Gauss' Law
Study Guide for Chapter 24
Outline
1. Electric Flux
Electric flux represents the total number of electric field lines crossing through a surface. The
basic equation for flux is
F œ IE
where I is the magnitude of the electric field, and E is the area of the surface. This equation
only applies if the electric field has constant magnitude and is perpendicular to the surface.
If the electric field E is at an angle to the surface, the flux is
F œ E † A œ IE cos )
where A is the area vector (pointing perpendicular to the surface), and ) is the angle between E
and A. In particular, the flux is zero if E is parallel to the surface.
If a surface has several parts, then you must add together the flux from each part. The most
general formula for flux involves an integral of the electric field over the surface.
2. Gauss' Law
Gauss's law states that the net electric flux out of a closed surface is proportional to the amount
of charge inside:
Fout  Fin œ
Uinside
%!
You may use Gauss's law on any closed surface, even a surface that you make up.
3. Applications of Gauss' Law
This section shows how to use Gauss' law to find the electric field due to certain continuous
charge distributions. As with the previous chapter, this section uses the notion of charge density:
- œ chargeÎlength
(linear charge density)
5 œ chargeÎarea
(surface charge density)
3 œ chargeÎvolume (volume charge density)
Using Gauss' law, they re-derive several previously derived formulas, and show that they apply in
a slightly more general context:
" ;
%1%! <#
" I œ
#1%! <
5
I œ
#% !
I œ
(point charge, sphere of charge)
(line of charge, cylinder of charge)
(plane of charge, slab of charge)
4. Superposition of Electric Fields
Remember that electric fields from more than one source add together.
5. Conductors and Electric Fields
This section uses Gauss' law to explore the electric field and charge distribution inside a
conductor. The main results of this section are:
1. The electric field inside a conductor is always zero.
2. When a conductor is charged, the charge distributes itself over the surface of the
conductor, in such a way that the electric field inside is zero.
3. A hollow cavity inside a conductor is shielded from outside charges.
4. If you place a charge in the cavity, it induces an equal and opposite charge on the inside
surface of the conductor.
Answers
1. FI œ )Þ! ‚ "!$ N † m# ÎC
2. FI œ "Þ(# ‚ "!$ N † m# ÎC; FI œ "Þ%* ‚ "!$ N † m# ÎC; FI œ !
3. ; œ !Þ*$ µC
6. 3 œ $Þ& pCÎm$
9. I œ %Þ& kNÎC
4. each FI œ "Þ" ‚ "!# N † m# ÎC
7. ; œ (Þ& µC
5. 3 œ !Þ%( µCÎm$
8. ; œ $" nC; I œ $Þ) kNÎC
10. each has 5 œ !Þ(" µCÎm#