Maxwell’s equation for electrostatics
E 
Ex E y Ez

,



x
y
z  r   o
remember   xˆ 



 yˆ   zˆ  , and E  xˆ  E x  yˆ  E y  zˆ  E z . xˆ , yˆ , zˆ are the unit
x
y
z
vectors in the x, y, and z directions.
E is the electric field, it starts on a positive charge and ends up on a negative charge. The
electric field has units of volts/m. In ECE 346 we typically use m or cm instead of m,
1m  104 cm  106 m .
 is the charge density, charge is quantized in units of e=1.602*10-19 coulombs, and  has
units of coulombs/m3, or coulombs/cc, a cc=cm3.
  8.854  1014
o
Fd
is the permittivity of vacuum. It is a constant that relates charge to electric
cm
field.
 is the relative permittivity of material. Is silicon   11.7 , in oxidized silicon SiO2   3.9 .
r
r
r
The relative permittivity measures how much more charge can be stored on a capacitor for a
given electric field.
Since   E 

r 
, then
o
  E  dV   
V
V

r
 o
 dV 
Q
, where Q is the total
r  o
charge enclosed in the volume.
Gauss was pretty clever and he found a shortcut for the math. He realized that instead of doing
a volume integral he could do a surface integral, since he was integrating the divergence of a
vector. It’s much easier to do the integral on a surface where the electric field is constant then
the integral in the volume where the electric field is changing.
  E  dV   E  dA  
V
S
Q
, where
r  o
dA is a small piece or area that points out
of the enclosed surface. Usually when
Gauss’ law is taught the examples used
often have spherical symmetry, and in
spherical coordinates the directions are r̂ ,
̂ , and ˆ , and the normal to a sphere points
in the r̂ direction.
i
When the charge distribution only changes
as a function of radius (not with  or ) then
the electric field only points away from or to
the origin, in other words we always have
E  E  0 , and only Er  0 when charges
are present. When the charge distribution
only changes with x then, and we always
have E y  Ez  0 , and E x  0 when
charges are present.
ii
i
http://mathworld.wolfram.com/SphericalCoordinates.html
ii
https://www.physics.byu.edu/faculty/christensen/Physics%20220/FTI/24%20Electric%20Flux%20and%20
Gauss%27s%20Law/24.6%20A%20spherical%20Gaussian%20surface%20Surrounding%20a%20point%
20charge.htm
In ECE 346 we will see a negative charge density   e  N A for  x p  x  0 , and a positive
charge density of   e  N D for 0  x  xn . There will be just as many negative charges as
positive charges, so N A  x p  N D  xn . Since the charge density only changes with x we have
E x  0 , and E y  Ez  0 . We have Ex  0 for x < -xp or x > xn. since there are no more charges
for the electric field to start or end on.
Using Gauss’ law when x > 0,
we put the right side of our cube
for x > xn (where Ex=0) and the
left side of the cube at x. The
charge inside is
  xn  x   S 2  e  N D . On the
left side of the cube we have
iii
 Ex  S 2 
 e  N D   xn  x 
e  N D   xn  x   S 2
, or E x 
for 0  x  xn . In a similar manner
r  o
r  o
we can put another cube for x < 0. We put the left side of our cube where x < -xp (where Ex=0)


and the right side of the cube at x. The charge inside is  x p  x  S 2  e  N A . On the right side
we have E x  S 2 
iii
 e  N A  x p  x   S 2
r  o
, or E x 
 e  N A  x p  x 
r  o
https://www.slideshare.net/avocado1111/lecture-3-27649499/16
for  x p  x  0 .