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PHYSICS 2
ELECTROMAGNETIC PROPERTIES OF MATTER

Coulomb’s Law – force between point charges
F

Electric field from point charge q2
E

q1 q2
4 o r 2
1
q2
4 o r 2
1
Force on charged particle in electric field
F qE

Electric flux and Gauss’s Law
E 

 E  dA 
qenclosed
o
Electrical potential V or V
W  q V
f
work done in moving a charge in an electric field W   F  ds
i
f
V  V f  Vi   E  ds
i
V
x
V represented by equipotential lines or surfaces: E from gradient of V
E  V
q
potential due to a point charge q with V = 0 at 
4 o r
Electric dipole (-q  +q)
V

 Ex  
1
p  qd
V
electric dipole moment: direction from negative to positive charge
p cos 
4 o r 2
1
potential of dipole r >> d
  p E
 tends to line p up parallel to E
U  pE
need work to rotate dipole away from equil (stable &
unstable)
potential energy of a dipole (angle is measured between
direction of electric field & direction of dipole)
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
Parallel plate capacitors (vacuum between plates, A >> d2 ignore edge effects)
Q  CV
E
C
Q
V

A o d
o A
d
Q

surface charge density
A
QV CV 2 Q 2
U


2
2
2C
u

 E2
U
U

 o
Vol A d
2
potential energy stored
energy density stored in electric field
Dielectrics
Cd  K C
K = dielectric constant or relative permittivity

K  d  1 K(air) = 1.00054 K(paper) = 3.5
o
Dipoles in dielectric line up  dielectric has become polarized  bound charges
at ends of dielectric Qbound  electric field reduced
Charges on plates of capacitor called Qfree

Polarization of materials
+
-
+
+
+
+
free
N
V
p  qd
n
-
-
+
+
+
-
-
-
+
-
bound
number of dipoles / volume
electric dipole moment
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Pnp
Vector sum all elementary dipoles / volume to give polarization
vector (C.m-2) - Region of uniform polarization is where all
dipoles point in the same direction.
Qbound
  bound
A
P  e  o E electric susceptibility,  e
linear dielectric material
P
permittivity of dielectric (K  r)
To include effects of dielectric  o  
  d  K o
D
Qfree
  free
A
electric displacement vector
 total   free   bound  D  P   o E
D  o E  P
line of D connect free charges
line of P connect bound charges
line of E connect total charges
D   o (1  e ) E  K  o E
K  1  e
dielectric constant
adding dielectric E  E / K

Polarization of molecules (non-polar) – electrons move in response to external
electric field
p  qd
electric dipole
p  E
(atomic) polarizability, 
  4 o a 3 polarizability for atom: radius a ~10-10 m

o
n
 K  1
3  K  1
 o
n  K  2
very dilute gas (non polar)
  linking microscopic quantity,  to macroscopic quantity, K
Clausius- Mossotti equation (non polar molecules)
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

1 
 pE 

(polar molecules)
P  n p coth 


 kT  pE 
kT 

Langevin equation – polarization decreases with increasing temperature


i
Conductors
dq
dt
J
i  n q vdrift A
EJ


R

i   J  dA
J  n q vdrift
n = number of charges particles / vol
resistivity,  (.m)
m
e n
1
time between electron collisions , 
2

L
A
1
G
R
i
A
conductivity
resistance (ohms, )
conductance (Siemens, S)
Magnetism
F  qv B
Right hand open palm rule: B  fingers; v ( q)  thumb;
F  open palm of hand
m v2
F  qv B 
r
Motion in a circle (microwaves from magnetron , mass spectrometer)
mv
qB
fC 
 r
radius of orbit
cyclotron frequency
2 m
qB
Bi
VH 
Hall voltage n = charge carriers / vol
l = thickness of strip
nel
dFB  i dL  B force on a current element
  N i A n magnetic dipole moment Direction – right hand screw rule
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B    B
U B    B potential of magnetic dipole in magnetic field
Note: do not be confused with the symbol 
 = magnetic dipole moment or
o = permeability of a vacuum
 = permeability of magnetic medium
dB 
o i ds  r
4
r3
 B  ds  
B  o ni
o
Ampere’s Law
i
Magnetic field inside a solenoid (n = number of turns / length)
Ni
L
B  o n i  o
Bz 

magnetic field produced by current element Biot-Savart Law
o i a 2
2(a 2  z 2 )3/ 2
Magnetic field inside a toroidal coil
Magnetic field along the axis of a current loop
Magnetic materials
B  o ( H  M )   H
   r o
o = permeability of vacuum
 = magnetic permeability
r = relative permeability
  o 1  m  m = magnetic susceptibility
M  m H
Ferromagnetic materials:  or m are not constants
M is not proportional to H

Electromagnetic Induction
e
 E  ds  
e  L
di
dt
d B
d

dt
dt
  B  d A
induced emf
Faraday’s Law
emf of a coil (inductor)
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U  12 L i 2 energy stored by inductor
L  o n 2 l A
B2
u
2 o
Vs 

self inductance of a long solenoid
energy density for long solenoid
Ns
Vp
Np
is 
Np
Ns
ip
Transformers
Maxwell’s Equations
E all charges (free and bound)
P bound surface charges
D free charges
D   o E  P  K o E   E
B all currents (free and surface)
M bound surface currents
H free currents
Gauss’s Law
Gauss’s Law
M   B  dA  0
qtotal
E 
 E  dA 
D 
 D  dA  q
o

free
Faraday’s Law
 E  ds  

B  o H  M   r o H   H
Modified Ampere’s Law
dB
dt
 B  ds  
 H  ds  i
o
i  o  o
free

dE
dt
dD
dt
displacement current
id   o

d E
dt
Free space propagation in X direction for a linear polarized electromagnetic
direction
2 Ey
x 2
 o  o
2 Ey
t 2
0
wave equation for electric field ( E polarized in Y direction)
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E y  Eo cos( t  k x)
plane wave solution
 2 Bz
 2 Bz



0
o o
x 2
t 2
wave equation for magnetic field
( B polarized in Y direction)
Bz  Bo cos( t  k x)
v

k

vm 
1

k
o  o
1

plane wave solution
c
phase velocity of wave (speed of light in vacuum)
 cm phase velocity in a non-conductive medium

E cB
S
1

S 
EB  EH
1
o
E y Bz 
U = S A dt
1
c o
Poynting vector – rate of energy transfer per area
Ey 2 
c
o
Bz 2  I
intensity of plane EM wave (vacuum)
energy
1
1 2
1
utotal   E 2 
B  2 E2
2
2
cm 
total energy density for EM wave in non-conducting medium
Prad 
I
c
Prad 
2I
c
radiation pressure for absorption
p 
U
c
momentum transfer by EM wave for absorption
p 
2 U
c
radiation pressure for reflection
momentum transfer by EM wave for reflection
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