ELECTRIC
FIELD
LINES …..
Electric field lines
Recall that we defined
the electric field to be the
force per unit charge at a
particular point:
For a point charge
Electric field lines
IF q IS POSITIVE,
THEN THE FIELD IS
DIRECTED RADIALLY
AWAY FROM THE
CHARGE.
q>0
Electric field lines
IF q IS NEGATIVE,
THEN THE FIELD IS
DIRECTED RADIALLY
TOWARDS THE
CHARGE.
q<0
Electric field lines
Field lines were introduced by Michael Faraday to help visualize the direction
and magnitude of he electric field. The direction of the field at any point is
given by the direction of the field line, while the magnitude of the field is given
qualitatively by the density of field lines. In the above diagrams, the simplest
examples are given where the field is spherically symmetric. The direction of
the field is apparent in the figures. At a point charge, field lines converge so
that their density is large - the density scales in proportion to the inverse of
the distance squared, as does the field. As is apparent in the diagrams, field
lines start on positive charges and end on negative charges. This is all
convention, but it nonetheless useful to remember.
Electric field lines
The field lines for
an electric dipole:
x
|q+| = |q-|
This figure portrays several useful concepts. For example, near
the point charges (that is, at a distance that is small compared to
their separation), the field becomes spherically symmetric. This
makes sense - near a charge, the field from that one charge
certainly should dominate the net electric field since it is so large.
Along a line (more accurately, a plane) bisecting the line joining
the charges, we see that the field is directed along the -x direction
as shown.
Electric field lines
Field with two positive charges of equal magnitude
+
+
In this case, we see the zero-field region precisely between the
two charges, and we also see a fairly rapid convergence on a
spherically symmetric distribution of field lines.
Electric field lines
The figure shows the electric
field lines for a system of two
point charges.
(a) What are the relative
magnitudes of the charges?
(b)What are the signs of the
charges?
(c) In what regions of space is
the electric field strong? weak
Electric field
The figure shows the electric field lines for a system of two
lines
point charges.
(a) What are the relative magnitudes of the charges?
(b) What are the signs of the charges?
(c) In what regions of space is the electric field strong?
weak?
Answer
a)
b)
c)
There are 32 lines coming from the charge on the left,
while there are 8 converging on that on the right. Thus,
the one on the left is 4 times larger than the one on the
right.
The one on the left is positive; the one on the right is
negative.
The field is large near both charges. It is largest one a
line connecting the charges. Few field lines are drawn
there, but this is for clarity. The field is weakest to the
right of the right hand charge.
Electric field lines
Electric field lines
IN GENERAL, FIELD LINES FOLLOW A SIMPLE SET OF RULES:
•Electric field lines begin on positive charges and end on negative
charges, or at infinity
•Lines are drawn symmetrically leaving or entering a charge
•The number of lines entering or leaving a charge is proportional to the
magnitude of the charge
•The density of lines at any point (the number of lines per unit length
perpendicular to the lines themselves) is proportional to the field
magnitude at that point
•At large distances from a system of charges, the field lines are equally
spaced and radial as if they came from a single point charge equal in
magnitude to the net charge on the system (presuming there is a net
charge)
•No two field lines can cross since the field magnitude and direction must
be unique.
Electric Flux
How would you
measure 'the density of
electric field lines' in a
vicinity of space? First
think only of a discrete
set of electric field lines.
One obvious answer to
the question is that you
would count the number
of lines passing through
an imaginary
geometrical (not real!)
surface.
Electric Flux
The number of field lines passing through a geometrical surface of given area
depends on three things: the field strength (E), the area (A), and the orientation of
the surface (). The first two are obvious, and the following diagram will indicate
the last:
Electric Flux
It is useful also to represent the area A by a vector. The length
of the vector is given by the area (a scalar quantity), while the
orientation is perpendicular to the area. With this definition,
the flux can be defined as:
=
*
Electric Flux
[E]-Electric field; Newton/Coulomb {N/C} or {V/m}
[A]-Surface area; meter2 {m2}
[]-Angle; degrees or radians {o} or {radians}
[E]-Electric flux; Newton meter2/Coulomb {Nm2/C} or {Vm/C}
Electric Flux
Flux for Non-Uniform Fields / Flux for Non-Uniform Surface
You might have noticed that all these equations really only
work for uniform electric fields.
We can use them here provided we make them pertain to
differential area elements, and over a differential area the field
is uniform.
We then need to integrate to get the total flux through an
extended surface in a non-uniform field.
Electric Flux
Flux for Non-Uniform Fields
Flux for Non-Uniform Surface
Element of surface area dA

E


E

n
Enlarged view
q
Surface
The number of field lines passing through a geometrical surface of given area
depends on three things: the field strength, the area, and the orientation of the
surface.
Electric Flux
Flux for Non-Uniform Fields
Flux for Non-Uniform Surface
The differential electric flux passing through a differential
area is given by:
Element of surface area dA


E

n
As before, dA is a vector oriented perpendicular to the area, and the area itself
is differential (i.e., it's infinitesimally small and it's shape doesn't usually
matter). The total electric flux can be evaluated by integrating this differential
flux over the entire surface.
Electric Flux
Flux for one surface element
Surface

E
Total flux through surface
q
The integral is taken over
the entire surface.
Divide entire surface into small elements dA
Electric Flux
The simplest example is a spherical surface
centered on a point positive charge.
 E  ????
Electric Flux
 E  ????
Recall that the field lines radiate
outward from the charge in this case.
The density of field lines needs to
reflect the magnitude of the electrical
field. Since the field magnitude
decreases by the inverse square law, so
must the density of field lines.
Now consider the area of the spherical
surface. This increases like the square
of the radius. The total number of field
lines passing through the surface,
which is the product of their density
and the area of the surface, must then
be independent of the size of the
sphere. The number of field lines is
directly related to the size of the
charge at the center of the sphere.
Electric Flux
 kq 
E 2r
r
Electric field
  2
dA  nr sin  dd Surface element
 
 E   E  dA  4kq
S
The simplest example is a
spherical surface centered
on a point positive charge.
E 
q

Electric Flux
E 
q

Gauss’ law
Gauss' law is a generalization of the results discussed above
for the single charge and spherical surface.
It relates the electric flux passing through any surface
enclosing a charge distribution to the net charge enclosed by
the same surface.
E 
q

 
 E   E  dA
S
Gauss' Law
q2, q3 and q5 enclose by
surface
E

q2  q3  q5 

E


qtotal enclosed 


Gauss' Law
External charges don’t
contribute to the flux
Gauss’ law
  qtotal enclosed 
 E  dA 
S

It relates the electric flux passing through
any surface enclosing a charge distribution
to the net charge enclosed.
Useful when charge distribution has a high degree of symmetry
sphere, line, flat surface, …
Charged surface
S
y
Electric Flux
Gaussian surface (imaginary)
x
P
A
S
Charge density on surface
• E Perpendicular to charged
surface.
•No flux through sides of
Gaussian cylinder.
A
Electric Flux
 
 E  2 E  A  2 EA
A
Only have flux
through top and back
surfaces
  qtotal enclosed 
Gauss’ Law  E  dA 
S
Total charge
enclosed

S A
Electric Flux  E    S dA 
S

1
Electric Flux
 E  2EA
A
Electric Field
S
E
2
Charged surface
 S
 S
+Q
-Q
V
Area of plates A
air inside
Battery redistributes
charge between plates
but system remains
overall neutral

E0
P
PLATE
(1)
 S

Q
E
o A
 s
E
(2)
 S
o

E1
P

E2

E1
P

E2
+Q

E0

E1

E2
-Q
V
E
S
2
Real
Ideal
R
L
  qtotal enclosed 
 E  dA 

S
 
 E  dA E (2RL)
S
𝑞𝑡𝑜𝑡𝑎𝑙 𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 = 𝜆𝐿
𝜆
Vector
𝐸=
2𝜋𝜖𝑅
ELECTRIC
FIELD
LINES …..