Friday, 9 March 2018

homework and exercises - How come divergence of vecE is zero in this case where vecE=xifracleft[x,yright]sqrt(x2+y2)3,?


I hope you could help me clearing some doubts about Gauss' law of the electric field that states ϵ0E=ρ. Take for instance the case of a point charge in the origin in empty space. The law states E=0 anywhere but the origin, but it's enough to compute E to get to a contradiction: E=ξ[x,y](x2+y2)3

where I dropped every constant to ξ because that's my favourite greek letter, and considered the 2D case since in my opinion it's a good compromise between complexity and meaningfulness E=x2+y2(x2+y2)5=(x2+y2)152
which is clearly not zero except at infinity.



Answer




As pointed out in the comment, Gauss' law has a precise geometric meaning and so is strongly dependent on the dimension of the space. Let's work in 2D as in your example and apply Gauss' law: the flux of the electric field through a circle centered in the point charge is 2πrE. This must be equal to the total charge which is e/ϵ0. This gives for the modulus of the electric field: E=e2πϵ0r

and for the cartesian components: E=e2πϵ0(xx2+y2,yx2+y2)
As you can verify the divergence of this is zero: E1x2+y22x2(x2+y2)2+1x2+y22y2(x2+y2)2=0


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