Find charge distribution of a point charge at origin.

  • Thread starter yungman
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  • #1
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This is part of the problem in the exercise to find charge distribution of a point charge at origin. I know [tex]\nabla \cdot \vec E = \frac {\rho}{\epsilon_0}[/tex]

[tex] \rho = \epislon \nabla \cdot \hat r \frac q {4\pi\epsilon_0 r^2} \;\hbox { where } \vec r = \hat x x + \hat y y + \hat z z [/tex]

[tex]\nabla \cdot \vec E = \frac q {4\pi\epsilon_0}\left [ \frac {\partial }{\partial x} \left ( \frac x {r^3} \right ) + \frac {\partial }{\partial y} \left ( \frac y {r^3} \right ) + \frac {\partial }{\partial z} \left ( \frac z {r^3} \right ) \right ] [/tex]

[tex]\frac {\partial }{\partial x} \left ( \frac x {r^3} \right ) = \frac { r^3 - x d(r^3)}{r^6} = \frac { r^3 - 6x^2 r^2}{r^6} = \frac 1 {r^3}-\frac {6x^2}{r^4} [/tex]

The other two can be worked out as above for y and z.

[tex]\Rightarrow\; \nabla \cdot \vec E = \frac q {4\pi\epsilon_0} \left [ \frac 3 {r^3} - \frac {6 (x^2 + y^2 + z^2)}{r^4} \right ] = \frac q {4\pi\epsilon_0} \left [ \frac 3 {r^3} - \frac 6 {r^2} \right ] = \frac {\rho}{\epsilon_0}[/tex]

[tex] \rho = \frac q {4\pi} \left [ \frac 3 {r^3} - \frac 6 {r^2} \right ] [/tex]

The book gave:

[tex]\rho = q \delta^3(\vec r) [/tex]

I don't know what is [itex]\delta[/itex] in the book!!!

Can anyone verify my work?

Thanks
 

Answers and Replies

  • #2
tiny-tim
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hi yungman! :smile:

sorry, that's completely wrong :redface:

ρ is zero except at the origin, where it's infinite

δ is the Dirac delta function (strictly, a distribution rather than a function, since it only "lives" inside integrals), which you'll have to look up :wink:
 
Last edited:
  • #3
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hi yungman! :smile:

both ρ and E are zero except at the origin, where they're infinite

[tex]\rho[/tex] is zero except at the origin, but E can't be. Surely it must be the E-field of a point charge, [tex]E=\frac{q}{4\pi \epsilon_0 r^2}[/tex] ?
 
  • #4
tiny-tim
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oops!

oops! i've no idea why i wrote that! :redface:

(i've edited it now)

thanks for the correction, kloptok :smile:
 
  • #5
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hi yungman! :smile:

sorry, that's completely wrong :redface:

ρ is zero except at the origin, where it's infinite

δ is the Dirac delta function (strictly, a distribution rather than a function, since it only "lives" inside integrals), which you'll have to look up :wink:

Hi Tiny Tim

I don't know what I am thinking, I should know δ!!! And I should know that. This is a problem in Gauge Transformation and I never even stop and think!!! The problem started out as [tex] \vec A = \frac { qt}{4\pi\epsilon_0 r^2} \hat r [/tex]

And asked to find E, B, J and current distribution. I just blindly go through the steps and solve the problem without stop and think!!!!:surprised.

How can I avoid making the same mistake because without stop and think, I just apply the divergence and get the wrong answer!!!!??

Thanks!!! Today is my 58th birthday and I sure acted stupidly!!!!
 

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