Gauss

1
Gauss Divergence òòD.ds òòò r dv Ñ.D
r(r)

• By
• Engr. Mian Shahzad Iqbal
• Lecturer, Telecom Department
• University of Engineering and Technology
• Taxila

2
GaussDivergence why, oh why??

• òòD.ds òòòr(r)dv is clearly a useful means of
  calculating D and E from a macroscopic
  (i.e.sizeable!) distribution of charge
• It relates charge density in a volume of space to
  the field that it creates
• We will want a relationship between charge
  density at a point r(r) and the fields E(r) and
  D(r) that it creates at that point
• (honest, we will!)

Cut to the chase
3
Proof (non-examinable)
Surface for Integration
D,E
4
Proof (non-examinable)
Argh!! I am shrinking!!!
5
Proof (non-examinable)
6
Proof (non-examinable)
D,E
7
Proof (non-examinable)
dy
dx
D,E
dz
8
Proof (non-examinable)
dy
dx
D,E
dz
9
Proof (non-examinable)
dy
dx
D,E
dz
10
Proof (non-examinable)
dy
SKIP MATHS
dx
D,E
dz
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Now the maths

• Assume that, for example,D (Dx,Dy,Dz) over the
  entire left hand face, the back face and the
  bottom face all the faces that meet at the
  origin
• D is different on the other 3 faces
• Front face D (DxDx,Dy,Dz)
• Right face D (Dx,DyDy,Dz)
• Top face D (Dx,Dy,DzDz)

12
Now the maths

• Left face D (Dx,Dy,Dz)
• ds (0, -dxdz, 0)
• Right face D (Dx,DyDy,Dz)
• ds (0, dxdz, 0)
• Bottom face D (Dx,Dy,Dz)
• ds (0, 0, -dxdy)
• Top face D (Dx,Dy,DzDz)
• ds (0, 0, dxdy)
• Back face D (Dx,Dy,Dz)
• ds (-dydz, 0, 0)
• Front face D (DxDx,Dy,Dz)
• ds (dydz, 0, 0)

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Now the maths

• òòD.ds (Dx,Dy,Dz).(0, -dxdz, 0)
  (Dx,Dy,Dz).(0, 0, -dxdy) (Dx,Dy,Dz).(-dydz,
  0, 0) (DxDx,Dy,Dz).(dydz, 0, 0)
  (Dx,DyDy,Dz).(0, dxdz, 0) (Dx,Dy,DzDz).(0,
  0, dxdy)

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Now the maths

• òòD.ds -Dydxdz Dzdxdy Dxdydz
  (DxDx)dydz (DyDy)dxdz (DzDz)dxdy
• -Dydxdz Dzdxdy Dxdydz (Dx
  dxDx/x) dydz (Dy dxDy/y)dxdz (Dz
  dxDz/z)dxdy

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Now the maths

• òòD.ds (dxDx/x) dydz (dxDy/y)dxdz
  (dxDz/z)dxdy
• (Dx/x) dxdydz (Dy/y)dxdydz
  (Dz/z)dxdydz
• (Dx/x) dv (Dy/y)dv (Dz/z)dv
• òòD.ds (/x, /y, /z). (Dx,Dy, Dz) dv

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Now the maths

• òòD.ds (/x, /y, /z) .(Dx,Dy,Dz)dv
• òòD.ds Ñ.D dv charge enclosed
• Ñ.Ddv òòò r dv
• for an infinitesimally small volume dv, r is
  constant
• Ñ.Ddv r òòò dv rdv
• Ñ.D r(r)
• This is the differential, or at-a-point version
  of Gausss law, often called the Divergence
  Theorem
• (/x, /y, /z) Ñ is the Divergence Operator

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Gausss Law/Divergence Theorem

• òòD.ds òòò r(r) dv
• Ñ.D r(r)
• These are equivalent
• Ñ (/x, /y, /z)