WHAT DOES ELECTRIC FLUX THROUGH A CLOSED SURFACE TELL US
ABOUT THE CHARGE ENCLOSED BY THE SURFACE
Electric field lines:
 start on positive charges
 end on negative charges
For a CLOSED surface around a volume of space:
 the NET number of Electric Field Lines entering/leaving through the surface
(i.e. the electric flux):
o must depend on the amount of positive/negative charge INSIDE
No charge inside:
 net number of lines leaving = 0
 all lines go through
Positive charge inside:
 non-zero net number of lines leaving
 lines start on charge inside sphere
GAUSS’S LAW
Connects:
 charge enclosed by closed surface to NET Electric Flux through surface
Look at positive charge +q at centre of a sphere of radius r


 At every point on surface: E || Ai


A
o
i is surface normal at that point

 Component of E perpendicular to surface is
 kq
E n | E | e2
(at every point on surface)
r
 So for every point on surface:
 
kq
E  Ai  E n Ai  e2 Ai
r
Now calculate total flux through whole spherical surface around q
 e  lim
Ai  0
 E A   E dA 
n
over
surface
n
i
over
surface
ke q
dA
2 
r
 Factored En out of integral since constant everywhere on spherical surface
 Remaining surface integral  dA :
o sum of area elements over sphere surface (i.e. surface area of sphere)
2
o So:  dA  4r
ke q
2


So flux is: e r 2  4r  4 ke q
1
12
2
2
k

 But e 4  where  0  8.85 10 C / N  m is permittivity of free space
0
RESULT (Gauss’s Law):
e 
qinside
0
 Says that flux through a closed surface is:
o proportional to the charge inside
o independent of the radius (for a spherical surface)
Why does this work?
 Says number of electric field lines through
spherical surface is the same for all radii
1
2
E


2 but A  r so:
r
 
o  E  dA is independent of r
IMPORTANT: Gauss’s Law applies to any closed surface
 For a given charge q, the net flux through a closed surface around it is:
o Independent of the shape of the surface
o Independent of the location of the charge inside the surface
 Can choose surface that best suits the problem
GAUSS’S LAW (general statement):
  qenclosed
 e   E  dA 
closed
surface
0
 Powerful way to calculate electric field if we can factor En out of integral
o Trick is to choose surface so that En is uniform over all or part of surface
Definition: Gaussian Surface
 A closed surface selected for use in a Gauss’s law calculation
 Careful choice can simplify calculation of electric field
Using Gauss’s Law:
 1st step: must know direction of electric field from symmetry of problem
o radial (spherical symmetry) for point
charge
o radial (cylindrical symmetry) for a long line
of charge
o uniform for a large flat sheet of charge
 2nd step: must choose Gaussian surface that allows one to factor electric
field out of flux integral
Two important cases

E
uniform and perpendicular to part or all of Gaussian surface

 
o then flux is  E  dA  En A for that part of the surface

 E parallel (tangent) to part of the Gaussian surface
 
o then E  dA  0 for that part of the surface and there is no
contribution to the total flux
Example: Use Gauss’s law to find the electric field around a point charge q.
We will return to applications of Gauss’s law for continuous charge
distributions (lines and sheets) later.