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

Equation3
Review exercises 26 865
curlE=∇×E=− ∂B
∂t
where E is the electric field strength. This equation is a statement of Faraday’s law. A
time-varying magnetic field produces a space-varying electric field.
Equation4
curlH=∇×H=J+ ∂D
∂t
where H is the magnetic field strength and J is the free current density. This equation
statesthatatime-varying electric field gives risetoaspace-varying magnetic field.
The derivation of these equations is beyond the scope of this text but can be found in
manybooksonelectromagnetism.Thepoweroftheseequationsliesintheirgenerality.
Thebrevitywithwhichthemainlawsofelectromagnetismcanbeexpressedisatribute
tothe utility ofvector calculus.
REVIEWEXERCISES26
1 Find ∇φif
(a) φ = 3xyz
(b) φ =x 2 yz+xy 2 z+xyz 2
(c) φ =xy 2 z 2 +x 2 yz 2 +x 2 y 2 z
2 Ifφ=1/ √ x 2 +y 2 +z 2 ,showthat
∂ 2 φ
∂x 2 + ∂2 φ
∂y 2 + ∂2 φ
∂z 2 = 0
3 Functions satisfyingLaplace’sequationare called
harmonicfunctions.Showthat the following
functionsare harmonic:
(a) z =x 4 −6x 2 y 2 +y 4
(b) z = 4x 3 y −4xy 3
4 IfA=xi+yj+zkand
B = cosxi −sinxj,find
(a) A×B
(b) ∇·(A×B)
(c) ∇×A
(d) ∇×B
Verifythat
∇·(A×B)=B·(∇×A)−A·(∇×B).
5 Forarbitrary differentiablescalar fields φ and ψ show
that∇(φψ) = ψ∇φ +φ∇ψ.
6 Ifψ=x 2 yanda=xi+yj+zk,find∇ψ,
∇ ×a,∇ × (ψa).Showthat
∇×(ψa)=ψ∇×a+(∇ψ)×a.
7 Ascalarfield φ isafunctionofx,zandt only.Vectors
E and Hare defined by
( )
E = 1 ∂φ
ε ∂z i − ∂φ
∂x k H = − ∂φ
∂t j
where ε isaconstant.
(a) Show that ∇ ·E = 0.
(b) Show that ∇ ·H = 0.
Given that ∇ ×E = −µ ∂H ,where µ isaconstant,
∂t
show that φ satisfiesthe partial differential equation
∂ 2 φ
∂x 2 + ∂2 φ
∂z 2 = φ
µε∂2 ∂t 2
8 Ifv = (2x 2 y+3x 5 )i+e xy j+xyzkfind ∂v
∂x and ∂2 v
∂x 2.
9 An electrostatic potential isgiven byV = 5xyz.Find
(a) the associatedelectric field E,
(b) |E| atthe point (1,1,1).
10 An electrostatic field is given by
E = 3(x+y)i+2xyj.Findthedirectionofthisfieldat
(a) the point (2,2)
(b) the point (3,4).
11 Find the curlofA =yi +2xyj +3zk.
12 Find the curlofthe vector field
F = (x 2 −y)i + (xy −4y 2 )j.
13 If φ = 5e x+2y coszfind ∇φ atthepoint (0,0,0).