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(→r−→r′)ρ
But if I evaluate the total charge, I get
q=∫dq=∫∞−∞δ(→r−→r′)ρ dV
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(→r−→r′)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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