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Outline
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Area vector
Vector flux
More problems
Solid angle
Proof of Gauss’s Law
Electric field lines
These are fictitious lines we sketch which point in
the direction of the electric field.

1) The direction of E at any point is tangent to the
line of force at that point.
2) The density of lines of force in any region is
proportional to the magnitude of E in that region
Lines never cross.
Density is the number of lines going through an area (N)
divided by the size of the area
At R1
R1 R
2
At R2
N
density 
2
4R1
N
density 
2
4R2
N
At any r density 
4r 2
1
q
For a charge q located at the origin E 
40 r 2
density  E
It is important that the force is proportional to 1
r2
Gauss’s Law
The total flux of electric field out of any
closed surface is equal to the charge
contained inside the surface divided by  0 .
  Qenclosed
 E  dS 
S
0
 
E  dS
What is
water flow?
or flux of any vector, e.g. velocity of a

Consider a flow with a velocity vector v .
Let S be a small area perpendicular to v .
v
v
S
a)
b)

S  Sn
S

n
Area
 vector
Flux:
 
  vS cos  v  S
a) The volume of water flowing through S per unit time is
vS

b) Now S is tilted with respect to v . The volume of water flowing
through S per unit time is vS cos


 is the angle between velocity vector v and unit vector n
normal to the surface S.

Flux of electric field E
S
S

The flux of E
 
d  E  dS
 
 s   E  dS
S
 
   E  dS
More on area vector ….
n
Suppose there was an electric field given by


E  E0ix .
What would be the electric flux through a plane with area A if
the plane were oriented in the y-z plane at x  x0?
What would be the electric flux if the plane
were oriented in the x-z plane at y  y0 ?
Quiz
Suppose there was an electric field given by


E  E0 i y
What would be the electric flux through a plane with area
A if the plane were oriented in the xz plane at y = y0?
What would be the electric flux if the plane were oriented
in the yz plane at x = x0?
Quiz
Suppose there was an electric field given by
What would be the electric flux through a front side of a
cube of side L?
Have a great day!
Hw: All Chapter 4 problems
and exercises
Reading: Chapter 4