Mathematical Methods for Physics and Engineering : A Comprehensive Guide

7.10 EXERCISES

V 0 cosωt

V 1

V 2

V 3

V 4

R 1 =50Ω

R 2

I 1

I 2

I 3

L

C=10μF

Figure 7.18 An oscillatory electric circuit. The power supply has angular

frequencyω=2πf= 400πs−^1.

7.25 Define a set of (non-orthogonal) base vectorsa=j+k,b=i+kandc=i+j.

(a) Establish their reciprocal vectors and hence express the vectorsp=3i− 2 j+k,

q=i+4jandr=− 2 i+j+kin terms of the base vectorsa,bandc.

(b) Verify that the scalar productp·qhas the same value,−5, when evaluated

using either set of components.

7.26 Systems that can be modelled as damped harmonic oscillators are widespread;
pendulum clocks, car shock absorbers, tuning circuits in television sets and radios,
and collective electron motions in plasmas and metals are just a few examples.
In all these cases, one or more variables describing the system obey(s) an
equation of the form
x ̈+2γ ̇x+ω 02 x=Pcosωt,
where ̇x=dx/dt, etc. and the inclusion of the factor 2 is conventional. In the
steady state (i.e. after the effects of any initial displacement or velocity have been
damped out) the solution of the equation takes the form
x(t)=Acos(ωt+φ).
By expressing each term in the formBcos(ωt+), and representing it by a vector
of magnitudeBmaking an anglewith thex-axis, draw a closed vector diagram,
att= 0, say, that is equivalent to the equation.
(a) Convince yourself that whatever the value ofω(>0)φmust be negative
(−π<φ≤0) and that

φ=tan−^1

(

− 2 γω

ω^20 −ω^2

)

.

(b) Obtain an expression forAin terms ofP,ω 0 andω.

7.27 According to alternating current theory, the currents and potential differences in
the components of the circuit shown in figure 7.18 are determined by Kirchhoff’s
laws and the relationships

I 1 =

V 1

R 1

,I 2 =

V 2

R 2

,I 3 =iωCV 3 ,V 4 =iωLI 2.

The factori=

√

−1 in the expression forI 3 indicates that the phase ofI 3 is 90◦

ahead ofV 3. Similarly the phase ofV 4 is 90◦ahead ofI 2.

Measurement shows thatV 3 has an amplitude of 0. 661 V 0 and a phase of

+13. 4 ◦relative to that of the power supply. TakingV 0 =1V,andusingaseries