Divergence Theorem and E-field1 The Divergence Theorem and Electrical Fields © Frits F.M. de Mul

Divergence Theorem and E-field2 Gauss’ Law for E-field (1) volume V; surface A dA ┴ surface A volume V; surface A dA ┴ surface A dA Gauss’ Law: Goal of this integral expression : to calculate E from  Q or  provided symmetry present !!!!! to calculate E from  Q or  provided symmetry present !!!!! QUESTION: does an inverse expression, to locally calculate  (xyz) from E(xyz), exist ?? E E E-field arbitrary

Divergence Theorem and E-field3 Gauss’ Law for E-field (2) volume V surface A volume V surface A E dA Gauss’ Law: dV to look locally: observe local volume element dV at (xyz) X Y Z volume element dV has sides dx, dy and dz Question: calculate local  -distribution from E-field

Divergence Theorem and E-field4 The E-flux through dV (1) E Point P in dV at x,y,z. Y dy dz dx dV X Z P d   through dV = d   through left & right sides + d   through top & bottom sides + d   through front & back sides d   through left & right sides + d   through top & bottom sides + d   through front & back sides

Divergence Theorem and E-field5 The E-flux through dV (2) dV X Y Z dy dz dx E P Point P in dV at x,y,z. 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)

Divergence Theorem and E-field6 The E-flux through dV (3) Net flux d   through left - right side of dV : Point P in dV at x,y,z. P P yy+ 1/2 dyy- 1/2 dy d   = E y.dxdz at (x,y + dy/2,z) - E y.dxdz at (x,y - dy/2,z) d   = E y.dxdz at (x,y + dy/2,z) - E y.dxdz at (x,y - dy/2,z) EyEy E y = f (y)

Divergence Theorem and E-field7 The E-flux through dV (4) dV X Z dy dz dx E P Y Net flux d   : left/right: analogously: top/bottom: and front/back:

Divergence Theorem and E-field8 The E-flux through dV (5) dV X Z dy dz dx E P Y Net flux d   through dV: DEFINITION DIVERGENCE div : DEFINITION DIVERGENCE div :

Divergence Theorem and E-field9 Local expression for Gauss’ Law volume V surface A volume V surface A E dA dV element dV : enclosed charge in dV :  dV Gauss’ Law in local form: where E and  are f (x,y,z)

Divergence Theorem and E-field10 From “local” to “integral” volume V surface A volume V surface A E dA dV element dV : summation over all elements: all “internal” d  E ’s cancel d  E ’s at surface A remain only !

Divergence Theorem and E-field11 How to use the laws ? volume V surface A volume V surface A E dA 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).

Divergence Theorem and E-field12 Physical meaning of div volume V surface A volume V surface A E dA Divergence = local “micro”-flux per unit of volume [m 3 ]

Divergence Theorem and E-field13 “Gauss” in general volume V surface A volume V surface A E dA in accordance with general relation for a vector X : the end