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Homework Help: Electromagnetism Finding E-fields

  1. Feb 6, 2010 #1
    1. The problem statement, all variables and given/known data
    the electric potential of some configuration of charge is given by the expression:

    V(r)= (Ae^(-pr))/(r)

    where A and p are constants. Find the E-field E(r), the charge density rho(r) and total charge Q. The answer for rho is given in book as:
    epsilon-not * A * ((1/r)4* pi* dirac delta^3 (r) - p^(2) e^(-p*r))
    (The stuff inside parentheses is all divided by r


    2. Relevant equations
    -del V= E; del dot E = rho/epsilon-not ;



    3. The attempt at a solution
    I the took gradient of V using sperical coordinates using the formula
    -del V= E

    to obtain E.

    Then I used the formula

    del dot E = rho/epsilon-not

    to find rho. I am not getting the same rho as in the book. I dont understand why the dirac delta function is in the answer. Do you have to perform an integration or am I taking a wrong apprach all together?
     
  2. jcsd
  3. Feb 6, 2010 #2

    kuruman

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    Can you show how you got your volume charge density? What equation did you strart from?
     
  4. Feb 6, 2010 #3
    the formular for obtaing rho goes thru this eqn:

    del dot E = rho/epsilon-not

    Del dot E in sperical coords. is : (1/r^2) d/dr [ r^2 V(r)] (the (1/r^2) and r^2 are built into the formula.) once i did this multipled both sides by epsioln-not to obtain rho. This question is from Grifiths electrodynamics book ed. 5 problem 2.46
     
  5. Feb 6, 2010 #4

    kuruman

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    Then what did you do? When you took the derivatives, did you say

    [tex]\frac{r^2}{r^2}\frac{dV}{dr}=\frac{dV}{dr}\;?[/tex]

    How justified are you in canceling out the r2 terms at r = 0? That's where the Dirac delta function comes in.
     
  6. Feb 6, 2010 #5
    http://s22.photobucket.com/albums/b317/richard7893/?action=view&current=emag.jpg" [Broken]
    This is what I have so far if you click on the link. Can I cancel out the r^2 terms in the differential? Or do I distribute the r^2 in the numerator and then perform the differential? How can I insert a dirac delta function? If I cancel out the r^2 in the differential I dont get the same answer as the book.
     
    Last edited by a moderator: May 4, 2017
  7. Feb 7, 2010 #6

    kuruman

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    If the charge distribution has no singularity (does not blow up) as r goes to zero, then you can cancel the r2 in the numerator and denominator and proceed merrily on your way. If there is a singularity, then you essentially have "zero divided by zero". You know that in this case there is a singularity because the electric field goes to infinity as r goes to zero. This means a point charge at the origin which requires a Dirac delta function in the volume charge density.
     
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