Clausius-Mossotti–p.1/17 - University of Richmondggilfoyl/em/slides/clausius.pdfEffect of a...
Transcript of Clausius-Mossotti–p.1/17 - University of Richmondggilfoyl/em/slides/clausius.pdfEffect of a...
The Electromagnetic Properties of Materials
1. What are the types of electromagnetic matter?
2 1 0 1 22
1
0
2
z
Clausius-Mossotti – p.1/17
The Electromagnetic Properties of Materials
1. What are the types of electromagnetic matter?
Conductors.
2 1 0 1 22
1
0
2
z
Clausius-Mossotti – p.1/17
The Electromagnetic Properties of Materials
1. What are the types of electromagnetic matter?
Conductors.
Insulators/Dielectrics.
2 1 0 1 22
1
0
2
z
Clausius-Mossotti – p.1/17
The Electromagnetic Properties of Materials
1. What are the types of electromagnetic matter?
Conductors.
Insulators/Dielectrics.
2. What happens when you put a neutral object in an electric field?
2 1 0 1 22
1
0
1
2
z
Clausius-Mossotti – p.1/17
The Electromagnetic Properties of Materials
1. What are the types of electromagnetic matter?
Conductors.
Insulators/Dielectrics.
2. What happens when you put a neutral object in an electric field?
-2 -1 0 1 2-2
-1
0
1
2
y
z
Clausius-Mossotti – p.1/17
Mechanisms for Polarizing Particles
Induced Polarization: Shift electrons and nuclei.
Clausius-Mossotti – p.2/17
Mechanisms for Polarizing Particles
Induced Polarization: Shift electrons and nuclei.
Polar Objects: Some nuclei, molecules, etc. have a permanentdipole moment. They get rotated by the applied field.
Clausius-Mossotti – p.2/17
What is a dielectric?
The dielectric constant of a materialmeasures how the material respondsto an applied external electric field.
If the atoms in the material have adipole moment they will tend to orientthemselves in the applied field so thenet field in the material is reduced.Internal electric field reduced fromthe vacuum value by the dielectricconstant ǫr.
Dielectric are non-metallic substances (gas, liquid, or solid).
Many practical applications like storing energy in capacitors,piezoelectric for making measurements, accumulating charge in anaccelerator, etc.
Clausius-Mossotti – p.3/17
The Problem
In a linear dielectric,the polarization is proportional to the field. If thematerial is a gas of atoms, the induced dipole moment of each one is alsoproportional to the applied field. What is the relationship between theatomic polarizability α and the dielectric constant ǫr? How well do yourresults agree with the data in the table below?
Gas α/4πǫ0 (10−30 m3) ǫ†r
Hydrogen 0.667 1.00025
Helium 0.205 1.000065
Neon 0.396 1.00013
Argon 1.64 1.00052
Clausius-Mossotti – p.4/17
An Example
A thin, disk-shaped block of dielectric of thickness t and radius R has auniform polarization ~P = P z as shown in the figure.
1. What are the bound charges σb and ρb?
2. What is the field inside the dielectric along the axis of the disk?
3. What is the field outside the dielectric along the axis of the disk?
tP
y
z
Clausius-Mossotti – p.5/17
Problem 2.6
Find the electric field a distance z above the center of a flatcircular disk of radius R (see figure below), which carries auniform surface charge σ. What does your formula give inthe limit R → ∞? Also check the case z >> R.
charge densityσPlane with surface
x
y
z
R
~E =1
4πǫ02πσz
(
1
z−
1√
z2 + R2
)
z
Clausius-Mossotti – p.6/17
Results for the Example
Red - t=0.01R
Blue - t=0.5R
Green - t=2R
Gray - t=6R
-4 -2 0 2 4-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
z�R
EHzL
Clausius-Mossotti – p.7/17
Results for the Example
Red - t=0.01R
Blue - t=0.5R
Green - t=2R
Gray - t=6R
-4 -2 0 2 4 6 8 10-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
z�R
EHzL
Clausius-Mossotti – p.8/17
Effect of a Dielectric On a Capacitor
A capacitor is a device for storing electromagnetic energy and has many uses in electronics,power generation and storage, etc. A parallel plate capacitor consists of two metal surfaces ofarea A held a distance d apart (see figure). The capacitance C is defined as
C =Q
V
where Q is the amount of charge on the positive plate, −Q is the amount on the negativeplate, and V is the electric potential between the two plates. Assume d2
≪ A.
1. What is the capacitance in terms ofpurely geometric parameters or con-stants?
2. Suppose you fill the space between theplates with an insulating material of di-electric constant ǫr . What happens tothe capacitance?
3. How is the energy stored in the capac-itor effected?
Clausius-Mossotti – p.9/17
The Problem
In a linear dielectric,the polarization is proportional to the field. If thematerial is a gas of atoms, the induced dipole moment of each one is alsoproportional to the applied field. What is the relationship between theatomic polarizability α and the dielectric constant ǫr? How well do yourresults agree with the data in the table below?
Gas α/4πǫ0 (10−30 m3) ǫ†r
Hydrogen 0.667 1.00025
Helium 0.205 1.000065
Neon 0.396 1.00013
Argon 1.64 1.00052
Clausius-Mossotti – p.10/17
A Picture of the Atomic Environment
1. The microscopic atomic environment is one of rapid change andsensitive dependence on distance, i.e. the electric fields changerapidly in time and space.
2. We are interested in under-standing the bulk, macro-scopic behavior of materials.
3. Go after the average proper-ties of the material and theatomic environment.
4. Divide space into the region‘inside’ where the atom is lo-cated and the ‘outside’ whichis the average product of allthe other atoms.
Inside
Outside
FieldExternal
Clausius-Mossotti – p.11/17
Average Field of Our Spherical Atom
What is the average field inside a sphere of radius R due to all the chargewithin the sphere?
dτr − r’
x R
q
y
z
r’r
Clausius-Mossotti – p.12/17
The Field Inside a Uniform, Spherical, Charge Distribution
What is the electric field ~E at a point ~r inside a uniform sphere of chargewith radius R, centered at the origin, and with r < R?
r − r’
x R
P
y
z
rr’
dq
Clausius-Mossotti – p.13/17
The Field Inside a Uniform, Spherical, Charge Distribution
What is the electric field ~E at a point ~r inside a uniform sphere of chargewith radius R, centered at the origin, and with r < R?
r − r’
x R
P
y
z
rr’
dq dτr − r’
x R
q
y
z
r’r
Clausius-Mossotti – p.13/17
Problem 2-12
Clausius-Mossotti – p.14/17
The Electromagnetic Response of Atoms
For a gas of atoms the relationship between the atomic polarizability α and the dielectricconstant ǫr is the following.
ǫr =1 +
2Nα
3ǫ0
1 −Nα
3ǫ0
Below is a comparison of the dielectric constants calculated from the measured α’s.
Gas α/4πǫ ∗0
ǫ†r (measured) ǫr (calculated)
Hydrogen 0.667 1.00025 1.00023
Helium 0.205 1.000065 1.000071
Neon 0.396 1.00013 1.00014
Argon 1.64 1.00052 1.00056
∗ Units of 10−30 m3.† For 1 atm, 20◦ C.
Clausius-Mossotti – p.15/17
The Electromagnetic Response of Atoms
For a gas of atoms the relationship between the atomic polarizability α and the dielectricconstant ǫr is the following.
ǫr =1 +
2Nα
3ǫ0
1 −Nα
3ǫ0
Below is a comparison of the dielectric constants calculated from the measured α’s.
Gas α/4πǫ ∗0
ǫ†r (measured) ǫr (calculated)
Hydrogen 0.667 1.00025 1.00023
Helium 0.205 1.000065 1.000071
Neon 0.396 1.00013 1.00014
Argon 1.64 1.00052 1.00056
∗ Units of 10−30 m3.† For 1 atm, 20◦ C.
ITW
ORKS!!
Clausius-Mossotti – p.15/17
Electrostatics 4 Homework
0 50 100 1500.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Θ HdegL
Èr�dÈ
Lower limit for Vquad�Vmono > 0.01
Clausius-Mossotti – p.16/17
Electrostatics 4 Homework
Red -Multipole Expansion
Blue - Exact Result
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
1.5
2.0
2.5
3.0
z
VHzL
Comparison of Line Charge Electric Potentials
Clausius-Mossotti – p.17/17