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

26.6 Combining the operators grad, div and curl 863
Engineeringapplication26.3
Poisson’sequation
Recall from Engineering application 26.1 that engineers sometimes need to solve
complex electrostatic problems when designing electrical equipment. For example,
this need arises when calculating the electrical field strength in a high-voltage electrical
distribution station to ensure there is no danger of electrical discharge across
the air gap between components that have different voltages. Poisson’s equation is
an equation thatcan beusefulwhen carrying outthis work.Consider the following.
If E is an electric field andV an electrostatic potential, then the two fields are
related by
E=−∇V
FromExample26.2weknowGauss’slaw: ∇ · E = ρ ε 0
.Combiningthesetwoequations
we can write
that is
∇·(−∇V)= ρ ε 0
∇ 2 V = − ρ ε 0
ThispartialdifferentialequationisknownasPoisson’sequationandbysolvingitwe
coulddeterminetheelectrostaticpotentialinaregionoccupiedbycharges.Notethat
inacharge-freeregion, ρ = 0andPoisson’sequationreducestoLaplace’sequation
∇ 2 V = 0.
EXERCISES26.6
1 Ascalar field φ is given by
φ = 3x +y−y 2 z 2 .Show that φ satisfies
∇ 2 φ = −2(y 2 +z 2 ).
2 If φ =2x 2 y−xz 3 showthat ∇ 2 φ =4y−6xz.
3 Ifv =xyi −yzj + (y +2z)k findcurl(curl(v)).
4 If φ =xyzandv =3x 2 i+2y 3 j+xykfind ∇φ, ∇·v,
and ∇ · (φv).Show that
∇·(φv)=(∇φ)·v+φ∇·v.
5 Verifythat φ =x 2 y +y 2 z +z 2 x satisfies
∇·(∇φ)=2(x+y+z).
6 IfAis an arbitrarydifferentiablevectorfield show
thatthe divergence ofthe curlofAis always 0.
7 Expresseachofthe following in operator notation
using‘∇’, ‘∇ ·’ and‘∇×’:
(a) grad (divF)
(b) curl(grad φ)
(c) curl(curlF)
(d) div (curlF)
(e) div (grad φ)
8 Scalar fields φ 1 and φ 2 are givenby
φ 1 =2xy+y 2 z φ 2 =x 2 z
(a) Find ∇φ 1 .
(b) Find ∇φ 2 .
(c) State φ 1 φ 2 .
(d) Find ∇(φ 1 φ 2 ).
(e) Find φ 1 ∇φ 2 +φ 2 ∇φ 1 .
(f) Whatdo you conclude from(d)and(e)?