1. 21. Gauss’s Law “The Prince of Mathematics” Carl Friedrich Gauss
(1777 – 1855)
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2. Topics Electric Field Lines Electric Flux Gauss’s Law
Using Gauss’s Law
Gauss’s 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
Let’s compute the flux through a spherical surface about a point charge: +
We see that the flux is proportional
to the enclosed charge

11. Gauss’s Law

12. Gauss’s 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 C2/Nm2
is called the permittivity constant

13. Gauss’s Law
Gauss’s 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 Gauss’s Law

15. A Uniformly Charged Sphere
The enclosed charge is Q. Gauss’s 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 Gauss’s 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.
Gauss’s 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 Gauss’s 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 E+ E- 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: P dq r x q

26. Gauss’s 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 Gauss’s law.

30. A Hollow Spherical Conductor+q
E = 0
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
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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

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 Gauss’s 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