Electric Charge Deformation and Polarization

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Electric Charge / Deformation and Polarization

• Matt Pharr
• ES 241
• 5/21/09

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Electric Charge
Total charge is conserved
B
A
QA 0 QB 0 Qnet 0
10 protons, 10 electrons Net charge 10 - 10 0
QA 4 QB -4 Qnet 0
B
A
SI Units 1 Coulomb 6.242 1018 elementary
charges
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Capacitor
SI Units 1 Volt 1.602e-19 Joule
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Measurement of Electric Potential
R
V?
I
Current measured with galvanometer Ohms Law V
IR gives the potential
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A Capacitor, a Weight, and a Battery
In equilibrium
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F(l,Q) and Stress
Experimental Relation
a
-Q
F
Recall
l
Q
oil
Maxwell Stress
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Deformable Dielectrics
L
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Definition of Stress
No weight, no stress???
Nominal stress
How is there deformation due to voltage change?
Analogous to thermal expansion
Stress-free deformation
Stress generated due to constraint
Small a Very stiff
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3D Homogeneous Deformation
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Field Theory Recovers Maxwell Stresses in a Vacuum
Electric energy per current volume
Recall
Maxwell Stresses
P
11
Ideal Dielectric Elastomers
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Electrostriction
Close to extension limit High cross-link
density Deformation affects polarization
Well below extension limit Low cross-link
density Polarization unaffected by deformation
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Deformation Affects Polarization
A model quasi-linear dielectrics
Ideal dielectric elastomer
Quasi-linear dielectrics
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Pull-in Instability
Experimental Observation for oil
This can lead to electrical breakdown
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Pull-in Instability
Exercise Find critical electric field for
instability subject to a biaxial force in the
plane of membrane
Assume ideal dielectric elastomer and
incompressibility
Choose a free energy of stretching function
Neo-Hookean law
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Pull-in Instability
In equilibrium
For equal biaxial stress, s1 s2 s and ?1 ?2
?
Combining these two equations gives the following
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Pull-in Instability
If s/µ 0
If s/µ 1
Larger stretch before breakdown
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Solder Bumps
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Solder Relation to Class

• Multiple forces
• Chemical potential
• Electric current
• Package Warpage
• Temperature gradient
• Multiple phases

Ideas from Paper Covered in Class Kinetic laws
chemical potential, diffusion flux Principle of
virtual work work conjugates Traction Deformatio
n Rate Eulerian vs. Lagrangian
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Backup Slides
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Maxwell stress in vacuum (1873)
P
P
A field of forces needed to maintain equilibrium
of a field of charges
Electrostatic field
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Include Maxwell stress in a free-body diagram
Free-body diagram
h
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Work-conjugate, or not
Current State
Reference State
Nominal electric field and nominal electric
displacement are work-conjugate
Battery does work
True electric field and true electric
displacement are NOT work-conjugate
Battery does work