Download 3318 Homework 8

ECE 3318
Applied Electricity and Magnetism
Spring 2017
Homework #8
Date assigned: March 28, 2017
Date due: April 4, 2017
Do Probs. 1-8. (You are welcome to do the other problems as well for extra practice, but only
these should be turned in.)
1) Under normal conditions, can the relative permittivity of a material ever be less than 1.0? If
not, give a convincing explanation of why. Draw a picture showing how the dipoles line up
with the field, to help in your explanation.
2) A water molecule is shown below. Note that the angle between the two hydrogen ions is
about 104.45 degrees, and the distance between the centers of the hydrogen and oxygen
atoms is about 0.9584 Angstroms (one Angstrom is 10-10 meters), or 95.84 [pm]. Calculate
the magnitude of the dipole moment p for a single water molecule.
Each hydrogen atom has donated one electron to the oxygen atom, and hence each hydrogen
atom now has a positive charge that is -qe, where qe is the charge of a single electron. The
oxygen atom has a net charge of 2qe. Note that the total vector dipole moment of the
molecule p is the vector sum of two vector dipole moments, each one coming from one of the
hydrogen atoms and one of the donated electrons on the oxygen atom. This vector dipole
moment can also be thought of as the dipole moment coming from a single dipole that has a
charge of 2qe that is located where the oxygen atom is, and a charge of -2qe that is located at
the midpoint of the line segment that connects the two hydrogen atoms.
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3) Water has a relative permittivity of approximately 81. If an electric field E  xˆ [V/m] is
inside the water, what is the vector P (dipole polarization per unit volume)?
4) As a continuation of the previous problem, assume that the density of water is 1000 [kg/m3]
and that there are 3.343021  1025 molecules per kg of water. (This comes from assuming an
atomic weight of 18.01528, so that one mole of water equals 18.0152 grams. A mole contains
a number of molecules equal to Avogadro’s number of 6.02252  1023.) Determine the
average dipole moment pxave of each water molecule when
E  xˆ 1.0 [V/m].
5) As a continuation of the previous problem (and Prob. 2), calculate the average dipole
displacement angle  (the angle between the x axis and the angle of the dipole moment
inside the water) when the electric field inside the water is
E  xˆ 1.0 [V/m].
6) A parallel-plate capacitor has a plate separation of h [m]. The bottom plate is at zero [V]
while the top plate is at V0 [V]. Assume that x is measured vertically down from the top plate.
Also assume that between the plates is a dielectric with a relative permittivity of r. Calculate
the vectors D and E inside the capacitor.
7) An infinite uniform line charge density l0 [C/m] is along the z axis. Surrounding this line
charge is a cylindrical shell of dielectric, having an inner radius a and an outer radius b. The
relative permittivity of the dielectric is r. Find the electric field vector in all regions.
8) A point charge is in an infinite medium of dielectric material having a relative permittivity r.
Find the electric field vector and the potential function at any point in space, assuming that
the potential is zero volts at infinity.
9) Assume that an arbitrary charge density exits inside of an infinite medium of dielectric
material having a relative permittivity r. Give a convincing explanation of why the formulas
for the electric field and the potential function must be the same as if the charge density was
in free space, provided 0 is replaced by  = 0r in the free-space formulas. (Hint: Use
superposition, and note how the field of the single point charge is affected by the dielectric
medium, according to the problem above.)
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