
  Force (USAF).
• It contains a giant Faraday cage that houses a
• Sensitive
• Compartmented
• Information
• Facility (SCIF)

38
SCIFs

• Any externally generated electric field causes
  electrons in the Faraday cage to migrate in the
  direction opposite the field.

39
SCIFs

• The induced field exactly cancels the externally
  generated fields. Consequently, any electronic
  equipment inside is immune from an
  electromagnetic attack.

40
Summary

• Electric Flux
• Gausss Law
• The electric flux through a closed surface is
  determined by the enclosed charge

41
Summary

• Conductors
• The electric field within a conductor, in
  electrostatic equilibrium, is zero because the
  charge rushes to, and distributes itself on, the
  surface of the conductor