Page 159 - Engineering Electromagnetics, 8th Edition
P. 159
CHAPTER 5 Conductors and Dielectrics 141
5.23 A dipole with p = 0.1a z µC · mis 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
2
germanium are given as 0.43 and 0.21 m /V · s, respectively. If the electron
and hole concentrations are both 2.3 × 10 m , find the conductivity at this
19
−3
temperature.
5.25 Electron and hole concentrations increase with temperature. For pure
3
silicon, suitable expressions are ρ h =−ρ e = 6200T 1.5 −7000/T C/m .
e
The functional dependence of the mobilities on temperature is given by
2
5
5
2
µ h = 2.3 × 10 T −2.7 m /V · s and µ e = 2.1 × 10 T −2.5 m /V · s, where the
temperature, T ,isindegrees Kelvin. Find σ 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 × 10 18
2
−3
and 3.0 × 10 15 m , respectively. If µ e = 0.082 m /V · s and µ h = 0.0021
2
m /V · s, find the resistance offered between the end faces of the sample.
3
5.27 Atomic hydrogen contains 5.5 × 10 25 atoms/m at 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 −19 m.
(a) Find P.(b) Find r .
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 = 0.8mmand b = 3mm and a polystyrene
2
dielectric for which r = 2.56. If P = (2/ρ)a ρ nC/m in the dielectric, find
(a) D and E as functions of ρ;(b) V ab and χ e .(c)If there are 4 × 10 19
molecules per cubic meter in the dielectric, find p(ρ).
5.30 Consider a composite material made up of two species, having number
3
densities N 1 and N 2 molecules/m , respectively. The two materials are
uniformly mixed, yielding a total number density of N = N 1 + N 2 . The
presence of an electric field E induces molecular dipole moments p 1 and p 2
within the individual species, whether mixed or not. Show that the dielectric
constant of the composite material is given by r = f r1 + (1 − f ) r2 , where
f is the number fraction of species 1 dipoles in the composite, and where r1
and r2 are 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, let r =
r1 = 3, while r2 = 5 where x < 0. If E 1 = 80a x − 60a y − 30a z V/m, find
(a) E N1 ;(b) E T 1 ;(c) E 1 ;(d) the angle θ 1 between E 1 and a normal to the
surface; (e) D N2 ;( f ) D T 2 ;(g) D 2 ;(h) P 2 ;(i) the angle θ 2 between E 2 and a
normal to the surface.