(Pr. 1) In an isotropic medium, Poisson's ratio v relates Young's modulus to either the shear modulus G or the bulk modulus K (inverse of the compressibility) as: Y=2(1 + v)G-3(1-2v)K (see your theory slides). (a) A certain material has K-8 10% Pa and Y-10° Pa. What is its Poisson's ratio? What is its shear modulus? A vulcanized natural rubber (cis-polyisoprene) has K-2 10° Pa and Y= 1.3 106 Pa. What is Poisson's ratio for this vulcanized natural rubber? What is its shear modulus? (b) Show that in the limit where K is much larger than G, Poisson's ratio is equal to 1/2. This limit is valid for most liquids (including polymer melts and solutions), which is why they are termed "incompressible". (c) Show that the bulk modulus of an ideal gas is equal to its pressure. Assuming that a water molecule is a sphere of diameter 0.3 nm, estimate the bulk modulus of water assuming that the ideal gas result is valid even for liquid water. Compare your value with the actual bulk modulus of water (2 GPa).

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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(Pr. 1) In an isotropic medium, Poisson's ratio v relates Young's modulus to either the shear modulus G or the
bulk modulus K (inverse of the compressibility) as: Y=2(1 + v)G=3(1-2v)K (see your theory slides).
(a) A certain material has K = 8 -10³ Pa and Y = 10° Pa. What is its Poisson's ratio? What is its shear modulus?
A vulcanized natural rubber (cis-polyisoprene) has K = 2. 10° Pa and Y = 1.3 106 Pa. What is Poisson's ratio
for this vulcanized natural rubber? What is its shear modulus?
(b) Show that in the limit where K is much larger than G, Poisson's ratio is equal to 1/2. This limit is valid for
most liquids (including polymer melts and solutions), which is why they are termed "incompressible".
(c) Show that the bulk modulus of an ideal gas is equal to its pressure. Assuming that a water molecule is a
sphere of diameter 0.3 nm, estimate the bulk modulus of water assuming that the ideal gas result is valid even
for liquid water. Compare your value with the actual bulk modulus of water (2 GPa).
(Pr. 2) A simple viscometer consists of two square plates, of side 10 cm, separated by a distance of 1 mm. The
lower plate is fixed while the upper one can slide.
(a) If the space between the plates is filled with an oil whose viscosity is 1 Pa s, find the force that must be
applied to the upper plate to keep it moving at a speed of 0.5 cm s ¹.
(b) If the gap between the plates is filled with a rubber with a shear modulus of 5 105 Pa, and the upper plate is
shifted from the rest position of the rubber yielding an angular deformation of 0.01 radians, what force must be
applied to keep the upper plate in this position?
(c) A parallel-plate rheometer consists of a motor able to rotate a disk of radius R parallel to a flat horizontal
surface, and a heighth above it. The space between the disk and the surface is filled with a rather viscous liquid
of viscosity 7. Show that, for the disk to rotate at a constant angular velocity about its centre, the torque
applied to the disk must be proportional to R¹, and find the constant of proportionality. Why can the moment of
inertia of the disk be neglected under steady conditions (constant ) ? Hint: integrate the definition of viscosity.
(Pr. 3) Consider a phase made of molecules with permanent dipole moment p, in an applied electric field
Eexternal. Each dipole has energy W=-p. E= μ E cose, where E is the local electric field at a molecule's
position, 0 is the angle between E and p, and is the modulus of the vector p. Assume that E is parallel to the
z axis.
(a) Explain why the average molecular dipole moment is given by: (p)
Spexp
: (p₂)2 =
Sexp ΦΩ
L-E
μ(cose) 2, where do do sinode = do d(cose) is the differential of the solid-angle. Why is the solid angle used
here? How is the elementary probability dw that the dipole moment points in a given direction, related to the
differential d?
(b) By integrating by parts the numerator of the above expression for (p), show that its modulus is equal to:
[exp(a)+exp(-a)
(P₂) = μ(cose):
=H*exp(x)dx
[exp(a) exp(-a)], where a = µ E /(kB7)
a
The function L(x) =
=
μl
exp(x)dx
ex+e-x
ex-e-x
-= coth(x) is known as Langevin function. (coth = hyperbolic cotangent)
(c) Use the approximate expression ex1 + x (Taylor expansion to linear order), valid for small x, inside both
integrals in the expression for expansion (p) given in (a), to show that (p)
show that (p) = 37. Call K the effective
3kgT
3kgT
polarizability, deff.
(d) Explain the meaning of local electric field, and why is it different, in general, from the externally applied
field Eexternal. Show that, if the molecules are in the gas phase, the local field is simply equal to E = Eexternal. Show
that, for a gas of polar molecules, the polarization of the sample, defined as P = n (p), where n (=N/V) is the
number density of molecules, can be written as P = (-1) Eo E, where to is the vacuum permittivity, and give
the expression for & for this gas of polar molecules, as a function of the effective polarizability. Substitute the
value you found in (c) to obtain the expected temperature dependence of &
(e) If instead the phase is a polar liquid, combine the result of point (c) with either the Clausius-Mossotti cavity
field, or the "reaction" field (see theory slides), to obtain two different expressions for &. Compare the two
expressions with one another, and with the Kirkwood formula in the theory slides.
Transcribed Image Text:(Pr. 1) In an isotropic medium, Poisson's ratio v relates Young's modulus to either the shear modulus G or the bulk modulus K (inverse of the compressibility) as: Y=2(1 + v)G=3(1-2v)K (see your theory slides). (a) A certain material has K = 8 -10³ Pa and Y = 10° Pa. What is its Poisson's ratio? What is its shear modulus? A vulcanized natural rubber (cis-polyisoprene) has K = 2. 10° Pa and Y = 1.3 106 Pa. What is Poisson's ratio for this vulcanized natural rubber? What is its shear modulus? (b) Show that in the limit where K is much larger than G, Poisson's ratio is equal to 1/2. This limit is valid for most liquids (including polymer melts and solutions), which is why they are termed "incompressible". (c) Show that the bulk modulus of an ideal gas is equal to its pressure. Assuming that a water molecule is a sphere of diameter 0.3 nm, estimate the bulk modulus of water assuming that the ideal gas result is valid even for liquid water. Compare your value with the actual bulk modulus of water (2 GPa). (Pr. 2) A simple viscometer consists of two square plates, of side 10 cm, separated by a distance of 1 mm. The lower plate is fixed while the upper one can slide. (a) If the space between the plates is filled with an oil whose viscosity is 1 Pa s, find the force that must be applied to the upper plate to keep it moving at a speed of 0.5 cm s ¹. (b) If the gap between the plates is filled with a rubber with a shear modulus of 5 105 Pa, and the upper plate is shifted from the rest position of the rubber yielding an angular deformation of 0.01 radians, what force must be applied to keep the upper plate in this position? (c) A parallel-plate rheometer consists of a motor able to rotate a disk of radius R parallel to a flat horizontal surface, and a heighth above it. The space between the disk and the surface is filled with a rather viscous liquid of viscosity 7. Show that, for the disk to rotate at a constant angular velocity about its centre, the torque applied to the disk must be proportional to R¹, and find the constant of proportionality. Why can the moment of inertia of the disk be neglected under steady conditions (constant ) ? Hint: integrate the definition of viscosity. (Pr. 3) Consider a phase made of molecules with permanent dipole moment p, in an applied electric field Eexternal. Each dipole has energy W=-p. E= μ E cose, where E is the local electric field at a molecule's position, 0 is the angle between E and p, and is the modulus of the vector p. Assume that E is parallel to the z axis. (a) Explain why the average molecular dipole moment is given by: (p) Spexp : (p₂)2 = Sexp ΦΩ L-E μ(cose) 2, where do do sinode = do d(cose) is the differential of the solid-angle. Why is the solid angle used here? How is the elementary probability dw that the dipole moment points in a given direction, related to the differential d? (b) By integrating by parts the numerator of the above expression for (p), show that its modulus is equal to: [exp(a)+exp(-a) (P₂) = μ(cose): =H*exp(x)dx [exp(a) exp(-a)], where a = µ E /(kB7) a The function L(x) = = μl exp(x)dx ex+e-x ex-e-x -= coth(x) is known as Langevin function. (coth = hyperbolic cotangent) (c) Use the approximate expression ex1 + x (Taylor expansion to linear order), valid for small x, inside both integrals in the expression for expansion (p) given in (a), to show that (p) show that (p) = 37. Call K the effective 3kgT 3kgT polarizability, deff. (d) Explain the meaning of local electric field, and why is it different, in general, from the externally applied field Eexternal. Show that, if the molecules are in the gas phase, the local field is simply equal to E = Eexternal. Show that, for a gas of polar molecules, the polarization of the sample, defined as P = n (p), where n (=N/V) is the number density of molecules, can be written as P = (-1) Eo E, where to is the vacuum permittivity, and give the expression for & for this gas of polar molecules, as a function of the effective polarizability. Substitute the value you found in (c) to obtain the expected temperature dependence of & (e) If instead the phase is a polar liquid, combine the result of point (c) with either the Clausius-Mossotti cavity field, or the "reaction" field (see theory slides), to obtain two different expressions for &. Compare the two expressions with one another, and with the Kirkwood formula in the theory slides.
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