Page 254 - Engineering Electromagnetics, 8th Edition

236 ENGINEERING ELECTROMAGNETICS

D8.3. The semiconductor sample shown in Figure 8.1 is n-type silicon, hav-
ing a rectangular cross section of 0.9 mm by 1.1 cm and a length of 1.3 cm.
2
Assume the electron and hole mobilities are 0.13 and 0.03 m /V · s, respectively,
at the operating temperature. Let B = 0.07 T and the electric field intensity in
the direction of the current flow be 800 V/m. Find the magnitude of: (a) the
voltage across the sample length; (b) the drift velocity; (c) the transverse force
per coulomb of moving charge caused by B;(d) the transverse electric field
intensity; (e) the Hall voltage.

Ans. 10.40 V; 104.0 m/s; 7.28 N/C; 7.28 V/m; 80.1 mV

8.3 FORCE BETWEEN DIFFERENTIAL
CURRENT ELEMENTS
The concept of the magnetic field was introduced to break into two parts the problem
of finding the interaction of one current distribution on a second current distribution.
It is possible to express the force on one current element directly in terms of a
second current element without finding the magnetic field. Because we claimed that
the magnetic-field concept simplifies our work, it then behooves us to show that
avoidance of this intermediate step leads to more complicated expressions.
The magnetic field at point 2 due to a current element at point 1 was found to be

I 1 dL 1 × a R12
dH 2 =
2
4πR 12
Now, the differential force on a differential current element is

dF = IdL × B

and we apply this to our problem by letting B be dB 2 (the differential flux density at
point2causedbycurrentelement1),byidentifying IdLas I 2 dL 2 ,andbysymbolizing
the differential amount of our differential force on element 2 as d(dF 2 ):

d(dF 2 ) = I 2 dL 2 × dB 2

Because dB 2 = µ 0 dH 2 ,we obtain the force between two differential current
elements,

I 1 I 2
d(dF 2 ) = µ 0 dL 2 × (dL 1 × a R12 ) (13)
2
4πR 12
EXAMPLE 8.2
As an example that illustrates the use (and misuse) of these results, consider the
two differential current elements shown in Figure 8.3. We seek the differential force
on dL 2 .