3. (a) (b) Consider the electric field of a positive point charge: (r) V. A =...

  

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3. (a) (b) Consider the electric field of a positive point charge: (r) V. A = (r Ar) + r r = 1 q 4 r2 where the symbols have their usual meaning. Find the divergence of this field (i) at all points away from the origin = 0 and (ii) at the origin. Hence show the differential form of Gauss's law. You may use that in spherical polar coordinates 1 a 1 rsine 20 VE lim and that the formal definition of divergence is 1 AV-0 AV , (Aosine) + E n da. 1 Ap rsine do Find the charge density in a region for which the electric field in spherical coordinates is given by r bcose E = ar + +c. (3 marks) (3 marks) (1 mark) (3 marks) 3. (a) (b) Consider the electric field of a positive point charge: (r) V. A = (r Ar) + r r = 1 q 4 r2 where the symbols have their usual meaning. Find the divergence of this field (i) at all points away from the origin = 0 and (ii) at the origin. Hence show the differential form of Gauss's law. You may use that in spherical polar coordinates 1 a 1 rsine 20 VE lim and that the formal definition of divergence is 1 AV-0 AV , (Aosine) + E n da. 1 Ap rsine do Find the charge density in a region for which the electric field in spherical coordinates is given by r bcose E = ar + +c. (3 marks) (3 marks) (1 mark) (3 marks)