Conductors and Dielectrics
CHAPTER 5 PROBLEMS
3. Fink, D. G., and H. W. Beaty. Standard Handbook for Electrical Engineers. 15th ed. New York: McGraw-Hill, 2006.
4. Maxwell, J. C. A Treatise on Electricity and Magnetism. New York: Cambridge University Press, 2010.
5. Wert, C. A., and R. M. Thomson. Physics of Solids. 2nd ed. New York: McGraw-Hill, 1970. This is an advanced undergraduate-level text that covers metals, semiconductors, and dielectrics.
which this solid cone is constructed has a conductivity of 2×106S/m, use some good approximations to determine the resistance between the two circular faces.
5.9 (a) Using data tabulated in Appendix C, calculate the required diameter for a 2-m-long nichrome wire that will dissipate an average power of 450 W when 120 V rms at 60 Hz is applied to it. (b) Calculate the rms current density in the wire.
5.10 A large brass washer has a 2-cm inside diameter, a 5-cm outside diameter, and is 0.5 cm thick. Its conductivity isσ =1.5×107S/m. The washer is cut in half along a diameter, and a voltage is applied between the two rectangular faces of one part. The resultant electric field in the interior of the half-washer isE=(0.5/ρ)aφV/m in cylindrical coordinates, where the z axis is the axis of the washer. (a) What potential difference exists between the two
rectangular faces? (b) What total current is flowing? (c) What is the resistance between the two faces?
5.11 Two perfectly conducting cylindrical surfaces of lengthare located at ρ =3 andρ=5 cm. The total current passing radially outward through the medium between the cylinders is 3 A dc. (a) Find the voltage and resistance between the cylinders, andE in the region between the cylinders, if a conducting material havingσ =0.05 S/m is present for 3< ρ <5 cm.
(b) Show that integrating the power dissipated per unit volume over the volume gives the total dissipated power.
5.12 Two identical conducting plates, each having area A, are located at z=0 and z=d. The region between plates is filled with a material having z-dependent conductivity,σ(z)=σ0e−z/d, whereσ0is a constant. Voltage V0is applied to the plate at z =d; the plate at z=0 is at zero potential. Find, in terms of the given parameters, (a) the resistance of the material; (b) the total current flowing between plates; (c) the electric field intensityE within the material.
5.13 A hollow cylindrical tube with a rectangular cross section has external dimensions of 0.5 in. by 1 in. and a wall thickness of 0.05 in. Assume that the material is brass, for whichσ =1.5×107S/m. A current of 200 A dc is flowing down the tube. (a) What voltage drop is present across a 1 m length of the tube? (b) Find the voltage drop if the interior of the tube is filled with a conducting material for whichσ =1.5×105S/m.
5.14 A rectangular conducting plate lies in the xy plane, occupying the region 0<x<a,0<y<b.An identical conducting plate is positioned directly above and parallel to the first, at z=d. The region between plates is filled with material having conductivityσ(x)=σ0e−x/a, whereσ0is a constant.
Voltage V0is applied to the plate at z =d; the plate at z=0 is at zero potential. Find, in terms of the given parameters, (a) the electric field intensityE within the material; (b) the total current flowing between plates;
(c) the resistance of the material.
5.15 Let V =10(ρ+1)z2cosφV in free space. (a) Let the equipotential surface V =20 V define a conductor surface. Find the equation of the conductor surface. (b) FindρandE at that point on the conductor surface whereφ= 0.2π and z=1.5. (c) Find|ρS|at that point.
5.16 A coaxial transmission line has inner and outer conductor radii a and b.
Between conductors (a< ρ <b) lies a conductive medium whose
conductivity isσ(ρ)=σ0/ρ, whereσ0is a constant. The inner conductor is charged to potential V0, and the outer conductor is grounded. (a) Assuming dc radial current I per unit length in z, determine the radial current density fieldJ in A/m2. (b) Determine the electric field intensityE in terms of I and other parameters, given or known. (c) By taking an appropriate line integral ofE as found in part (b), find an expression that relates V0to I . (d) Find an expression for the conductance of the line per unit length, G.
5.17 Given the potential field V =100xz/(x2+4) V in free space: (a) FindD at the surface z=0.(b) Show that the z=0 surface is an equipotential surface.
(c) Assume that the z=0 surface is a conductor and find the total charge on that portion of the conductor defined by 0<x<2,−3<y<0.
5.18 Two parallel circular plates of radius a are located at z=0 and z=d. The top plate (z=d) is raised to potential V0; the bottom plate is grounded.
Between the plates is a conducting material having radial-dependent
conductivity,σ(ρ)=σ0ρ, whereσ0is a constant. (a) Find theρ-independent electric field strength,E, between plates. (b) Find the current density, J between plates. (c) Find the total current, I , in the structure. (d) Find the resistance between plates.
5.19 Let V =20x2yz−10z2V in free space. (a) Determine the equations of the equipotential surfaces on which V =0 and 60 V. (b) Assume these are conducting surfaces and find the surface charge density at that point on the V =60 V surface where x =2 and z=1. It is known that 0≤V ≤60 V is the field-containing region. (c) Give the unit vector at this point that is normal to the conducting surface and directed toward the V =0 surface.
5.20 Two point charges of−100π µC are located at (2,−1, 0) and (2, 1, 0). The surface x =0 is a conducting plane. (a) Determine the surface charge density at the origin. (b) DetermineρSat P(0,h,0).
5.21 Let the surface y=0 be a perfect conductor in free space. Two uniform infinite line charges of 30 nC/m each are located at x =0,y=1, and x=0,y=2.(a) Let V =0 at the plane y =0, and find V at P(1,2,0).
(b) FindE at P.
5.22 The line segment x=0,−1≤ y≤1,z=1, carries a linear charge density ρL =π|y|µC/m. Let z=0 be a conducting plane and determine the surface charge density at: (a) (0,0,0); (b) (0,1,0).
5.23 A dipole with p=0.1azµC·m is located at A(1,0,0) in free space, and the x=0 plane is perfectly conducting. (a) Find V at P(2,0,1).(b) Find the equation of the 200 V equipotential surface in rectangular coordinates.
5.24 At a certain temperature, the electron and hole mobilities in intrinsic germanium are given as 0.43 and 0.21 m2/V·s, respectively. If the electron and hole concentrations are both 2.3×1019m−3, find the conductivity at this temperature.
5.25 Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions areρh = −ρe=6200T1.5e−7000/T C/m3. The functional dependence of the mobilities on temperature is given by µh=2.3×105T−2.7 m2/V·s andµe=2.1×105T−2.5 m2/V·s, where the temperature, T , is in degrees Kelvin. Findσ at: (a) 0◦C; (b) 40◦C; (c) 80◦C.
5.26 A semiconductor sample has a rectangular cross section 1.5 by 2.0 mm, and a length of 11.0 mm. The material has electron and hole densities of 1.8×1018 and 3.0×1015m−3, respectively. Ifµe=0.082 m2/V·s andµh=0.0021 m2/ V·s, find the resistance offered between the end faces of the sample.
5.27 Atomic hydrogen contains 5.5×1025atoms/m3at a certain temperature and pressure. When an electric field of 4 kV/m is applied, each dipole formed by the electron and positive nucleus has an effective length of 7.1×10−19m.
(a) Find P. (b) Findr.
5.28 Find the dielectric constant of a material in which the electric flux density is four times the polarization.
5.29 A coaxial conductor has radii a=0.8 mm and b=3 mm and a polystyrene dielectric for whichr =2.56.IfP=(2/ρ)aρnC/m2in the dielectric, find (a)D and E as functions ofρ; (b) Vabandχe. (c) If there are 4×1019 molecules per cubic meter in the dielectric, findp(ρ).
5.30 Consider a composite material made up of two species, having number densities N1and N2molecules/m3, respectively. The two materials are uniformly mixed, yielding a total number density of N =N1+N2. The presence of an electric fieldE induces molecular dipole moments p1andp2
within the individual species, whether mixed or not. Show that the dielectric constant of the composite material is given byr = fr1+(1− f)r2, where
f is the number fraction of species 1 dipoles in the composite, and wherer1 andr2are the dielectric constants that the unmixed species would have if each had number density N .
5.31 The surface x =0 separates two perfect dielectrics. For x>0, letr = r1=3, whiler2=5 where x <0. IfE1 =80ax−60ay−30azV/m, find (a) EN1; (b)ET1; (c)E1; (d) the angleθ1betweenE1and a normal to the surface; (e) DN2; ( f ) DT2; (g)D2; (h)P2; (i) the angleθ2betweenE2and a normal to the surface.
5.32 Two equal but opposite-sign point charges of 3µC are held x meters apart by a spring that provides a repulsive force given by Fsp =12(0.5−x) N.
Without any force of attraction, the spring would be fully extended to 0.5 m.
(a) Determine the charge separation. (b) What is the dipole moment?
5.33 Two perfect dielectrics have relative permittivitiesr1 =2 andr2=8. The planar interface between them is the surface x−y+2z=5. The origin lies in region 1. IfE1=100ax+200ay−50azV/m, findE2.
5.34 Region 1 (x ≥0) is a dielectric withr1=2, while region 2(x <0) has r2=5. LetE1=20ax−10ay+50azV/m. (a) FindD2. (b) Find the energy density in both regions.
5.35 Let the cylindrical surfacesρ=4 cm andρ=9 cm enclose two wedges of perfect dielectrics,r1=2 for 0< φ < π/2 andr2=5 forπ/2< φ <2π.
IfE1=(2000/ρ)aρV/m, find (a)E2; (b) the total electrostatic energy stored in a 1 m length of each region.