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PHYS2012
ASSIGNMENT 2010
ELECTROMAGNETIC PROPERTIES OF MATERIALS
DUE FRIDAY 15 OCT 2101
Question 1
On a macroscopic scale, when the polarization P is proportional to the electric field E
within the dielectric (dielectric constant r)
P=np
p=qd
P  e o E   r 1   o E
where n is the number density (molecules/volume), p is the dipole moment, q is the
charge separated and d is the charge separation.
On an microscopic scale an induced electric dipole moment p is proportional to an
external electric field E and the constant of proportionality  is called the polarisability of
the molecule
p  E
For an atom (assume atom spherical volume of radius a),   4   o a 3
For non-polar molecules (no permanent electric dipole moments)

 For a dilute medium (r ~ 1)   o ( r  1)
n
 A better model that can be applied to more dense media, is the Clausius-Mossotti
3   1
 o r
equation
n r  2
(a)
What is a simple way to measure r?
n
(b)
Plot
vs. r (r from 1 to 1.5) for a dilute and dense media.
o
(c)
What is the approximate value of r when the equation for the polarisability of
non-polar molecules for a dilute medium fails?
(d)
The dielectric constant for oxygen gas is rg = 1.000523 and its molar mass is 32.0
g. One mole of oxygen gas at STP (273 K, 1 atm) has a volume of 22.4 L. In the
liquid state, the density of oxygen is 1.190103 kg.m-3.
(NA = 6.021023)
(i)
Calculate the number densities for oxygen in the gas ng and liquid nl states.
(ii)
Estimate the radius of an oxygen atom. Is your answer reasonable?
(iii) Assume that the atomic polarizability  is the same in the gaseous and liquid
states. Estimate its dielectric constant for liquid oxygen r l. Comment on your
answer
(e)
The oxygen gas was placed into an external electric field E = 12.5 kV.m-1.
(i)
Calculate the induced electric dipole moment p.
(ii) Calculate the separation d between the centre of positive and negative charge for
the oxygen atoms. Comment on your answer.
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Question 2
A parallel plate capacitor has a square plate area of 0.200 m2 and a plate separation of
0.0100 m. The potential difference between the plates is 3.00 kV. A dielectric slab with a
dielectric constant of 3 completely fills the space between the plates of the capacitor.
While the capacitor is still connected to the battery, the dielectric slab is partially
removed. The slab is withdrawn in a direction parallel to the plates through a distance x
(slab fully inserted, x = 0 and half space filled, x = L/2, slab fully removed, x = L).
Assume the dielectric slab is removed very slowly so that the kinetic energy of the slab is
zero and that edge effects at the ends of the capacitor can be ignored.
(a)
Show and justify the following relationship as functions of x
Capacitance
Charge
Energy stored by capacitor
 L
C   0   x   r  L  x 
 d 
  LV 
Q 0
  x   r  L  x 
 d 
  LV 2 
U cap   0
  x   r  L  x 
 2d 
U cap  U cap ( x)  U cap (0)
  LV 2 
U cap   0
  x  r  L  x  r L
 2d 
Change in energy stored by battery
qbattery   Q( x)  Q(0) 
  LV 2 
U battery    0
  x  r  L  x  r L
d


2
  LV 
Work done by external force Wme    0
  x  r  L  x  r L
 2d 
External force in removing dielectric
  LV 2 
Fme   0
  r  1 independent of x
 2d 
(b) Calculate the above quantities when x = 0, L/2, L. Comment on the changes.
x=0
x = L/2
x = L Comments
x
m
x increasing
-10
C
10 F
Q
10-6 C
Ucap
10-3 J
Is energy to or from capacitor?
Ucap
10-3 J
-3
Is energy to or from battery?
Ubattery 10 J
Wme
10-3 J
Fme
10-3 N
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Question 3
A metal sphere of radius a and charge +Q is surrounded by a thick spherical layer of
dielectric material with a dielectric constant r and it has an outer radius b as measured
from the centre of the metal sphere. Starting with Gauss’s Law in the form
 D dA  q f show and/or justify the following relationships inside the dielectric
material (a < r < b):
(a)
(b)
(c)
(d)
(e)
(f)
Q
radial outward
4 r 2
Q  Qb
Q

electric field E 
radial outward
2
4  r 0 r
4  0 r 2
electric displacement D 
  1 
Qb  Q  r

 r 
Q   r 1 
polarization P 

 radial outward
4 r 2   r 
  1  3
 rˆ 
3
bound charge density b  Q  r
  (r ) use   2   4   r 
r 
 r 
Bound charge within the interior of dielectric material
  r 1  3
  r 1 
  r 1 
3

d



Q

(
r
)
d



Q

(
r
)
d



Q






b

   r 

 r 
 r 
surface charge density
Q   r  1 
 b (r  a) 


4 a2   r 
bound charge (at each surface)
 b (r  b) 
Q   r  1 


4  b2   r 
(g)
total charge of dielectric material Qdielectric = 0.
(h)
Sketch the electric field lines inside and outside the dielectric material. Explain
why the number of electric field line change at the boundaries of the dielectric
material.
(i)
Sketch the distribution of the free and bound charges for the metal sphere and
dielectric material.
(j)
How does the electric field surrounding the metal sphere (charge +Q) and
dielectric layer compare with the electric field surrounding a point charge Q?
Explain.
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Question 4
A hysteresis curve for a sample of iron is shown in the diagram below.
(a)
What are the important features of the hysteresis curve?
Hysteresis Curve for an Iron sample
2.0
1.5
1.0
B (T)
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-100
-50
0
H
50
100
(A.m-1)
(http://www.whitefang.net/Academics/Physics/I3-Hysteresis/hysteresis.htm)
A Rowland ring is a donut shaped ring or torus of a given ferromagnetic material with
two coils around it. The first long coil is used to set up the H-field inside the ring by a
current i. As the current i in this coil changes, an induced emf will be set up in the second
coil to give a value for the B-field. Consider a ring that is evenly wound with N = 200
turns of wire and has an average circumference, L = 0.50 m and cross sectional area, A =
4.0010-4 m2.
It is required that the magnetic flux through the ring to be, m = 4.0010-4 T.m2.
(b) What is the required value for the B-field?
(c) If the torus has an air core, what current I is required to produce the above
magnetic flux m ?
(d) If the torus has an iron core whose hysteresis curve is given above (for the
calculation, only use the magnetization part of the curve M  H), what current is
needed to give the required magnetic flux m ?
(e) What are the permeability µ and the relative permeability µr of the iron used in
your calculation for part (d)?
(f)
If an air gap, a = 1 mm in length is cut in the ring, what current I is required to
maintain the same magnetic flux?
(g) Compare the current through the coil windings for the three cases.
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