1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Recovering a vector field from the divergence/flux

  1. Nov 21, 2012 #1
    1. The problem statement, all variables and given/known data

    (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).

    2. Relevant equations

    I think the divergence theorem is the way to go

    3. The attempt at a solution

    Here is what i've done:


    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 ?

    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.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted