Understanding Engineering Mathematics

whereVis the capacitor voltage. Such equations will be solved in Chapter 15, but for

now note that in doing so it is necessary to solve the associated quadraticauxiliary

equation

λ^2 +

R

L

λ+

1

LC

= 0

for the quantityλ, Greek ‘lambda’. Write down the solutions to this quadratic for general

values ofR,L,C. Solve the quadratic for the following values of these parameters:

(i) R= 10 (,C= 1 / 9 F,andL= 1 H (ii) R= 6 (,C= 1 / 9 F,andL= 1 H

(iii) R= 2 (,C= 1 / 9 F,andL= 1 H

In the case of (iii) you will obtaincomplex roots– such numbers are studied in

Chapter 12, but the next application gives you a preview.

2.Acomplex numberis one of the formz=a+jbwhereaandbare ordinary, real
numbers andjis a symbol for the ‘imaginary number’

√

−1. All you need to know

is thatj^2 =−1 and that otherwise the normal rules of algebra apply. Investigate the

‘algebra’ of complex numbers, that is the rules for adding, subtracting, multiplying and

dividing complex numbers. You will see this in Chapter 12, and as a taster you can try

to derive the results:

(i) (a+jb)(c+jd)=ac−bd+j(bc+ad)(ii)

a+jb

c+jd

=

ac+bd

c^2 +d^2

+j

bc−ad

c^2 +d^2

In (ii) multiply top and bottom of the left-hand side byc−jd.

3.Emperor Yang Sun (ninth century CE) had to choose between two equally qualified
clerks for a promotion, He set them the following problem to decide between them.
A man walking in the woods heard thieves arguing over the division of rolls of cloth
which they had stolen. They said that if each took six rolls there would be five left
over; but if each took seven rolls, they would be eight short. How many thieves were
there, and how many rolls of cloth?
The candidates performed the calculation in the ancient Chinese way, by using
bamboo rods on the tiled floor. This technique led to the early development of adding
and subtracting rows and columns of arrays to solve linear equations. Would you get
the job?

4.An open box is made from a 12 cm square card by cutting squares of equal size from
the four corners and folding the edges up. Calculate the size of the squares to be cut
to yield the box with the largest volume.

5.Vapour deposition methods are often used for thin film layers in the fabrication of
semiconductor devices. Such methods involve exposing a semiconductor wafer to a
mixture of gases in a heated vacuum chamber. The chemical composition of the surface
layer, and the deposition time depend on the relative concentrations of the vapours in the
chamber. If three gasesA,B,Care to be combined to form a number of overlayers of
new moleculesABCthen the time dependence of the deposition of the new molecules