Page 190 - Engineering Electromagnetics, 8th Edition

172 ENGINEERING ELECTROMAGNETICS

Because the total charge is a function of the potential difference, we have to be careful
in defining a capacitance. Thinking in “circuit” terms for a moment,
dQ dV 0
I = = C
dt dt
and thus
dQ
C =
dV 0
By differentiating Eq. (48), we therefore have the capacitance
ρ ν0
S
C = S = (49)
2πV 0 2πa
The first form of Eq. (49) shows that the capacitance varies inversely as the square
root of the voltage. That is, a higher voltage causes a greater separation of the charge
layers and a smaller capacitance. The second form is interesting in that it indicates
that we may think of the junction as a parallel-plate capacitor with a “plate” separation
of 2πa.In view of the dimensions of the region in which the charge is concentrated,
this is a logical result.
Poisson’s equation enters into any problem involving volume charge density.
Besides semiconductor diode and transistor models, we find that vacuum tubes, mag-
netohydrodynamic energy conversion, and ion propulsion require its use in construct-
ing satisfactory theories.

D6.7. In the neighborhood of a certain semiconductor junction, the volume
3
6
6
charge density is given by ρ ν = 750 sech 10 πx tanh 10 πx C/m . The di-
electric constant of the semiconductor material is 10 and the junction area is
2
2 × 10 −7 m . Find: (a) V 0 ;(b) C;(c) E at the junction.
Ans. 2.70 V; 8.85 pF; 2.70 MV/m

√
7
3
D6.8. Given the volume charge density ρ ν =−2 × 10 0 x C/m in free
space, let V = 0at x = 0 and let V = 2Vat x = 2.5 mm. At x = 1 mm, find:
(a) V ;(b) E x .
Ans. 0.302 V; −555 V/m

REFERENCES
1. Matsch, L. W. Capacitors, Magnetic Circuits, and Transformers. Englewood Cliffs, NJ:
Prentice-Hall, 1964. Many of the practical aspects of capacitors are discussed in
Chapter 2.
2. Ramo, S., J. R. Whinnery, and T. Van Duzer. Fields and Waves in Communication
Electronics. 3rd ed. New York: John Wiley and Sons, 1994. This classic text is primarily
directed toward beginning graduate students, but it may be read by anyone familiar with
basic electromagnetics concepts. Curvilinear square plotting is described on pp. 50–52. A
more advanced discussion of methods of solving Laplace’s equation is given in Chapter 7.