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# Gauss Divergence D.ds r dv .D rr - PowerPoint PPT Presentation

Gauss ® Divergence òò D . ds = òòò r dv ® Ñ . D = r ( r ). . Alan Murray. Gauss ® Divergence … why, oh why??. òò D . ds = òòòr ( r )dv is clearly a useful means of calculating D and E from a macroscopic (i.e.sizeable!) distribution of charge

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

Alan Murray

Gauss®Divergence …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

Alan Murray – University of Edinburgh

y

x

Proof (non-examinable)

Surface for Integration

D,E

Alan Murray – University of Edinburgh

Surface for Integration

D,E

y

x

Proof (non-examinable)

Cut to the chase

Argh!! I am shrinking!!!

Alan Murray – University of Edinburgh

z

dx

D,E

dz

y

D =D(0,0,0)

x

Proof (non-examinable)

Alan Murray – University of Edinburgh

z

dx

dz

y

D =D(0,0,0)

x

Proof (non-examinable)

D,E

Alan Murray – University of Edinburgh

y

D =D(0,0,0)

x

Proof (non-examinable)

dy

dx

D,E

dz

Alan Murray – University of Edinburgh

y

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

D =D(0,0,0)

x

Proof (non-examinable)

dy

dx

D,E

dz

Alan Murray – University of Edinburgh

y

D =D(0,0,0)

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

x

Proof (non-examinable)

dy

dx

D,E

dz

Alan Murray – University of Edinburgh

y

D =D(0,0,0)

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

x

Proof (non-examinable)

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

Now the maths …

• 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

Now the maths …

• 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

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.

So what does Divergence “mean”

E

Alan Murray – University of Edinburgh

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.

So what does Divergence “mean”

Alan Murray – University of Edinburgh