Question

How to find volume charge density when elecric field and a point and volume is given?

Answer 1

The fundamental equation here is

∇⃗ ⋅(r^r2)=4πδ3(r⃗ ).

This can be proven through Fourier analysis, but you can intuitively understand that this works in the case of a single charge q: take the field of a single charge at the origin

E⃗ =14πϵ0qr2r^.

We know that this field is sourced by a point charge at the origin: this is consistent with our previous equation, as

∇⃗ ⋅E⃗ =q4πϵ0∇⃗ ⋅(r^r2)=1ϵ0qδ3(r⃗ ).

Also, by direct computation, you have

∇⃗ r=r⃗ r=r^.

Now that you have those rules, you can simply calculate the divergence of your electric field.

∇⃗ ⋅E⃗ =q4πϵ0∇⃗ ⋅(e−brr^r2)=q4πϵ0(∇⃗ e−br⋅r^r2+e−br∇⃗ ⋅(r^r2)).

Applying the rules that we have found, we get

∇⃗ ⋅E⃗ =qϵ0(δ3(r⃗ )−b4πr2)e−br,

as expected.