082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

10 Chapter 1 Review of algebraic techniques
densityhasdecayedtoapproximately37%ofthatattheedge.Skindepthisimportant
because it affects the resistance of wires and other conductors: the smaller the skin
depth, the higher the effective resistance and the greater the loss due toheating.
At low frequencies, such as those found in the domestic mains supply, the skin
depthissolargethatoftenitcan beneglected; however, invery large-diameter conductors
and smaller conductors at microwave frequencies it becomes important and
has tobe taken into account.
The skin depth, δ, isgiven by
( ) 2 1/2
δ =
ωµσ
where µisamaterialconstantknownasthepermeabilityoftheconductor, ω isthe
angular frequency of the signal and σ isthe conductivity of the conductor.
d
Figure1.1
Cross-sectionofaradialconductor
illustratingaskindepth δ.
EXERCISES1.2
1 Evaluate
(a) 2 3 (b) 3 2 5 13
(c)
5 12
19 −11
(d)
19 −13 (e) (2 1/4 ) 8 (f) (−4) −2
(g) 4 −1/2 (h) (9 1/3 ) 3/2 √ √
(i) 32 2
√
(j) 0.01 (k) 81 3/4
2 Useascientificcalculator to evaluate
(a) 10 1.2 (b) 6 −0.7 (c) 6 2.5
(d) (3 −1 4 2 ) 0.8
3 Expresseachofthe followingexpressions usinga
single positiveindex:
(a) x 4 x 7 (b) x 2 (−x)
(c)
x 2
x
(d)
x −2
x −1
(e) (x −2 ) 4 (f) (x −2.5 x −3.5 ) 2
4 Simplify asmuch aspossible
(a)
(c)
x 1/2
x 1/3 (b) (16x 4 ) 0.25
( ) 1/3
27 2xy 2
y 3 (d)
(2xy) 2
(e) √ a 2 b 6 c 4 (f) (64t 3 ) 2/3