Lecture 4 Electric Potential - Welcome to CUNY

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Lecture 4
Electric Potential
Conductors
Dielectrics
Electromagnetics
Prof. Viviana Vladutescu
Electric Potential
Electric Potential
The electric field intensity is
acting as a force on any charges
it arrives upon.
Therefore in moving a unit
charge from P1 to P2, work must
be done against the field.
When force is applied to move an object, work is
the product of the force and the distance the
object travels in the direction of the force
W 
P2
P2
P2
P1
P1
P1
 F  d l   Q E  d l Q  E  d l
but since t heforcem ovest hecharge
P2
against t he field  W  Q  E  d l
P1
P2
Therefore
W
  E  dl
Q
P1
without specifying the path
P1
E
The scalar line integral of an
Irrotational (conservative) E
field is path-independent
P2
E

d
l

0

Equipotential surfaces
A set of points with same potential forms equipotential
surface. For a point charge, equipotentials are spheres
at fixed radius r.
Consider the plot of the
electrostatic potential contours
forming equipotential surfaces
around the point charge
superimposed over the field
lines for the point charge
As we can notice the field goes into the direction of
decreasing potential
If the behavior of the potential is unknown, the electric
intensity field can be determined by finding the maximum
rate and direction of the spatial change of the potential
field
E   V
P2
W
  E  dl
By using the above in the following equation
Q
P1
we get
P2
P2
P1
P1
   E  d l     V  d l 
P2
P2
 (V )  a dl   dV  V
l
P1
2
P1
Potential difference
 V1
P2
P2
 V21    E  d l   
P1

P2
Q
4 0R P1
V
Q
4 0R
W
V 
Q
P1
Q
4 0R
2
a R dRa R 
Q  1
1
    VP1  VP2

4 0  R2 R1 

Absolute potential at some finite
radius from a point charge fixed at
the origin (reference voltage of zero
at an infinite radius)
Work per Coulomb required to pull a
charge from infinity to the radius R
For a collection of charges of continuous distribution
V
V 
dQ

4 0R
V
V
1
4 0
1
4 0
1
4 0
v
 R dv (V)
v
s
 R ds (V)
s
l
 R dl (V)
l
Review
If the electrical force moves a charge a certain distance, it
does work on that charge. The change in electric potential
over this distance is defined through the work done by this
force:
Work done=Charge on Q*Potential
where potential is shorthand for change in electric potential,
or potential difference. This is analogous to the definition of
the gravitational potential energy through the work done by
the force of gravity in moving a mass through a certain
distance. The units of potential difference, or simply potential,
are Joules / Coulomb, which are called Volts (V). Physically,
potential difference has to do with how much work the electric
field does in moving a charge from one place to another.
• Batteries, for example, are rated by the
potential difference across their terminals.
In a nine volt battery the potential
difference between the positive and
negative terminals is precisely nine volts.
On the other hand the potential difference
across the power outlet in the wall of your
home is 110 volts.
Conductors
Are caractherized by ε, μ and σ
The conductivity σ (S/m or 1/Ω*m or mhos/m)
-depends on the charge density ρ
-depends on the temperature
Ex of superconductors: yttrium-barium-copper-oxide
Current and Current Density
• Current
• Current
density
amountof charge(C) 1C

given time(s)
1s
current(A) A
J
 2
2
area(m ) m
I   J ds
Types of current
-conduction currents: present in conductors and
semiconductors and caused by drift motion of conduction
e- or holes in a media in response to an applied field ex:
J=σ* E (conduction current density)
-displacement or electrolytic currents: is the
result of migration of positive and negative ions as well
known as time-varying field phenomenon that allows
current to flow between plates of a capacitor.
-convection currents: involve the movement of
charged particles through vacuum, air or other
nonconductive media (e- in a cathode ray tube)
J&E
V=I*R
V   R I
j
k k
j
Conservation
of charge

 J  
t
(V)
k
I
j
j
 0 (A)
Conduction currents
Q  Nqu  an st
Q
I 
 Nqu   s
t
 A
J  Nqu  2 
m 
For most conducting materials the average drift velocity is
directly proportional to el field intensity
u  e Em / s  J  e e E  E
Conductors in static electric field
Inside a conductor
ρ=0
E=0
Under static conditions the E
field on a conductor surface is
everywhere normal to the
surface (the surface of a
conductor is an equipotential
surface under static conditions)
Charactheristics of E on conductor
/free space interfaces
-The tangential component of the E field on a conductor
surface is zero
-The normal component of the E field at a conductor /free
space boundary is equal to the surface charge density on
the conductor divided by the permittivity of free space
Boundary Conditions at a Conductor/
/Free Space Interface
Et=0
En=ρs/ε0
E

d
l

E

w

0

E

0
t
t

abcda
 s S
s
s E  d s  En S   0 or En   0
Dielectrics
-Ideal dielectrics do not contain free charges
-contain bound
charges
Induced electric dipoles
The material is polarized
Polar molecules (Permanent dipole moment)
Nonpolar molecules
Ex: By aligning the molecules during the fabrication of a
material (use E field when the material is melted and maintain
it until it solidifies) we can obtain electrets
The volume density of the electric
dipole moment
nv
P  lim
v 0
p
k 1
k
v
Polarization vector
n-#of molecules per unit volume
Vector sum of the induced
dipole moments
Homogeneous & linear & isotropic media
D = εE
D=ε0E+P
Polarization charge densities
 ps  P  an
-surface
 p    P
-volume
A polarized dielectric may be replaced by an equivalent
polarization surface charge density and an equivalent
polarization volume charge density for field calculation
V
1
4 0

s
 ps
R
ds 
1
4 0

v
p
R
dv
Total Charge
Q    ps ds    p dv 
s
v
  P  an ds     Pdv  o
s
v

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