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


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

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Gauss divergence d ds r dv d r r l.jpg

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

Alan Murray


Gauss divergence why oh why l.jpg

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


Proof non examinable l.jpg

z

y

x

Proof (non-examinable)

Surface for Integration

D,E

Alan Murray – University of Edinburgh


Proof non examinable4 l.jpg

z

Surface for Integration

D,E

y

x

Proof (non-examinable)

Cut to the chase

Argh!! I am shrinking!!!

Alan Murray – University of Edinburgh


Proof non examinable5 l.jpg

dy

z

dx

D,E

dz

y

D =D(0,0,0)

x

Proof (non-examinable)

Alan Murray – University of Edinburgh


Proof non examinable6 l.jpg

dy

z

dx

dz

y

D =D(0,0,0)

x

Proof (non-examinable)

D,E

Alan Murray – University of Edinburgh


Proof non examinable7 l.jpg

z

y

D =D(0,0,0)

x

Proof (non-examinable)

dy

dx

D,E

dz

Alan Murray – University of Edinburgh


Proof non examinable8 l.jpg

z

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


Proof non examinable9 l.jpg

z

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


Proof non examinable10 l.jpg

z

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


Now the maths l.jpg

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 = (Dx+Dx,Dy,Dz)

  • Right face : D = (Dx,Dy+Dy,Dz)

  • Top face : D = (Dx,Dy,Dz+Dz)

Alan Murray – University of Edinburgh


Now the maths12 l.jpg

Now the maths …

  • 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


Now the maths13 l.jpg

Now the maths …

  • 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


Now the maths14 l.jpg

These cancel

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 maths15 l.jpg

dxdydz = dv

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


Now the maths16 l.jpg

Now the maths …

  • 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


Gauss s law divergence theorem l.jpg

Gauss’s Law/Divergence Theorem

  • òòD.ds= òòò r(r)dv

  • Ñ.D= r(r)

  • These are equivalent

  • Ñ = (¶/¶x, ¶/¶y, ¶/¶z)

Alan Murray – University of Edinburgh


So what does divergence mean l.jpg

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.

So what does Divergence “mean”

E

Alan Murray – University of Edinburgh


So what does divergence mean19 l.jpg

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.

So what does Divergence “mean”

Alan Murray – University of Edinburgh


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