Gauss ® Divergence òò D . ds = òòò r dv ® Ñ . D = r ( r )

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Gauss ® Divergence òò D . ds = òòò r dv ® Ñ . D = r ( r )

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Gauss ® Divergence òò D . ds = òòò r dv ® Ñ . D = r ( r )

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Gauss® DivergenceòòD.ds = òòò r dv®Ñ.D = r(r)

Alan Murray

- 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

Alan Murray – University of Edinburgh

z

y

x

Surface for Integration

D,E

Alan Murray – University of Edinburgh

z

Surface for Integration

D,E

y

x

Cut to the chase

Argh!! I am shrinking!!!

Alan Murray – University of Edinburgh

dy

z

dx

D,E

dz

y

D =D(0,0,0)

x

Alan Murray – University of Edinburgh

dy

z

dx

dz

y

D =D(0,0,0)

x

D,E

Alan Murray – University of Edinburgh

z

y

D =D(0,0,0)

x

dy

dx

D,E

dz

Alan Murray – University of Edinburgh

z

y

D =D(0,0,0+dz)

D =D(0,0,0)

x

dy

dx

D,E

dz

Alan Murray – University of Edinburgh

z

y

D =D(0,0,0)

D =D(0,0+dy,0)

x

dy

dx

D,E

dz

Alan Murray – University of Edinburgh

z

y

D =D(0,0,0)

D =D(0+dx,0,0)

x

dy

SKIP MATHS

dx

D,E

dz

Alan Murray – University of Edinburgh

- 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 = (Dx+Dx,Dy,Dz)
- Right face : D = (Dx,Dy+Dy,Dz)
- Top face : D = (Dx,Dy,Dz+Dz)

Alan Murray – University of Edinburgh

- Left face : D = (Dx,Dy,Dz)
- ds = (0, -dxdz, 0)

- Right face : D = (Dx,Dy+Dy,Dz)
- ds = (0, +dxdz, 0)

- Bottom face : D = (Dx,Dy,Dz)
- ds = (0, 0, -dxdy)

- Top face : D = (Dx,Dy,Dz+Dz)
- ds = (0, 0, +dxdy)

- Back face : D = (Dx,Dy,Dz)
- ds = (-dydz, 0, 0)

- Front face : D = (Dx+Dx,Dy,Dz)
- ds = (+dydz, 0, 0)

Alan Murray – University of Edinburgh

- òòD.ds= (Dx,Dy,Dz).(0, -dxdz, 0)+ (Dx,Dy,Dz).(0, 0, -dxdy)+ (Dx,Dy,Dz).(-dydz, 0, 0)+ (Dx+Dx,Dy,Dz).(dydz, 0, 0)+ (Dx,Dy+Dy,Dz).(0, dxdz, 0)+(Dx,Dy,Dz+Dz).(0, 0, dxdy)

Alan Murray – University of Edinburgh

These cancel

These cancel

These cancel

- òòD.ds= -Dydxdz – Dzdxdy – Dxdydz+(Dx+Dx)dydz+(Dy+Dy)dxdz+(Dz+Dz)dxdy
- = -Dydxdz – Dzdxdy – Dxdydz+(Dx+ dx¶Dx/¶x) dydz+(Dy+ dx¶Dy/¶y)dxdz+(Dz+ dx¶Dz/¶z)dxdy

Alan Murray – University of Edinburgh

dxdydz = dv

- òòD.ds= (dx¶Dx/¶x) dydz+ (dx¶Dy/¶y)dxdz+ (dx¶Dz/¶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

Alan Murray – University of Edinburgh

- òòD.ds= (¶/¶x, ¶/¶y, ¶/¶z) .(Dx,Dy,Dz)dv
- òòD.ds= Ñ.Ddv = charge enclosed
- Ñ.Ddv = òòò r dv
- for an infinitesimally small volume dv, r is constant

- Ñ.Ddv = ròòò dv = rdv
- Ñ.D = r(r)
- This is the differential, or “at-a-point” version of Gauss’s law, often called the Divergence Theorem
- (¶/¶x, ¶/¶y, ¶/¶z) = Ñ is the Divergence Operator

Alan Murray – University of Edinburgh

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

Alan Murray – University of Edinburgh

Not the same as “to diverge”

Q

The electric field here diverges, but its DIVERGENCE is zero at this point, as there is no charge present at the point, ρ=0.

The electric field here diverges, and its DIVERGENCE is non-zero at this point, as there is charge present at the point, ρ>0.

E

Alan Murray – University of Edinburgh

Not the same as “to diverge”

E

The electric field does not diverge and its DIVERGENCE is zero at this point, as there is no charge present at the point, ρ=0.

The electric field here does not diverge, and its DIVERGENCE is non-zero at this point, as there is charge present at the point, ρ>0.

Alan Murray – University of Edinburgh