Tuesday, September 13, 2016

homework and exercises - Delta Dirac Charge Density question



I have to write an expression for the charge density ρ(r) of a point charge q at r, ensuring that the volume integral equals q.


The only place any charge exists is at r. The charge density ρ is uniform:


ρ(r)=δ(rr)ρ


But if I evaluate the total charge, I get


q=dq=δ(rr)ρ dV

=ρδ(xx)dxδ(yy)dyδ(zz)dz


The Dirac delta functions integrate to one each, but what becomes of the charge density ρ? For that matter, how does one integrate a zero dimensional point over 3 dimensinal space? Any help greatly appreciated.


EDIT: So it seems that the charge density is just the charge itself (ρ=q)?



Answer



First equation is wrong, it should say ρ(r)=δ(rr)q. (Note that you had two errors).
You treat it like a normal charge density ρ(r), if you integrate the density over any volume you get the total charge within that volume.



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