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

164 Chapter 4 Coordinate systems
y
u = 45°
y
3
3
x
O
45°
Figure4.15
When θ isfixed andrvaries, a straight
line from the origin is traced out.
x
Figure4.16
Whenrisfixedand θ varies,acircleis
sweptout.
Q
y
3
O
Figure4.17
As θ varies from0 ◦ to 180 ◦ a
semicircle is traced out.
P
x
y
S
R
O
P
Q
Figure4.18
Surface forExample4.8.
x
Example4.8 Describethe surfacedefined by 1 r2,0 ◦ θ 90 ◦ .
Solution Herervariesfrom1to2and θ variesfrom0 ◦ to90 ◦ .Figure4.18illustratesthesurface
so formed.
AtP,r=1,θ=0 ◦ ;atQ,r=2,θ=0 ◦ ;atR,r=1,θ=90 ◦ ;atS,r=2,θ=90 ◦ .
We have seen some simple polar curves in Figures 4.16 and 4.17. In general a polar
curve isgiven by the equationr = f (θ),where the radiusrvaries with the angle θ.
Engineeringapplication4.3
Two-dimensionalantennaradiationpattern
Polarcurves areoftenusedtodepictradiation patternsfromantennas.Itisoftenthe
casethattheelectricfieldstrengthatafixeddistancefromanantennasuchasshown
inFigure 4.19 dependsupon the angle θ.
A typical expression for field strength at a particular angle θ could be
∣ cos( π cosθ)
2 ∣
∣.
sinθ