University Electromagnetism: Derivation of Gauss'Law (in integral and differential form)

1. The Divergence Theorem and Electrical Fields © Frits F.M. de Mul

2. Gauss’ Law for E- field (1) Goal of this integral expression : to calculate E from  Q or   provided symmetry present !!!!! QUESTION: does an inverse expression, to locally calculate  (xyz) from E (xyz), exist ?? volume V; surface A dA ┴ surface A dA dA Gauss’ Law: E E E -field arbitrary

3. Gauss’ Law for E- field (2) volume element dV has sides dx, dy and dz Question: calculate local  - distribution from E -field volume V surface A E dA Gauss’ Law: dV to look locally: observe local volume element dV at (xyz) X Y Z

4. The E -flux through dV (1) d   through dV = d   through left & right sides + d   through top & bottom sides + d   through front & back sides E Point P in dV at x,y,z. Y dy dz dx dV X Z P

5. The E -flux through dV (2) Calculate d   through right side: d   = E.dA = E y .dxdz at ( x,y+dy /2, z ) Calculate d   through left side: d   = E.dA = - E y .dxdz at ( x,y - dy /2, z ) dV X Y Z dy dz dx E P Point P in dV at x,y,z.

6. The E -flux through dV (3) Net flux d   through left - right side of dV : Point P in dV at x,y,z. P y y+ 1/2 dy y- 1/2 dy d   = E y .dxdz at ( x,y + dy /2, z ) - E y .dxdz at ( x,y - dy /2, z ) E y E y = f (y)

7. The E -flux through dV (4) Net flux d   : left/right: analogously: top/bottom: and front/back: dV X Z dy dz dx E P Y

8. The E -flux through dV (5) Net flux d   through dV : DEFINITION DIVERGENCE div : dV X Z dy dz dx E P Y

9. Local expression for Gauss’ Law enclosed charge in dV :   dV Gauss’ Law in local form: where E and  are f (x,y,z) volume V surface A E dA dV element dV :

10. From “local” to “integral” summation over all elements: volume V surface A E dA dV element dV : all “internal” d  E ’s cancel d  E ’s at surface A remain only !

11. How to use the laws ? Integral expression: from   to E, but in symmetrical situations only ! div E (x,y,z) =  (x,y,z) /   Differential (local) expression: from E (x,y,z) to  (x,y,z) . volume V surface A E dA

12. Physical meaning of div Divergence = local “micro”-flux per unit of volume [m 3 ] volume V surface A E dA

13. “ Gauss” in general in accordance with general relation for a vector X : the end volume V surface A E dA