Recovering a vector field from the divergence/flux

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Homework Statement



(Long and unappealing looking question - but could please use few hints with what is going wrong with my analysis :) )

In brief: the object is to work out what vector field causes a given divergence - I've managed to get the divergence & flux but recovering the field is causing me some issues & there's not much information around about this reverse process.


In this question there is a solid sphere that emits charged particles from within its volume (evenly) at a given rate. They pass out of the surface so it has a current density(J) kind of flux.

we are told that in this case: (which is obvious - but I just include this in case my description has lost anyone)

the divergence (of J) = the total charge crossing the spheres surface per second / total volume

and are given a number to check it is right when the expression is evaluated (mine is).


Homework Equations



I think the divergence theorem is the way to go

The Attempt at a Solution



Here is what i've done:

1)

rate of p'ticles out per second * charge of particle = charge out per second (= current out)

I will call this total current out by 'C' to keep things cleaner.

2)a) therefore the divergence is:

C / (volume of the sphere) = C / ((4/3)*pi*r^3)

2)b) ? I could also get the flux at this point by

C/surface area ?



3)
OK so this is very simple so far. But it now asks to use the calculated divergence of J to get the vector field.

I'm not used to the proper equation editor - but if we consider the diverence theorem for a solid and a vector field as:

the volume integral of divergence equated with the closed surface integral of the flux
through the surface area element.

I hope people will be able to understand clearly enough what I am saying.

So my view is that since the Div(J) = constant throughout the sphere.
therefore the triple integral simply becomes the product of the divergence and volume

so from my previous notation the triple integral becomes just 'C'

so now I have just:

C = the surface integral of the flux over the sphere

Thoughts that i'm not sure how to use to progress:

1) the vector field will be parallel to the surface area vector
1b) <vector field | surface normal > = |vector field|

2) the < vector field | surface normal vector > will be constant over the surface

3) the direction of the vector field is not constant over the surface as it is always normal

my plan was to take < vector field | surface normal > outside the integral ... then the value of the integral is the just the surface area.

then C/surface area = |vector field| .... which feels close but I don't know how to regain it's vector qualities.

Homework Statement





Homework Equations





The Attempt at a Solution

 

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