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

900 Chapter 27 Line integrals and multiple integrals
Equation1
∮ ∫
D·dS=
S
V
ρdV
whereD =electricfluxdensity,and ρ iselectricchargedensity.Notethatther.h.s.isthe
totalchargeenclosedbythevolumeV.Thisequationstatesthatthetotalfluxcrossinga
closedsurfaceSwhichenclosesavolumeV isequaltothetotalchargeenclosedbythe
surface.ThisisalsotheintegralformofGauss’slaw.(SeeEngineeringapplication27.7
which isobtained by lettingD = ε 0
E.)
Equation2
∮
B·dS=0
S
where B = magnetic flux density. This law states that the net magnetic flux crossing
any closed surface is zero. So whilst charges can be thought of as sources or sinks of
electric flux,there areno equivalent sources or sinks ofmagnetic flux.
Equation3
∮
CE·dr=− ∂ ∂t
∫
S
B·dS
Note that in the theory of electrostatics, differentiating
∮
partially with respect to time
always yields zero, and so this equation reduces to E·dr = 0. This is a condition
discussed inSection 27.5 and confirms thatan electrostatic field isaconservative field.
Equation4
∮
CH·dr= ∂ ∂t
∫
S
∫
D·dS+ J·dS
S
This isthe integral formof Ampère’s circuitallaw. The closed curveC bounds an open
surfaceS. Acurrentwith density ∂D +Jflows through the surfaceS.
∂t
C
EXERCISES27.9
1 Startingwith Maxwell’sequation ∇ ·D = ρ,by
integrating both sides over anarbitraryvolumeV and
usingthe divergence theoremobtain Equation1
above.
2 Startingwith Maxwell’sequation ∇ ·B = 0, by
integrating both sides over anarbitraryvolumeV and
usingthe divergence theoremobtain Equation2
above.
3 Startingwith the equation ∇ ×E = 0forstatic
electric fields, use Stokes’theoremto showthat
∮
E·dr=0.
4 Startingfrom ∇ ×H = ∂D +J andusingStokes’
∂t
theorem obtain Equation 4.
( )
∂
5 Inmagnetostatics
∂t = 0 ,Ampère’slaw (orthe
magnetic circuit law) states:
∮
H·dr=I
C
whereI is the currentenclosed bythe closedpathC.
Obtain
∫
thislaw from Equation 4 andusing
S J·dS=I.