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Molecular Model of Induced Charge
Electronic polarization of nonpolar molecules
Total charge Q   qi  0
i
But dipole moment d   qi ri may be nonvanishi ng
i
For nonpolar molecules d  0 in the absence
of the applied electric field E but they
acquire finite dipole moment in the field :
d   0 E
( is the polarizabi lity of a molecule/a tom)
Electronic polarization of polar molecules
In the electric field more molecular dipoles are oriented
along the field
Polarizability of an Atom
- separation of proton and electron cloud in the applied
electric field
P- dipole moment per unit volume, N – concentration of atoms
When per unit volume, this dipole moment is called
polarization vector P  Nqδ   0E

Property of the material: Dielectric susceptibility   N
Polarization charges induced on the surface:  ind  Pn  P  n
For small displacements: P~E; P= 0 E
The field inside the dielectric is reduced :
   ind
E 
E  free
 0  free
0
K 0K
K  1  ;
 ind  (
K 1
) free
K
Gauss’s Law in Dielectrics
EA 
(   i ) A


0
KEA 


 KE d A
Q free
0
1

 i   1  
K
A
0
Gauss’s Law inDielectrics
Forces Acting on Dielectrics
We can either compute force directly
(which is quite cumbersome), or use
relationship between force and energy
F  U
CV 2
Considering parallel-plate capacitor U 
2
Force acting on the capacitor, is pointed inside,
hence, E-field work done is positive and U - decreases
U V 2 C
Fx  

x
2 x
x – insertion length
Two capacitors in parallel
C  C1  C2 
0
d
w( L  x) 
V 2  0w
Fx 
( K  1)
2 d
K 0
wx
d
w – width of the plates
More charge here
constant force
Electric Current
Charges in Motion – Electric Current
Electric Current – a method to deliver energy
Very convenient way to transport energy
no moving parts (only microscopic charges)
Electric currents is in the midst of electronic circuits
and living organisms alike
Motion of charges in electric fields
Force on a particle : F  qE
Accelerati on : a  F / m
d 2r
Equation of motion : m 2  qE(r, t )
dt
When E is time - independen t, the total energy is conserved :
mv 2
 q (r )  const
2
Motion in a uniform electric field
For x - components :
a  qE / m
v(t )  v0  at
at 2
x(t )  x0  v0t 
2
Other components of v do not change
Deflection by a uniform electric field
x  vi t
qE 2
y
t
2m
y   x 2 : Parabolic trajector y
v fx  vi
qE l
v fy  
m vi
Application: Cathode Ray Tube
Electric Current in Conductors
In electrostatic situations – no E-field inside
There is no net current. But charges (electrons)
still move chaotically, they are not on rest.
On the other side, electrons do not move with
constant acceleration.
Electrons undergo collisions with ions. After
each collision, the speed of electron changes
randomly.
The net effect of E-field – there is slow net
motion, superimposed on the random motion
Vchaotic ~ 106 m / s
Vdrift ~ 104 m / s
Direction of the Electric Current
is associated with the rate of flow of charge
ΔQ
dQ
through surface A : I 

Δt
dt
1 Coulomb
Unit : 1 Ampere 
1 second
Current density is the current per unit area :
I
J
A
Current in a flash light ~ 0.5 A
In a household A/C unit ~ 10-20 A
TV, radio circuits ~ 1mA
Computer boards ~ 1nA to 1pA
Current, Drift Velocity, Current Density
Q  qnAvd t

J

I Q

 qn v d [ A / m 2 ]
A At
Concentration of mobile charge
carriers per unit volume: n
Average speed in the direction
of current (drift speed): vd
For a variety of charge carriers:


J   | qi | ni v d
i
Current density J, is a vector
while total current I is not


I   J d S
Example: An 18-gauge copper wire has nominal
diameter of 1.02 mm and carries a constant current
of 1.67 A to 200W lamp. The density of free electrons
is 8.5*1026 el/m3. Find current density and drift velocity
J
I
4I

A d2
J  nevd ;
2  106 A / m2
vd  1.5  104
m/ s
Why, then, as we turn on the switch, light comes
immediately from the bulb?
E-field acts on all electrons at once (E-field
propagates at ~2 108 m/s in copper)
Electric current in solution of NaCl is due to
both positive Na+ and negative Cl- charges flow
Ohm’s Law
Current density J and electric field E are established inside a conductor when a
potential difference is applied –
Not electrostatics – field exists inside and charges move!
In many materials (especially metals)
over a range of conditions:
J = σE or J = E/r
with E-independent conductivity σ=1/r
This is Ohm’s law
(empirical and restricted)
Conductors, Insulators and
Semiconductors