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Old Sep 12th 2010, 04:25 PM   #1
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calculating charge from a potential

Hi,

I'm given a potential V(r)=Ae^(-br)/r, where A and b are constants. I am asked to determine the electric field, the charge distribution, and the total charge.

The electric field is the negative gradient of V, and I calculated it to be Ae^(-br)(rb+1)/r^2 in the radial direction. This has been checked with wolfram alpha.

The charge distribution is epsilon-naught times the divergence of the electric field, and I calculated the div of E to be -(b^2)Ae^-(br)/r - 2Ae^(-br)(rb+1)/r^3. I haven't checked with wolfram yet, but I'm almost 100% certain this is correct.

However, I attempted to calculate the total charge two different ways, and in neither case am I getting anything meaningful.

First, I attempted to integrate the charge distribution over all space, and this integral I found to be nonconvergent.

Second, I used the divergence theorem, turning the integral of the charge distribution over space to a surface integral of the E field with r=infinity (I don't know if that's actually valid, I've never used the divergence theorem with any infinite bounds), and this returned 0.

I checked a few times and I don't see anything wrong with my calculations, but I grant there may be. Am I lacking some sort of insight into the charge calculation, or is this some sort of trick question, and the total charge is in fact indeterminant?

Thanks
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Old Sep 12th 2010, 04:39 PM   #2
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omg i forgot to multiply by r^2 when calculating the divergence. that was smooth
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