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

1. Gauss® DivergenceòòD.ds = òòò r dv®Ñ.D = r(r)  Alan Murray

2. 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

3. z y x Proof (non-examinable) Surface for Integration D,E Alan Murray – University of Edinburgh

4. z Surface for Integration D,E y x Proof (non-examinable) Cut to the chase Argh!! I am shrinking!!! Alan Murray – University of Edinburgh

5. dy z dx D,E dz y D =D(0,0,0) x Proof (non-examinable) Alan Murray – University of Edinburgh

6. dy z dx dz y D =D(0,0,0) x Proof (non-examinable) D,E Alan Murray – University of Edinburgh

7. z y D =D(0,0,0) x Proof (non-examinable) dy dx D,E dz Alan Murray – University of Edinburgh

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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